ICare-DT: a digital twin for continuous forecasting of intensive care patients

ICare-DT: a digital twin for continuous forecasting of intensive care patients Yanke Li1,2,* , Yahang Qi1,3 , Yumi Kim1 and Diego Paez-Granados1,2,* 1

SCAI Lab, Department of Health Science and Technology, ETH Zurich, Switzerland 2

arXiv:2611.09712v1 [cs.LG] 6 Nov 2026

3 *

Swiss Paraplegic Research, Nottwil, Switzerland

Department of Statistical Sciences, University of Toronto, Canada

Corresponding authors: yanke.li@hest.ethz.ch, diego.paez@hest.ethz.ch

Abstract Digital twins of intensive care patients should update forecasts from new data and predict treatment responses, but few studies test whether routine records support either. We propose ICare-DT, a Transformer-based twin that updates a latent patient state hourly from new measurements without retraining. We trained it on 140,473 eICU-CRD stays to forecast vital signs crossing their training-set means. On test data, later forecasts beat the twin’s own hour-6 forecasts of the same windows in all 12 settings (+0.073 to +0.153 AUROC). On validation data, controls that withheld or swapped the new hours traced the gain to the patient’s own measurements. Without refitting, the model kept its gain on a second database (8 of 9 settings, +0.025 to +0.114 AUROC) in a prespecified evaluation of the previously analyzed MIMIC-IV test split. Refitted within MIMIC-IV, a twin state fine-tuned on clinical endpoints gave gradient boosting a significant gain for acute kidney injury (+0.0035 AUROC; hypotension +0.0010, not significant). To test response prediction, we built a benchmark that replaces the treatments in real records with simulated doses and injects responses of known size, visible only at measurement times. Ridge models supervised with injected effects recovered them from real records only through frequently measured variables (heart rate) read at their measurement times. We asked standard learners to predict these effects for a dosing regime no patient had received. No standard learner outperformed predicting zero effect at an out-of-support query. When told how the simulated responses were generated, a separate learner recovered the effects from the same records without seeing their true values (R2 0.30 to 0.40 at one effect size). Routine records support forecasts that improve with each patient’s new data, and our benchmark is a testbed for models that learn how responses are generated.

Introduction Patients in intensive care can deteriorate within hours, and clinicians adjust vasopressors, fluids and ventilation as they watch each patient respond. Two questions recur at the bedside: where is this patient heading, and how would this patient respond if the treatment were changed? A digital twin of the patient has been proposed to help answer both 1–4 . In the National Academies’ definition, a digital twin is a model of one individual that is updated dynamically from that individual’s data, is personalized and informs decisions 5 . 1

For a machine-learning model, the patient is represented by the patient record: an irregularly sampled multivariate time series of vital signs, laboratory values and treatments that grows by one hour at a time. A twin therefore needs two capabilities. The first is continuous forecasting: the twin keeps a patient state, a learned summary of the record so far, updates it as each new hour arrives without retraining, and forecasts upcoming events from it. The second is treatmentresponse prediction: forecasting the patient’s course under alternative treatments, whose contrast is, in causal terms, a conditional treatment effect that changes over time 6–8 . Both must be learned from routine records, and two features of these records make this hard. First, what is measured and when depends on clinical practice and on the patient’s condition 9 ; in 400 stays of the eICU Collaborative Research Database (eICU-CRD) 10 studied here, the recorded heart rate changes in 57% of hours and creatinine, a laboratory marker of kidney function, in only 4%. Second, a record shows only the outcome under the treatment that was given, and because the sickest patients are the ones treated, a beneficial treatment can look harmful, a bias known as confounding by indication (Fig. 1a) 11,12 . Machine learning has advanced each capability separately. For prediction, foundation and sequence models of health records learn one representation from many patients and forecast many outcomes, and they are judged by their accuracy across tasks and hospitals 13–17 . Relational models capture dependencies among irregularly sampled intensive care variables 18 , and landmark models update a prediction at set times as data accrue 19 . For treatment effects, counterfactual sequence models estimate how outcomes would evolve under alternative treatment sequences 20–26 ; g-computation models among them, such as G-Net 23 , simulate every recorded variable forward under a chosen treatment sequence. Because true effects are never observed in real data, these and other causal methods are scored on simulators in which the effects are known 27–31 . Intensive care twins have also been built from expert knowledge: a twin of early sepsis with expert-specified causal structure was checked prospectively against observed responses to single interventions, although not against known individual effects 32 . These evaluations leave open two questions that must be answered before a twin can be tested at the bedside. The first is whether a twin’s forecasts improve because it reads the patient’s new data. Accuracy at fixed time points cannot show this, and neither can a gain in accuracy later in the stay, because a later forecast is closer to the events it predicts and can be more accurate without using anything new about the patient. A convincing test compares forecasts of the same future window made at the same lead time with and without the new hours, includes trivial references that need no learning, such as the patient’s last measured value, and is fixed before the results are seen 33,34 . Verification of any kind is rarely reported: in a scoping review of 149 studies of human digital twins, 2 mentioned verification, validation and uncertainty quantification 35 . The second question is whether routine records hold enough information to learn individual treatment responses at all. A simulator can answer it only if it reproduces which values are recorded and when, because an effect enters the record only in hours in which the affected variable is measured 9,36,37 . Yet the intensive care benchmark used for the Causal Transformer writes a simulated outcome at every hour 22 , generators that let the user set the effects have no time dimension 38,39 , and the realism of simulators is seldom audited 40,41 . Here we propose ICare-DT, a digital twin of intensive care patients built for continuous fore2

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Figure 1: ICare-DT and its evaluation. a, Association and intervention. Because the sickest patients are treated, a treatment can be associated with higher risk even when it lowers risk. The operator do(A) sets the treatment A by intervention. b, Data and design. Three harmonized databases are placed on an hourly grid of 161 input channels (41 physiological, 120 treatment; Methods), and bars give each database’s stays in the experimental cohort (Table 1). ICare-DT is trained on eICU-CRD (eICU in the panel), and every analysis uses eICU-CRD except one registered read of the previously analyzed MIMIC-IV test split. That read scored the two prediction results (synchronization and the clinical-endpoint probes), with disclosed secondary outcomes (Supplementary Note 7). MIMIC-III appears only in earlier transfer analyses that were not registered (Supplementary Note 8). A semi-synthetic generator writes injected responses onto real eICU-CRD records. Its generated dataset passed a locked realism audit (Fig. 4b). c, Inputs. A causally masked Transformer encodes the patient history Ht (physiology, laboratory values, treatments, measurement mask and static covariates) into the patient state zt . d, Readouts. The joint competingrisks forecast gives, for each of four vital signs, the probabilities that a crossing of the training-set mean, live discharge from the unit or death comes first, 1 to 48 hours ahead. Clinical-endpoint probes read the state for four clinical endpoints (in the panel, ICU death is death in the unit and kidney injury is acute kidney injury; Methods). The counterfactual rollout generates the next hour under a chosen action and was evaluated only on a historical benchmark (Fig. 4a). State synchronization re-reads newly recorded hours with frozen weights. e, Causal query, the intended use of a response module. Predicted courses under do(A = 1) and do(A = 0) are compared for the same history. In this paper, the twin’s rollout is not used for individual effects: on washout panels of real records, effects estimated by readers and treatment-effect learners are scored against the known individual effects. The panels share the generator’s code, and their realism audit is weak evidence (Figs. 3c, 4b and 5–7; Results). The scatter in e is schematic.

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casting, and we test whether routine records support both capabilities (Fig. 1). ICare-DT is trained on eICU-CRD (Table 1). A causally masked Transformer maps the record up to each hour to a latent patient state. Every hour, the twin runs the same frozen network on the record so far and so assimilates the patient’s new measurements without retraining, which we call synchronization. From the state, one head per vital sign forecasts which of three events comes first within 48 hours: a crossing of the vital sign’s training-set mean, live discharge from the unit or death. Because a crossing marks an excursion from the cohort average rather than a clinical event, we also probed the state on four clinical endpoints: death in the unit, hypotension (low blood pressure), acute kidney injury (a rise in creatinine) and hypoxemia (low blood oxygen saturation). To test forecasting, we compared each synchronized forecast with the twin’s own hour-6 forecast of the same window, fed the model control inputs that keep the lead time but withhold the patient’s new hours or replace them with another patient’s, and used the patient’s last measured value and gradient boosting as references. To test treatment-response prediction, we built a semi-synthetic generator that writes responses of known size onto real eICU-CRD records while reproducing how these records are measured, and audited its realism against a threshold fixed in advance. With the same code we built washout panels: real records with simulated dosing, a drug-free washout and then sustained treatment or none, so that each patient’s effect is known. On these panels we tested standard treatment-effect learners and our candidate response module, ICare-RH; the twin’s own counterfactual rollout, which generates the next hour under a chosen action, was not tested there. Registered analyses followed protocols committed to version control before they ran; these are internal registrations, not external preregistrations (Methods). They include a registered read of the Medical Information Mart for Intensive Care (MIMIC-IV) 42 , a single scoring of its previously analyzed test split. The first capability is supported on eICU-CRD. On its test set, synchronized forecasts beat the twin’s own hour-6 forecasts of the same window in all 12 settings (four vital signs, three forecast windows), by 0.073 to 0.153 in the area under the receiver operating characteristic curve (AUROC; Fig. 2c). On the validation split, control inputs traced the gain to the patient’s own measurements, though not to information beyond the target’s last measured value (Fig. 2b,d); on the same split, however, a gradient-boosted model reading the same hours was the stronger forecaster, so we make no claim of state-of-the-art accuracy. Judged as a forecast, the synchronized one nevertheless beat the patient’s last measured value in 11 of the 12 settings on eICU-CRD, in a disclosed comparison that was not registered. In the registered read of the previously analyzed test split of MIMIC-IV, the model, applied without refitting, kept the gain in 8 of the 9 settings that could be scored there (+0.025 to +0.114 AUROC), meeting the registered transfer criterion; against the patient’s last measured value it was better in 1 of 9 settings and worse in 2, not meeting the registered secondary criterion (Table 2). With all models refitted within MIMIC-IV, adding a twin state fine-tuned on the clinical endpoints to gradient boosting gave a significant gain for acute kidney injury only (+0.0035 AUROC). For the second capability, the generator passed its realism audit, a classifier test that scores 0.5 when generated and real records cannot be told apart, once it reproduced how values are recorded (audit AUC 0.572 against a pass threshold of 0.75; Figs. 3 and 4). On the washout panels, which reuse the generator’s code with separate parameters, ridge regressions trained 4

with the true effects as labels (readers, which cannot be deployed) recovered the injected responses at real measurement cadence on heart rate but not on creatinine, the two responding variables tested, and only when reading the hours in which the value changed (Fig. 5). By design, the benchmark asks whether a model can carry what each patient’s earlier responses reveal to a regime that the data never show. None of 16 standard treatment-effect learners, including the Causal Transformer and the g-computation model G-Net (Causal Transformer code), outperformed predicting zero effect at the decision query, a treatment regime that no patient had received. Neither registered hour inside the dosing period (30 or 42) met the registered overlap criteria for a query inside the treatment support. Unlike these standard learners, a learner given the generator’s response structure (the form of the injected response, the distribution of patient traits and how outcomes are recorded) recovered the effects from the same factual data (Fig. 6). ICare-RH recovered effects in simulation but failed label-free placebo checks, run before any true effect was read, and was therefore never scored on real records (Fig. 7). A twin can therefore already keep a patient state whose forecasts improve as the patient’s own measurements arrive, whereas predicting an individual patient’s response to treatment remains unproven. Our results suggest that a response module should learn the response structure from factual data, and we will release the generator and washout-panel code, with the registered protocols, as a testbed for such modules.

Results We first describe ICare-DT and test continuous forecasting on eICU-CRD and MIMIC-IV. We then build a benchmark with known treatment responses and ask whether individual responses can be recovered from routine records.

ICare-DT keeps an hourly patient state ICare-DT reads each stay’s record on an hourly grid of 161 channels (41 physiological, 120 for treatments such as drugs, fluids and ventilator settings), a mask of the hours in which each physiological variable was measured and four static covariates (Fig. 1c; Methods). A causally masked Transformer maps the record up to hour t to a 64-dimensional patient state zt . The tracked vital signs are heart rate, respiratory rate and two oxygen-saturation channels, SpO2 and SaO2 ; in eICU-CRD, the SaO2 channel combines monitor saturation readings with blood-gas values (Methods). Crossings are upward for the rates and downward for the saturations (Fig. 1d). Five models, differing only in random seed, were trained on the 140,473 stays of the persondisjoint eICU-CRD training split (Table 1).

Hour-48 forecasts discriminate threshold crossings and death on eICU-CRD We first checked that the twin’s forecasts are informative when issued at a fixed time in the stay. In a registered read of the eICU-CRD test set (30,099 stays of 20,886 patients; Supplementary Note 6), forecasts issued at hour 48 discriminated threshold crossings with AUROCs of 0.646 to

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0.820 across the four vital signs and horizons of 6 to 48 hours (five-seed means; 227 to 1,449 stays per setting; Fig. 2a). These stays were still in the unit at hour 48, measured and not yet past the threshold, and were followed until the outcome was known. Without recalibration, the mean forecast was 91% to 105% of the observed rate at horizons of 6 and 12 hours and 68% to 89% at 24 and 48 hours (Methods). The registered plan specified a descriptive 48-hour death forecast, the largest of the four heads’ death probabilities (Methods). Among the 11,498 stays still in the unit after hour 48, it discriminated death in the unit within 48 hours with an AUROC of 0.798 (95% interval across seeds 0.768 to 0.828). Its mean forecast was far below the observed rate (0.0023 against 0.026), in part by construction, because each target-specific head counts a death only if it precedes that target’s crossing.

Synchronizing the twin with the patient’s record improves its forecasts A twin should forecast better as it reads more of the patient’s record. We tested synchronization in 12 settings, which pair the four vital signs with a landmark L and horizon H of (12, 6), (18, 12) or (30, 24) hours. The synchronized forecast reads the record up to hour L and forecasts a crossing in the next H hours; the hour-6 forecast is the same frozen model’s forecast of that window from the first six hours of the stay. With model and window fixed, the gain measures the value of reading more of the record, which adds measurements but also shortens the lead time. A stay entered a setting if its target was measured every hour through the window (592 to 6,648 stays per setting; 42.9% of test stays entered at least one; Methods). In the registered analysis of the test set, synchronization improved on the hour-6 forecast in all 12 settings, by 0.073 to 0.153 AUROC, and every setting survived Holm correction over the 12 settings (Fig. 2c and Supplementary Table S6). Calibrated log-loss and Brier score also improved in all 12 settings (Fig. 2e; Brier score in the source data). In a disclosed comparison (reported in full but not registered), the synchronized forecast also beat the patient’s last measured value of the target, a persistence baseline, in 11 of 12 settings after Holm correction (median difference 0.044 AUROC, range 0.010 to 0.085; Fig. 2c). On the validation split, however, a multivariate gradient-boosted model reading the same hours (a fixed, untuned CatBoost recipe 43 ) was the stronger forecaster of threshold crossings. The twin’s forecasting heads, read from one checkpoint, trailed it by a median of 0.012 AUROC and were worse in 3 of 12 settings after Holm correction over the 120 contrasts of that comparison (Table 2; Methods). Adding the twin’s state to the boosted model raised its AUROC point estimate in 10 of 12 settings (median 0.004), but no AUROC gain survived correction.

Control inputs trace the gain to the patient’s own measurements Because a later forecast is also closer to its window, the gain could come from the shorter lead time rather than from the new data. To separate the two, the same frozen model read hours 7 to L of the validation records in three versions (Fig. 2b): as recorded (synchronized); held at their hour-6 values and marked as unmeasured, which keeps the lead time but adds no information;

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Figure 2: Synchronized forecasts beat the twin’s own hour-6 forecasts. All panels show eICU-CRD (MIMIC-IV: Table 2 and Supplementary Note 7). a, Forecasts on the test set: AUROC of the hour-48 forecasts of a crossing of each vital sign’s training-set mean, by horizon (227 to 1,449 stays per setting; mean of five seeds with 95% t intervals). b, Information arms (validation), the control inputs of the text: AUROC of the same frozen model when hours 7 to L are read as recorded (Synchronized), held at their hour-6 values (Held at hour 6) or taken from the other patient nearest in hour-6 state (Other patient), and of the model’s own forecast made at hour 6 (Hour-6 forecast). Thin lines show settings; thick lines, means per vital sign. c, Synchronization gain (test): AUROC gain of the synchronized forecast over the hour-6 forecast (circles; registered) and over the patient’s last measured value (squares; disclosed analysis). Rows give landmark L and horizon H as L → H. d, Source of the gain (validation): AUROC gain of the synchronized forecast over Held at hour 6 (circles) and over Other patient (diamonds), and the advantage of the patient’s own last measured value over the other patient’s, without a model (squares). e, Calibrated log-loss of the synchronized forecast minus that of the hour-6 forecast (Test) or of Other patient (Validation). Negative values favor the synchronized forecast. Probabilities were clipped before taking logs, so only the sign is interpreted. In c–e, filled markers survive Holm correction within their family, and bars are 95% bootstrap intervals (test: 2,000 stay-level replicates; validation: 10,000 whole-person replicates). In b–e, stays had their target measured every hour through the forecast window (592 to 6,648 stays per setting on the test set, 551 to 7,707 on validation). Both splits had been read in earlier analyses (Supplementary Note 2). AUROC, area under the receiver operating characteristic curve.

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or taken from the other patient whose hour-6 state was closest and spliced onto this patient’s first six hours, which adds hours that are not the patient’s own. The synchronized forecast beat the held input in all 12 settings, by 0.080 to 0.146 AUROC, after Holm correction over the 36 primary contrasts (three per setting; Fig. 2d). Part of this margin is a handicap of the held input: although it reads the same six hours as the hour-6 forecast, it fell below that forecast in all 12 settings, by 0.006 to 0.036 AUROC (significant in 5 after the same correction). The heads were never trained on records held constant, so the gain over this input is not purely a gain from new data (Proposition 1; Methods). The synchronized forecast also beat the other-patient input in all 12 settings after the same correction, by 0.111 to 0.185 AUROC, which met the registered personalization condition (Table 2). Splicing could itself degrade the forecast, but the synchronized forecast beat the other patient’s own forecast, made without a splice, by a similar margin (0.119 to 0.188 AUROC; Methods). The condition is necessary for personalization but not sufficient, because vital signs track their recent values: without any model, the patient’s own last measured value beat the other patient’s by a similar margin (0.115 to 0.188 AUROC). A difference-in-differences contrast, prespecified as descriptive, asked whether the model’s advantage over the other-patient input exceeds this model-free advantage. After Holm correction it did so in no setting and fell significantly short in two (Fig. 2d and Supplementary Table S6), although the contrast need not be zero even when the new hours inform the forecast only through the last measured value (Methods). Information beyond the target’s last measured value is therefore not shown (Proposition 2). As a forecast, however, the synchronized one beat the last measured value in 11 of 12 test settings.

Transfer to MIMIC-IV without refitting All analyses so far used eICU-CRD, on which the twin was trained. To test whether the gain holds in a second database, we applied the five frozen checkpoints, without refitting, to the persondisjoint test split of MIMIC-IV (14,211 stays) in one registered read. This split had been read before, including zero-shot transfers of earlier versions of the eICU-CRD-trained twin, fine-tunes of the same pretrained model and outcomes that overlap ours (Methods; Supplementary Note 7). The SaO2 target was not scored, because in MIMIC-IV that channel holds sparse blood-gas values. This left 9 settings, with thresholds at the MIMIC-IV training means and a SpO2 target recorded differently from that of eICU-CRD (Methods). The synchronized forecast outperformed the hour-6 forecast in 8 of 9 settings after Holm correction (estimates +0.025 to +0.114 AUROC; not significant in SpO2 at (30, 24)). This meets the registered transfer criterion of at least 5 of 9 settings with none significantly worse before correction, but unlike on eICU-CRD the gain was not significant in every setting. Against the patient’s last measured value, the synchronized forecast was better in 1 of 9 settings after Holm correction and, unlike on eICU-CRD, worse in 2: SpO2 at (12, 6) and heart rate at (30, 24) (median difference −0.011 AUROC, range −0.084 to +0.034). The registered secondary criterion was therefore not met (Table 2). The control inputs were not repeated on MIMIC-IV, so the attribution of the gain to the patient’s newly recorded measurements rests on the eICU-CRD validation split. As disclosed secondary outcomes with no registered criterion, the read also scored the frozen

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hour-48 forecasts of death and of leaving the unit. Because the harmonized records keep a hospital-death flag without its time, the death forecast was scored against hospital death, with AUROCs of 0.687 for death in a hospitalization whose unit stay ended within the next 48 hours and 0.726 for death at any time in the hospitalization; the AUROC for leaving the unit within 48 hours was 0.765. Both mean forecasts fell below the observed rates, in part by construction (Methods; Table 2).

A fine-tuned twin state approaches gradient boosting on eICU-CRD and adds to it for acute kidney injury Beyond its own forecasts, a twin’s state could serve as an input to risk models of clinical endpoints. Two registered analyses run during model development, the endpoint probes, asked whether the twin’s state predicts the four clinical endpoints within 24 hours of landmarks at 12, 24 and 48 hours, and whether it adds information to the boosting recipe above. Probe P0 read the frozen state, and probe P1 trained the encoder on the endpoints (Methods). They were scored on a tuning set of the eICU-CRD training split that the pretrained encoder had seen in training, which favors the twin’s states. Used alone, each representation trailed boosting: an encoder trained on the endpoints from random initialization by 0.004 to 0.021 AUROC, with every interval below zero, one fine-tuned from the pretrained twin by 0.010 to 0.029 and the frozen state by 0.020 to 0.086. The hybrid, which adds the fine-tuned state to the boosting features, improved AUROC by 0.0042 for hypotension and 0.0075 for acute kidney injury, with unadjusted intervals above zero (Supplementary Fig. S3). Neither probe met its registered success criterion (P1 missed it narrowly; Table 2), and the probes did not show a benefit of pretraining (Methods). We refitted the boosting comparator, both encoders and the hybrid within the MIMIC-IV training split (66,007 stays) with the recipes above, with five seeds for the encoders and early stopping on 10% of training patients, and the MIMIC-IV test split was read once. Death in the unit was not scored as an endpoint, because the harmonized records lack its time (Methods). Adding the fine-tuned state to the boosting features improved discrimination of acute kidney injury by +0.0035 (95% CI +0.0007 to +0.0065; one-sided Holm-adjusted P = 0.018), which was significant under the registered test, but not significantly that of hypotension (+0.0010; −0.0012 to +0.0033; P = 0.18). The encoder trained from random initialization trailed boosting by 0.0109 to 0.0294 AUROC across the three endpoints, and because no non-inferiority margin was registered, these deficits were not tested against one; the fine-tuned encoder was below the randomly initialized one for hypotension and acute kidney injury (intervals below zero; Methods).

Reproducing the measurement process lets the generator pass its audit We now turn to the second capability, predicting how a patient would respond to treatment. A response module, the part of a twin that makes this prediction, must be tested on records in which each patient’s effect is known, which real records never reveal. The twin’s counterfactual rollout, which generates the next hour under a chosen action, had been evaluated only on the semi-synthetic benchmark of earlier work, which we call the historical benchmark (Supplementary Fig. S2), and that benchmark could not show that the twin recovers effects. On it, an 9

earlier twin’s rollout estimated average effects with a much lower mean absolute error than a naive treated-versus-untreated comparison (0.021 against 0.213), but it was less accurate than predicting zero effect (0.017) in all five replicates, because the true average effect was zero in 71% of the benchmark’s outcome, treatment and horizon combinations (Fig. 4a; Supplementary Fig. S2; Supplementary Note 1). We therefore built a semi-synthetic generator that keeps each real eICU-CRD record and writes treatment responses onto 12 outcome channels, so that outcomes under both treatments are known (Fig. 3a,b; Methods). The analyses that follow do not score the twin’s rollout: they ask how much of each patient’s effect the records reveal and whether learners recover it from factual data. We checked realism with a classifier two-sample test 44,45 : a ridge logistic discriminator was trained to tell generated from real records of 400 matched patients from 644 summaries per patient, such as each channel’s mean and lag-1 autocorrelation. Its AUROC, the audit AUC, is 0.5 when the two cannot be told apart. A generator version passed at an audit AUC of at most 0.75, a threshold locked in advance, if its standardized-mean-difference, missingness and label checks were also met (Methods). Without corrections, the generator failed on a probe of 3,000 patients (audit AUC 0.963; Fig. 4b). An earlier version (audit AUC 0.9999) differed from real records mainly in when values were recorded: in the audit’s 400 real patients, creatinine repeated its previous value in 97% of hours, whereas that version wrote a new value every hour (Fig. 4c,d). Two parameter-free corrections reproduce this measurement process. The measurement hold records a new generated value only when the real value changed, and the missingness correction restores the gaps that real records leave in the treatment channels. Neither passed alone (audit AUC 0.956 and 0.767); together they gave 0.641 on the probe and 0.572 on all 200,825 generated stays, with every other check met (Table 2). Because the corrections changed only how values are recorded and no generator parameter was tuned, the earlier failures had mainly reflected the measurement process; only the 12 written channels still differ from the real record (Fig. 4e). The audit AUC bounds the distance between real and generated summaries only from below, so a pass cannot show that they agree in every respect (Proposition 3; Methods).

Washout panels carry individual responses of known size The generated dataset’s interventions last a single hour, so the individual-effect analyses use washout panels built with the same code, which sustain the intervention and define each patient’s effect against a copy with the injected response switched off (Fig. 3c; Methods). Each panel holds 140-hour windows of 400 real eICU-CRD records with simulated dosing of one treatment: a pattern that varies between patients in window hours 6 to 53 (natural dosing), a washout until hour 89 and, from hour 90, sustained dosing in one counterfactual arm, do(1), and none in the other, do(0). No dose occurs in the 36-hour washout before the decision, so no patient received the queried regime and it lies outside the treatment support by design (Proposition 6). The benchmark therefore asks whether a model can carry what each patient’s earlier responses reveal to a regime that the data never show. The injected response depends on decaying past exposure and on a patient-specific trait and is capped at an amplitude in reference standard deviations, a fixed

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Figure 3: Semi-synthetic data on real intensive care records. a, A real eICU-CRD history branches into simulated treatment arms, so the effect τ = Y (1) − Y (0) is known. b, A latent severity ut and mediator mt evolve hourly under the recorded treatments. The structural change δt combines a background route with a direct treatment route lagged by 2 hours and scaled by a per-stay responsiveness ηi . The generated outcome, the real value plus 0.6 δt , is recorded only in hours in which the real value changes (measurement hold). Dotted arrows mark links switched off in the factual record (the treatment policy acts only in the counterfactual arms, and the hidden confounder has weight λ = 0) or the real change times used by the hold. The generator’s latent state is labeled St in the panel and is distinct from the twin’s state zt . c, Washout panels, which carry every individual-effect result. Real windows supply the outcomes, and dosing is simulated: natural dosing, a washout, then do(1) versus do(0) from hour 90. A bounded response to a decaying memory of past doses, with patient-specific traits ζij , is injected at amplitude c. The target τi is the 12-hour contrast between the arms in the active frame minus that in a placebo frame with c = 0. d, Audits and controls: the realism audit of the generated dataset (Fig. 4b); the recovery bar (R2 = 0.2, or 0.106 on the scale 1 − RMSE/SD(τ ) of Figs. 5 and 6) against an oracle reference with a recovery of 0.80 (Fig. 5d); and the negative controls used as gates. The panels share the generator’s code and measurement hold but use a separate frozen parameterization, and their realism audit is weak evidence (Fig. 4b; Results). RMSE, root mean squared error.

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Figure 4: Reproducing the measurement process lets the generator pass its realism audit. a, Error in the average treatment effect on the historical benchmark, in five replicates (log scale). The twin rollout, from an earlier model version with a clock defect that shifted its generated physiology by at least one hour (Methods), has a far lower error than the naive treated-versus-untreated difference but a higher error than predicting zero in all five replicates, because the true average effect is zero in 71% of the outcome, treatment and horizon combinations. The true-mean-effect arm assigns the true mean effect to every combination. b, Audit AUC of the locked realism audit, a person-grouped ridge discriminator on 644 summaries of 161 channels in 400 matched patients (0.5, indistinguishable); teal bars pass its threshold of 0.75. Generator versions are shown on 3,000-patient probes (no correction, hold only, missingness correction only, both) and, with both corrections, on the full dataset of 200,825 stays. The last bars apply the same audit to the 140-hour windows of the three washout panels (discovery, replication, fixed schedule), each panel’s generated outcome values, set to missing wherever the real record is missing, with the real record for every other channel, and to the generated dataset on the same stays and windows (Methods). On these windows the audit did not flag measurement noise of up to 0.15 reference standard deviations added to the real records, and as stored the panels were separable from real records (Supplementary Table S29). c,d, Lag-1 autocorrelation (c) and share of hours that repeat the last recorded value (d) per channel, real against semi-synthetic records, for an earlier generator version that wrote a new value every hour (before; audit AUC 0.9999) and for the audited dataset (after). e, Standardized mean differences (semi-synthetic minus real) that remain in the audited dataset. Only the 12 written outcome channels differ; all others match to within 10−6 . AUC, area under the receiver operating characteristic curve.

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scale per outcome. The effect τi is the 12-hour mean difference between the arms caused by this response. Because the same trait shapes the responses recorded before the decision, the pre-decision record (window hours 0 to 89, stay hours 24 to 113) can carry information about τi , whereas measurement times come from the real record and do not depend on the effect (Supplementary Notes 4 and 5). The discovery panel holds the 400 stays of the exploratory analyses below, the replication panel 400 stays, 395 of them new, and the fixed-schedule panel instead doses every discovery patient in the same hours (window hours 6 to 24). Applied to each panel’s 140-hour window, with the panel’s generated outcome values set to missing wherever the real record is missing and the real record for every other channel, the locked audit passed all three panels (audit AUC 0.714, 0.727 and 0.720 in the discovery, replication and fixed-schedule panels; threshold 0.75; Fig. 4b; Methods), and it also passed them on window hours 0 to 89, which the estimators read (audit AUC 0.713 to 0.738). On the 140-hour windows, however, it did not flag measurement noise of up to 0.15 reference standard deviations added to the real records and did not flag a uniform heart-rate shift of 15 beats per minute (Supplementary Table S29). As stored, with filled cells and simulated dosing, all three panels were separable from real records (audit AUC 0.968 to 0.978). The pass is therefore weak evidence and does not show that the panels resemble real records.

Recovery of injected responses depends on measurement cadence Before testing estimators that see only factual data (the treatments given and the outcomes observed), we asked how much the pre-decision record reveals about each patient’s effect. We measured it with readers: ridge regressions fitted with the true per-patient effect as the label, which do not use the twin and cannot be deployed. Their cross-validated R2 estimates a lower bound on the effect information in their features (Proposition 6); the figures plot the equivalent √ recovery, 1 − RMSE/SD(τ ) = 1 − 1 − R2 . The history-blind reader uses the raw pre-decision values, with no model of how past doses accumulate. The occupancy-contrast reader projects each channel on a detrended, saturating summary of the patient’s recent doses (occupancy), and the cadence-aware reader uses only the hours in which the recorded value changes (the inferred measurement times); both take their exposure family from the generator (Methods). A reader clears the registered bar of R2 = 0.2 (recovery 0.106) when its bootstrap probability of lying below the bar is under 0.05. It separates when it clears the bar and, if the history-blind reader clears it too, exceeds that reader with a bootstrap interval of the difference above zero. We mapped recovery at several amplitudes for three treatment-response pairs on the same 400 stays (Fig. 5a–c). Heart rate, the target of the injected phenylephrine and norepinephrine responses, changes its recorded value in 57% of hours; creatinine, the target of the injected saline (NaCl) response, in 4%. On hourly records, with a new generated value every hour, the occupancy-contrast reader cleared the bar in all three pairs from half a standard deviation upward (R2 0.57 to 0.63 at one standard deviation). At real cadence, with a new value only when the real one changed, it cleared the bar in no pair at any amplitude; its highest R2 , 0.22, had an interval reaching below the bar. Of the mapped readers, only the cadence-aware reader cleared it at real cadence, on heart rate for both drugs at 0.75 and 1 standard deviation (R2 = 0.44 at one standard

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deviation; Fig. 5b,c). No reader cleared it on creatinine at real cadence at any amplitude tested (Fig. 5a; Supplementary Table S2). We then registered one cell of this map, one pair at one amplitude: phenylephrine acting on heart rate at 0.75 reference standard deviations, which caps each patient’s shift at 11.25 beats per minute. It was the smallest amplitude at which the cadence-aware reader had separated in an exploratory sweep on the discovery stays. At real cadence, the reader separated in the discovery panel (R2 = 0.33) and in the replication panel (R2 = 0.31), exceeding the historyblind reader in both (Fig. 5d; Supplementary Table S3; Supplementary Note 1). The replication panel reuses the discovery panel’s person-indexed generator draws, traits included, so it tests recovery of nearly identical effects (correlation 0.9998) from largely new records. Because the discovery run reproduces the exploratory run, the new evidence comes from the replication and fixed-schedule panels. On the fixed schedule, without differences in exposure between patients, the reader did not separate (R2 = 0.24, interval reaching below the bar), as the registered claim anticipated (Methods).

None of 16 standard learners beats predicting zero effect outside the treatment support Because readers see the true effects, we asked whether estimators that see only factual data extract the same information. Three registered benchmarks on the registered cell (Fig. 6) evaluated 13 static learners of the conditional average treatment effect (CATE; Methods), three sequence models, the Causal Transformer 22 , recurrent marginal structural networks (RMSN) 20 and the counterfactual recurrent network (CRN) 21 , and a g-computation sequence model, which simulates every recorded variable forward under the queried treatments: G-Net 23 , as implemented with the Causal Transformer code. CRN failed its synthetic positive control with a large effect and was excluded; every scored learner passed it. G-Transformer 24 , a second g-computation sequence model, was not run, because we found no released code (Methods). Most static learners were fitted to factual 12-hour treatment windows starting in window hours 6 to 42, before the washout; every model was then queried for dosing from hour 90 (the decision query). We scored learners by centered PEHE (precision in estimation of heterogeneous effects), the standard deviation across patients of the error in the predicted effect. It ignores a constant offset, so predicting zero, or any constant, scores the standard deviation of the true effects, and a learner beats this reference only if it captures differences between patients. None of the 16 scored learners outperformed predicting zero in any panel: no 95% bootstrap interval of a learner’s centered PEHE minus that of predicting zero lay wholly below zero (Fig. 6b and Supplementary Table S4). Most learners’ predictions were essentially uncorrelated with the true effects (|r| ≤ 0.124; Fig. 6c), so no rescaling could lower their centered PEHE below that of predicting zero by more than 1% (Proposition 6). The two random-slope lag models correlated 0.20 to 0.31 with the true effects, but in the discovery and replication panels their predictions spread 2.4 to 2.7 times as widely as the true effects (Fig. 6c; Supplementary Table S16). Because the decision query lay outside the treatment support, every estimator must extrapolate to it through assumptions (Proposition 6). Neither registered hour inside the dosing period

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Figure 5: Recovery of injected responses depends on measurement cadence and on how the record is read. a–c, Recovery of the individual memory-route effect from the pre-decision record against effect size (amplitude cap in reference standard deviations, SD) for NaCl acting on creatinine (a), phenylephrine on heart rate (b) and norepinephrine on heart rate (c), on the same 400 real eICU-CRD records. Each panel gives the share of hours in which the responding variable’s recorded value changes. Of the records shown, only the three real-cadence washout panels of the registered cell (d) were realism-audited, and that audit is weak evidence (Fig. 4b). Recovery is 1 − RMSE/SD(τ ), and the dashed bar of 0.106 corresponds to R2 = 0.2. Readers are ridge models fitted with the true per-patient effect, so they are not deployable estimators (Methods). Lines show the occupancy-contrast reader on hourly records and at real cadence, and the cadence-aware reader at real cadence. Bands are 95% whole-person bootstrap intervals, and filled markers clear the bar. Missing points: the cadence-aware reader at 0.75 SD for NaCl and hourly records at 0.75 SD for norepinephrine. d, The registered cell, phenylephrine acting on heart rate at 0.75 SD, in five registered runs: synthetic records, real records observed every hour, real records at real cadence in the discovery and replication panels (both shaded) and a fixed schedule. The history-blind reader uses the raw pre-decision record. The assimilating reader normalizes the occupancy contrast by the energy of the patient’s occupancy profile and adds exposure summaries and channel means; despite its name, it does not use the twin. An oracle reference that knows each patient’s trait reaches a recovery of 0.80 (R2 = 0.96) in every run. RMSE, root mean squared error.

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(30 or 42) met the registered overlap criteria on the recorded treatments (Methods), so no control inside the support was scored, and the benchmark cannot distinguish a failure of the learners from the lack of treatment variation at the query. Given the response structure, the factual record does hold information about each patient’s effect. The oracle-structure learner sees factual data only but is given the generator’s trait prior, bounded response map and measurement convention. It reached R2 of 0.33, 0.40 and 0.30 in the discovery, replication and fixed-schedule panels, as high as the cadence-aware reader on the discovery panel (Fig. 6a), and its predictions correlated 0.55 to 0.65 with the true effects. It was better than the standard learners in 95 of 102 registered comparisons after Holm correction within the static (78 comparisons), sequence (18) and g-computation (6) families, with no losses. Every win was over a comparator that did not itself beat zero. A linear response model given the same structure matched it on both metrics, so the advantage does not require the bounded response map.

ICare-RH recovers simulated responses but fails its placebo gate Because the oracle-structure learner and the linear response model are given the generator’s structure, neither is deployable. We built ICare-RH, a candidate response module for ICare-DT that is fitted to factual data alone and takes no numerical constant from the generator, although its design drew on the oracle-structure learner. The registered version does not yet read the ICareDT state (Methods). Each patient has a random response slope and a mean-reverting baseline, and the lag and decay of an exposure kernel are fitted by marginal likelihood. On 400 simulated patients from a generator matched to the registered cell, ICare-RH correlated 0.786 with the true effects and lowered centered PEHE below that of predicting zero by 0.086 (95% CI 0.073 to 0.100; Fig. 7a), passing all four of its synthetic gates. Outside its model family, it beat predicting zero in seven of eight variants and failed only when the response gain drifted (Fig. 7b; Methods). On the washout panels, ICare-RH failed its registered placebo gate and was therefore never scored on real records (Fig. 7c–e). The gate’s first checks are label-free and ran before any target was read, on each panel and on a placebo copy without the injected effect, on which predicted effects should vary far less than on the active panel. Two of them failed. The learned response lag fell outside the required 1 to 3 hours in all five discovery folds, and on placebo records predicted effects spread 1.05, 0.43 and 0.64 times as widely as on active records in the discovery, replication and fixed-schedule panels, against a limit of 0.25. The kernel decay sat at its upper bound in every fold of both natural-dosing panels (discovery and replication), on placebo and active records alike, as it had on synthetic panels with baseline drift or level shifts but no injected response (Supplementary Note 5); at that bound, the exposure feature can absorb slow drift in the baseline. These data cannot tell whether a misspecified baseline, an uncaptured response, a non-global optimum or finite-sample variation caused the failure (Proposition 7).

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Figure 7: ICare-RH recovers simulated responses but fails its placebo gate. ICare-RH is a hierarchical random-slope response model fitted to factual data (Methods). a, On the synthetic registered cell (simulated patients from a generator matched to the registered cell; Methods), correlation with the true effect against centered PEHE minus that of predicting zero, for ICare-RH (95% interval), 13 static learners and three sequence models (Causal Transformer, CT; RMSN; CRN, excluded from the registered benchmark after failing its synthetic positive control). This comparison is descriptive and spans separately registered analyses. b, ICare-RH across synthetic variants inside and outside its model family (95% intervals, filled when the interval lies below zero; correlations r at right). c–e, Label-free placebo gate on the washout panels, run before any target was read. c, Learned kernel decay per fold on synthetic records (true decay 0.8, dashed) and on the active and placebo (zero-effect) copies of the three washout panels; the parameter’s upper bound of 0.99 is dotted. The decay is a diagnostic outside the gate’s clauses. d, Learned response lag per fold on synthetic and active records, with the registered range of 1 to 3 hours shaded. e, Spread of predicted effects on placebo records relative to active records, against the registered limit of 0.25. ICare-RH was not scored on real records. Synth., synthetic; Disc., discovery; Repl., replication; Fixed, fixed schedule; PEHE, precision in estimation of heterogeneous effects; RMSN, recurrent marginal structural network; CRN, counterfactual recurrent network.

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Discussion Of the two capabilities that a patient’s digital twin needs, routine intensive care records already support one: forecasts that improve as ICare-DT assimilates the patient’s own measurements through synchronization. One is not yet supported: an estimator that sees only factual data and recovers individual effects without being given the response structure. Injected individual responses were recovered from real records only by models that a deployed twin cannot use, those given the true effects or the generator’s response structure, and at real measurement cadence only for a frequently measured variable read at its measurement times. For forecasting, our evidence for the twin concerns assimilation: its forecasts improved as it read the patient’s own new hours, although on eICU-CRD validation data a gradient-boosted model reading the same hours was the stronger forecaster. Our two control inputs test where this gain comes from and let twin builders check two of the National Academies’ three criteria 5,35 : the record held at hour 6 tests dynamic updating, and another patient’s hours test personalization (Fig. 2b,d); the third criterion, informing decisions, was not tested. The twin forecast better from the patient’s own hours than from another patient’s, a necessary but not sufficient condition for personalization, because the patient’s own last measured value held a similar advantage without any model; information beyond that value was not shown. In the registered MIMIC-IV read, the frozen model met its transfer criterion for the synchronization gain but not the secondary criterion against the last measured value, which outperformed it in 2 of 9 settings after Holm correction (Results). Evaluations of digital twins should therefore report such trivial references beside learned comparators. A fine-tuned twin state can also complement a gradient-boosted risk model. With all models refitted within MIMIC-IV, adding the fine-tuned twin state to gradient boosting gave a significant gain for acute kidney injury only (+0.0035 AUROC). This complementarity suggests hybrid designs in which a twin’s state feeds established risk models, although whether a gain of this size matters clinically was not assessed. A complete twin must also answer questions about treatment. The authors of ICareFM, an intensive care foundation model, name the separation of treatment effects from physiological deterioration an open problem 13 , and our results suggest how a twin that addresses it could be built and tested. In such a twin, synchronization would supply forecasts that improve with the patient’s own data, and a separate response module could be attached once it passes placebo checks and recovers injected effects at real measurement cadence. The twin’s own rollout has not yet been tested this way. For one injected response route in eICU-CRD, the recovery results of Fig. 5 suggest what such a response module needs. Plausible first targets are responses carried by frequently sampled vital signs, such as heart rate after vasoactive drugs, if the record is read at its measurement times (Proposition 5). The registered heart-rate amplitude, however, lies in the range of differences between phenylephrine and norepinephrine during spinal anesthesia 46,47 , whereas a randomized trial in septic shock found no such difference 48 . No reader recovered a creatinine response at real cadence. On the panels, only learners given the generator’s response structure recovered the effects from factual data, which suggests that a deployable module should learn that structure

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from the data; carrying it to a regime that no patient received still rests on assumptions (Proposition 6). It should also allow for slow drift in the patient’s baseline, which these data cannot rule out as the cause of ICare-RH’s placebo failure (Proposition 7). Finally, the panels copy measurement times from the real record, so sampling never responded to the injected effect. Clinicians often measure a variable more frequently after changing a treatment, and such treatment-responsive sampling 37 remains to be tested. These conclusions rest on a benchmark built to reproduce when values are recorded. Our results, from one database, suggest that benchmarks of irregularly sampled health records should reproduce their measurement process and report a realism audit with a preset threshold. The generated outcome values of the washout panels used here passed such an audit, but on their 140-hour windows it did not flag measurement noise of 0.15 reference standard deviations added to the real records, and as stored the panels were separable from real records, so the pass does not show that they resemble real records. Their dosing was simulated, and their stays were drawn without regard to the eICU-CRD split. Several limitations qualify these conclusions. All results come from data already used during model development: the eICU-CRD validation and test sets had been read before, the endpoint probes’ tuning set was scored twice, and several analysis choices followed earlier results (Supplementary Note 2). All data come from hospitals in the United States. Apart from one registered read of MIMIC-IV for synchronization and the clinical-endpoint probes, with hospital death and leaving the unit as disclosed secondary outcomes, the analyses used eICU-CRD. The MIMIC-IV test split had been read before, including zero-shot transfers of earlier versions of the twin, finetunes of the same pretrained model and overlapping outcomes such as hospital death and length of stay, and we knew those results when we designed the read (Supplementary Note 7). Because the harmonized records keep no death time, the death forecast, which targets death in the unit before each head’s crossing, was scored against hospital death. The frozen model was applied to MIMIC-IV without adaptation, although several inputs are recorded at different frequencies there, above all the SaO2 channel, which holds monitor saturation readings and blood-gas values in eICU-CRD but blood-gas values only in MIMIC-IV. The control inputs and the treatment-response analyses were not repeated on MIMIC-IV. The forecast targets are crossings of each vital sign’s training-set mean, which mark excursions and are not clinical events. The synchronization gain was scored in stays with hourly measurement of the target, together 42.9% of test stays. In descriptive subgroups of the hour-48 forecasts, Hispanic patients were evaluable in only five settings, where their median AUROC was 0.58, against 0.71 and 0.70 for White and Black patients in those settings (Methods). The gradient-boosting comparator was a fixed CatBoost recipe and was never tuned. The washout panels inject simulated responses, so they establish no real treatment effect, and because their treatment is a binary hourly exposure, they do not cover dose–response effects 49 . The decision query lies outside the treatment support, and no query inside it met the registered overlap criteria, so the benchmark tests extrapolation under structural assumptions as well as effect estimation. G-Net was run as implemented with the Causal Transformer code, with declared changes (Supplementary Note 9), a configuration published for other datasets and limited tuning, and its bootstrap intervals hold the fitted models fixed. G-Transformer was not run, because we found no released code. The propositions were derived after the results and rest on 20

idealizations (Methods). Within these limits, assimilating the patient’s own newly recorded measurements already improves ICare-DT’s forecasts over those it made from the stay’s first six hours. Only models given the true effects or the response structure were shown to recover injected individual responses from eICU-CRD records. The next step is a response module that needs neither: one that learns the response structure from factual data, passes placebo checks and recovers injected responses at real measurement cadence before it is tested against real treatment decisions, for example by emulating a randomized target trial 50,51 . Before clinical use, the twin’s forecasts would also need clinically defined targets, recalibration and prospective evaluation in the units where it is deployed. To support this work, we will release the generator and washout-panel code as a testbed for response modules, with the registered protocols and analysis records 33,52,53 .

Methods Data sources and cohorts We used three critical care databases that are available to credentialed users through PhysioNet 54 : the eICU Collaborative Research Database (eICU-CRD) 10 , MIMIC-IV 42 and the CareVue era of MIMIC-III 55 . They were harmonized with the BlendedICU pipeline 56 , which our code base extends to MIMIC-III. The experimental cohort of 323,677 stays comprised 200,825 eICUCRD stays of 139,362 patients from 208 hospitals, 94,458 MIMIC-IV stays of 65,366 patients and 28,394 MIMIC-III CareVue stays of 23,692 patients (Table 1). The MIMIC-III cohort was restricted to CareVue patients who do not appear in MIMIC-IV. Before this restriction, it held 61,532 stays of 46,476 patients. All analyses use eICU-CRD except the registered MIMIC-IV read of the two prediction results and its disclosed secondary outcomes. MIMIC-III enters this paper only through earlier transfer analyses, which were not registered and are disclosed in Supplementary Note 8. Records were placed on an hourly grid with 161 time-varying channels and four static covariates (age, sex, height and weight). Of the channels, 41 are physiological variables: 11 vital signs, 2 respiratory mechanics variables, 23 laboratory values, urine output and four Glasgow Coma Scale channels. The other 120 are treatment channels: drug, infusion and fluid exposures, and five ventilator settings. Medications were encoded as hourly exposures. A dose given over an interval was spread over the hours it overlapped, and a bolus was assigned to its hour. Because BlendedICU forward-fills values before export, its observation mask records availability after forward filling and does not mark measurement. From BlendedICU’s hourly data before forward filling, we therefore built a measurement record that marks the hours in which each of the 41 physiological variables was measured. The model uses this measurement record as the observation mask of the physiological variables, and BlendedICU’s availability mask as that of the treatment channels. For the four tracked vital signs, the measurement record also defines the forecast origins, labels and censoring (Model). The semi-synthetic generator and the washout panels instead infer measurement times from changes in the recorded value (Semi-synthetic generator and realism audit). Normalization statistics, and with them the crossing thresholds, were computed on training data only. They came from the eICU-CRD training split for the eICU-CRD analyses and

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from the MIMIC-IV training split for the MIMIC-IV read. eICU-CRD was split by person into training (140,473 stays of 97,553 patients), validation (30,213 stays of 20,904 patients) and test (30,139 stays of 20,905 patients) splits (inclusion flow in Supplementary Table S18). We removed from the test split all 40 stays of the 19 patients whose records had been opened in an earlier analysis. This left a test set of 30,099 stays of 20,886 patients. For the clinical-endpoint probes, a salted hash of the person identifier divided the training split further into a fitting set (119,330 stays of 82,946 patients) and a tuning set (21,143 stays of 14,607 patients). In the harmonized data, MIMIC-IV is split by person. The registered read used its training split (66,007 stays) and its test split (14,211 stays of 9,805 patients), and no MIMIC-IV validation data entered it. Sex was taken as recorded in each database. It was a static covariate of the model and also defined descriptive subgroups. Race and ethnicity are not part of the harmonized records, and the model never used them. We took ethnicity from the eICU-CRD patient table, as recorded by the contributing hospitals, and used it only for descriptive subgroup analyses of the eICU-CRD test set. Each patient was assigned the most frequent entry across their stays. The entries were grouped as White (recorded as Caucasian), Black (recorded as African American), Hispanic, Asian, Native American and other or unknown (Supplementary Table S7).

Model ICare-DT reads the hourly record of an intensive care stay: its 161 channels, their observation mask and the four static covariates (Fig. 1b,c). An encoder maps each hour’s physiology, treatments, mask and static covariates to a 64-dimensional embedding. It is a two-layer multilayer perceptron with sigmoid linear unit activations, and it adds learned random Fourier features of absolute time. A causally masked Transformer 57 then gives a 64-dimensional patient state zt that depends only on the record before the decision hour. The Transformer has two layers, four heads, feed-forward width 256, Gaussian error linear unit activations and dropout 0.1. Forecasts are made for future bins that end 1, 6, 12, 24 and 48 hours ahead. For each tracked target k and bin j, a target-specific two-layer head outputs four logits. Their softmax gives the probabilities of no event, threshold crossing, live discharge from the unit and death. The four tracked targets were heart rate, respiratory rate and the two oxygen-saturation channels of the harmonized data, SpO2 and SaO2 . In eICU-CRD, the SpO2 channel holds nursecharted pulse oximetry, and the SaO2 channel combines the bedside monitor’s periodic saturation readings with blood-gas values. In MIMIC-IV, the SpO2 channel holds hourly bedside pulse oximetry, and the SaO2 channel holds blood-gas values only. A threshold crossing is the first crossing of the standardized training mean (z = 0) in a patient not already past it. It is upward for heart and respiratory rate and downward for the saturations. Such a crossing marks an excursion of the vital sign and does not by itself define a clinical event. For oxygen saturation, the clinically defined analogue is the hypoxemia endpoint (Clinical endpoints). Forecasts start from origins. An origin for target k is an hour, before the patient left the unit, in which the target was measured. The target must not have met the crossing criterion in that hour or in any earlier measured hour. Every forecast is therefore conditional on this status. From an

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origin, the bin in which a crossing is first detected is labeled crossing, and earlier bins are labeled no event. Without a crossing, a bin is labeled no event only if the target was measured in every hour of it. Live discharge and death are taken from the unit discharge time and status recorded in eICU-CRD. Either is labeled in its bin when it falls within 48 hours and within the retained record, provided that all earlier bins are complete. Otherwise follow-up is censored at the first hour in which the target was not measured or at the end of the record. A crossing detected in the hour of discharge or death is left unresolved, and bins after the last labeled bin are masked. The loss is the origin-weighted categorical negative log-likelihood of these labels, summed within an origin P and averaged over origins. The cumulative incidence of crossing is Fk,J = j≤J Sk,j−1 pcross k,j with Sk,j = Sk,j−1 pnone k,j . Discharge and death therefore act as competing events. They rule out a later crossing and so remove probability mass from its cumulative incidence. Models were trained on the person-disjoint eICU-CRD training split (140,473 stays) with seeds 13, 17, 23, 29 and 31. Training used AdamW with learning rate 4 × 10−4 , batch size 256, weight decay 10−4 , gradient clipping at 1, at most 12 epochs and patience 3. Checkpoints were selected on validation negative log-likelihood balanced across origins and targets, with ties broken by the lower seed. Seed 29 (epoch 11) had the lowest validation loss and seed 23 the second lowest, and all five checkpoints were kept. No test data were used for training, early stopping, calibration or selection. The twin has three operations (Fig. 1d,e). Synchronization re-reads the patient’s growing record with frozen weights to update the state. The readouts predict from the patient state: they are the joint competing-risks forecast (Forecast evaluation) and the clinical-endpoint probes (Clinical endpoints). The counterfactual rollout generates the next hours under a chosen action through a transition adapter. It was evaluated only on a historical benchmark (Historical rollout). The individual-effect analyses do not use the rollout. In these analyses, readers fitted with the true effects as labels estimate a lower bound on the individual-effect information in the predecision record (Washout panels and recoverability map). Standard learners and a separately built response model, ICare-RH, were tested on whether they recover the effects from factual data (CATE learners and oracle-structure references; ICare-RH and its placebo gate).

Forecast evaluation The competing-risks forecasts were scored on the eICU-CRD test set in a single registered read. Its analysis plan was committed before the read (Table 2; Supplementary Table S1 and Supplementary Note 6). It fixed the landmark, the 17 forecast settings (four vital signs at four horizons, and death), the metrics and the rule that every setting is reported. Each of the five checkpoints was applied with frozen weights at hour 48 of the stay, and nothing was fitted, recalibrated or selected on test data. For target k and horizon H ∈ {6, 12, 24, 48} hours, the forecast is the cumulative incidence of crossing within H hours of an origin at hour 48 (Fk,J for the bin ending at H; Model). A stay entered a setting if hour 48 was an origin for the target and its outcome within H hours was resolved. An outcome was resolved as positive when a crossing was detected. It was resolved as negative when the target was measured in every hour of the horizon without a crossing, or when the patient left the unit alive or died before any crossing. We excluded stays

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without a valid unit discharge record, stays censored by an unmeasured hour or by the end of the record, and stays with a crossing in the hour of discharge or death. This left 227 to 1,449 stays of 220 to 1,349 patients per setting, with 61 to 481 crossings (Supplementary Table S19). The forecasts therefore describe patients who were still in the unit, measured and not past the threshold at hour 48, and who were then measured until the outcome was known. Without recalibration, the mean forecast was 91% to 105% of the observed rate at 6 and 12 hours and 68% to 89% at 24 and 48 hours (Supplementary Table S19). In descriptive subgroups, the median AUROC was 0.69 to 0.77 across age, sex and unit type (Supplementary Table S7). Hispanic patients were evaluable in only five settings, where their median was 0.58, against 0.71 and 0.70 for White and Black patients in those settings. The model has no separate mortality head. Its death forecast, fixed in the plan as a descriptive summary, is the largest of the four target-specific death cumulative incidences at 48 hours. Each head’s death class counts a death only if it precedes that head’s target crossing. This summary therefore targets death in the unit before a crossing, whereas the outcome counts every death in the unit in its window. The death forecast was evaluated in the 11,498 stays of 9,506 patients still in the unit after hour 48 with a valid discharge record. Of these stays, 294 ended in death in the unit within the following 48 hours. This population includes stays whose target had already been measured beyond its threshold by hour 48, so some heads were read outside the hours on which they were trained. A descriptive decision-curve analysis of the crossing and death forecasts is in Supplementary Table S8. For each seed and setting we computed AUROC, the area under the precision-recall curve, log-loss, Brier score, the mean forecast, the calibration intercept and slope, and a ten-bin reliability table. The calibration intercept and slope come from a logistic regression of the outcome on the logit of the forecast. Five-seed means carry Student t intervals with 4 degrees of freedom, which describe variation between training seeds only. Patient-level intervals came from 2,000 bootstrap replicates per seed and setting, each resampling patients with all their stays. A separate verification job reproduced all 85 seed-by-setting results exactly from the sealed prediction files (Supplementary Note 6). The same frozen forecasts were scored on MIMIC-IV as disclosed secondary outcomes of the registered read (Registered MIMIC-IV read). On the MIMIC-IV test split, the five checkpoints were applied without refitting at hour 48 to stays whose record reached hour 48 and whose unit stay lasted beyond it. The death forecast was the summary above. The forecast of leaving the unit within H hours (H = 6, 12, 24 or 48) applied the same summary to both exits. It was the largest, over the four target-specific heads, of the cumulative incidence of leaving the unit, alive or dead, before the head’s target crossed its threshold. The death forecast therefore targets death in the unit before the head’s crossing, and the exit forecast leaving the unit before the head’s crossing. Each head was trained only at hours when its target was measured and had not yet crossed. Where a head’s target had already been measured beyond its threshold by hour 48, that head was therefore read outside its training hours, as in the death summary on eICU-CRD. MIMIC-IV records the time of in-hospital death. The harmonized records used here, however, keep only a flag for death in the hospital, without its time, and no export of that time was added for this read. The death forecast was therefore scored against two outcomes. The first was death in a 24

hospitalization whose unit stay ended within 48 hours of the landmark. It includes every death in the unit in that window, and also stays discharged alive from the unit in that window whose patient died later in the hospital. The second was death at any time in the hospitalization. Leaving the unit was scored from the recorded unit length of stay, rounded up to whole hours. Each head counts a death or a unit exit only if it precedes the head’s crossing. The two death outcomes, in contrast, count every death in their windows, including deaths after a live unit exit, and the exit outcome counts every unit exit in its window. A mean forecast below the observed rate therefore arises in part by construction, as for the death forecast on eICU-CRD. The metrics were those above. Intervals are 95% whole-person bootstrap intervals of the five-seed mean (10,000 replicates), and the Supplementary Information also gives Student t intervals across seeds. These secondary outcomes were descriptive and had no registered criterion. They were scored in the 6,951 stays of 5,396 patients still in the unit after hour 48. The death forecast had an AUROC of 0.687 (95% whole-person bootstrap interval 0.657 to 0.718) for death in a hospitalization whose unit stay ended within the next 48 hours (324 stays), an outcome that includes every death in the unit in that window, and 0.726 (0.708 to 0.744) for death at any time in the hospitalization (972 stays). Its mean forecast, 0.0039, was below the observed rates of 0.047 and 0.14, in part by construction, because each target-specific head counts a death only if it precedes that target’s crossing, and both outcomes also count deaths after a live unit exit. For the exit forecast, which targets leaving the unit before a vital-sign threshold crossing, the AUROC for leaving the unit within 48 hours was 0.765 (0.754 to 0.776; 0.765 to 0.808 for horizons of 6 to 48 hours), and its mean forecast, 0.27, was below the observed rate of 0.51, in part by construction, because each target-specific head counts a unit exit only if it precedes that target’s crossing. Hospital death and length of stay had been scored on this split before, hospital death also by tabular models trained on MIMIC-IV, which reached AUROCs of 0.876 to 0.885 among stays of at least 48 hours with inputs on a corrected time grid (Supplementary Note 7). These are context only, and no comparison was made.

State synchronization and control inputs The synchronization analysis used the settings (L, H) ∈ {(12, 6), (18, 12), (30, 24)}. The synchronized forecast (S2) read hours 1 to L and predicted a crossing in (L, L + H]. The hour-6 forecast (S1) read hours 1 to 6 and predicted the same window as [F6 (2H) − F6 (H)]/S6 (H). Here F6 and S6 are the cumulative incidence of crossing and the probability of no event of any class, read at hour 6. Because L = 6 + H, this forecast conditions on no event of any class by hour L. The roster of each setting contained the stays that reached hour 30 and had the target measured in every hour through the label window. Rosters held 592 to 6,648 stays per setting on the test set, and 42.9% of test stays appeared in at least one setting. The estimate was the five-seed mean AUROC difference. The registered inference used 2,000 stay-level bootstrap replicates and Holm correction within the 12 settings 58 (Statistics and reproducibility). A re-run without the hour-30 condition (592 to 7,711 stays) gave the same result, with 12 of 12 settings surviving Holm correction and gains of 0.072 to 0.153 AUROC. Calibrated log-loss and Brier score used isotonic maps fitted on validation data. Their magnitudes depend on a probability clip, so we interpret only their direction.

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The last-value comparator ranked stays by the last measured value of the target at the landmark, in the direction of the crossing. A disclosed post hoc read scored a second hour-6 forecast as an alternative comparator: the forecast F6 (H) of a crossing within the H hours after hour 6. This earlier-window forecast exceeded S1 by 0.021 and 0.050 AUROC in two saturation settings and by 0.002 in one other, none surviving Holm correction. The synchronized forecast exceeded the better of the two hour-6 forecasts in all 12 settings (point estimates 0.073 to 0.121 AUROC). The control inputs were tested on the validation split. Each setting’s roster kept the records that reached its own landmark, rather than hour 30, and whose label was resolved, with stays censored by an unmeasured hour dropped (551 to 7,707 stays per setting). We call the four forecasts compared there the information arms (Fig. 2b; isolation arms in the protocols and the Supplementary Information). They are the synchronized forecast (S2), the hour-6 forecast (S1) and the forecasts from the two control inputs, labeled “Held at hour 6” and “Other patient” in Fig. 2b. Both control inputs changed only hours 7 to L of the record that the frozen model read. In the information-constant arm (IC), the input held at hour 6, these hours repeated the hour-6 values, and the observation mask was set to zero. In the donor arm (SH), the other-patient input, these hours were taken from a donor and spliced onto the patient’s first six hours. The donor is the other person whose hour-6 state in the frozen model is closest to the patient’s. Two sensitivity versions changed only the observation mask of the held input. IC-A repeated the hour-6 mask of the non-target variables and marked the target channels as unmeasured. IC-O did the same and also marked the forecast target as measured at hour L. A sensitivity version of the donor arm (SH-A) used the same donor. Wherever the donor’s value was missing, it kept each variable that the patient had available at hour 6, at its hour-6 value. A splice-free comparison used the donor’s own forecast. Contrasts used 10,000 whole-person bootstrap replicates, shared across arms and seeds, and add-one P values. Holm correction was applied over the 36 primary contrasts: S2 minus IC, IC minus S1 and S2 minus SH, each in 12 settings. The registered decision rules required at least 6 of 12 settings to survive correction, with no setting significantly negative before correction (unadjusted P < 0.05). Donor contrasts were also tested under a two-way bootstrap that resampled the patients and their donors separately. Log-loss and Brier score on validation used isotonic maps cross-fitted over two person folds. With these maps, the synchronized forecast had a lower log-loss than S1, IC and SH in all 12 settings. In each of these three comparisons, the difference survived Holm correction in 11 of 12 settings. Supplementary Note 5 compares ten hour-6 and donor arms. They are S1, BR (S1 with a training-fitted intercept per setting), IC, IC-A, IC-O, SH and SH-A, and three donor-only scores: the donor’s own forecast, last measured value and number of measured hours. The synchronized forecast exceeded IC-A by 0.090 to 0.150 and IC-O by 0.089 to 0.144 AUROC in all 12 settings, each after Holm correction within its own 12 settings. It exceeded the donor’s own forecast by 0.119 to 0.188 AUROC. Under the two-way bootstrap, its advantage over the donor arm survived Holm correction in all 12 settings. The donor arm itself did worse than the hour-6 forecast in all 12 settings, by 0.014 to 0.058 AUROC (point estimates outside the registered families). In SpO2 (18, 12), part of the advantage over the donor may reflect the measurement pattern rather than the measured values. In that setting, the number of measured 26

hours also separated the patient’s record from the donor’s. A difference-in-differences contrast, prespecified as descriptive, was computed on the AUROC scale. It compared the advantage over the donor arm with the corresponding advantage of the patient’s own last measured value over the donor’s. Suppose that the new hours inform the forecast only through the target’s last measured value (a current-value restriction). This contrast is then not zero in general, so zero is not its null value. The sign of its point estimates also depended on the scale on which it was computed (Proposition 2; Supplementary Note 5). The multivariate gradient-boosted comparison ran under a prospective protocol (Table 2). It fitted the fixed CatBoost recipe of the clinical-endpoint probes (Clinical endpoints) on the training split. The boosted model and the twin forecasts from the seed-29 checkpoint were scored on the same validation rows in the 12 settings. The protocol’s Holm family held 120 contrasts, ten per setting: AUROC, log-loss and Brier score of the twin against this model and against a boosted model of the target channel’s own history, the same three metrics for the multivariate against the target-history model, and the AUROC of the twin against the last measured value. A follow-up analysis on the same rows added the twin’s state to the gradient-boosted model. This raised the AUROC point estimate in 10 of 12 settings (median 0.004), but no AUROC gain survived Holm correction over its 36 contrasts. Among the log-loss gains, only that for respiratory rate at (18, 12) survived (Holm-adjusted P = 0.043). The twin’s weights were kept frozen during synchronization because, in an earlier model version, event-specific online adaptation of the forecasting heads had not improved forecasts (Supplementary Fig. S1). That evaluation used 1,895 test patients and five events at four horizons, in a configuration with one isolated head per event and in a joint configuration. A win required both AUROC and cross-fitted calibrated log-loss to improve after Holm correction over 20 settings. No setting met this criterion, and one setting in each configuration was significantly worse after correction. Updates were attempted at all 37,253 scheduled update times of each configuration and seed, and the campaign’s rollback rule reverted 31% to 37% of them. These results are a descriptive re-read of a completed campaign that used an earlier readout. Per-patient weight adaptation was set aside on this evidence before the analyses reported here.

Clinical endpoints Two registered development probes, P0 and P1 (Supplementary Fig. S3), predicted four clinical endpoints from the patient state, each within 24 hours of landmarks at 12, 24 and 48 hours. The endpoints were death in the unit, hypotension, acute kidney injury and hypoxemia. Hypotension was a mean arterial pressure below 65 mmHg in two consecutive measured hours, using invasive pressure when it was measured in that hour and non-invasive pressure otherwise. Acute kidney injury was creatinine stage 1 or higher of the Kidney Disease: Improving Global Outcomes (KDIGO) criteria, computed from in-unit values. Stage 1 was met by a rise of at least 26.5 µmol l−1 over the lowest value of the previous 48 hours. It was also met by a value of at least 1.5 times the lowest value of the previous 7 days. A stay was at risk of acute kidney injury only with a creatinine value before the landmark and a first creatinine below 353.6 µmol l−1 . Hypoxemia was SpO2 below 90% in two consecutive measured hours. A landmark row was positive if the first

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detected onset fell in the 24-hour window and negative if the record covered every in-unit hour of the window. Rows with unknown status were excluded (133 to 1,936 per endpoint). The tuning set contained 555, 3,136, 1,762 and 587 events of the four endpoints, in the order above, among 40,050, 26,318, 27,402 and 37,401 landmark rows. The comparator was CatBoost 43 with 800 trees, depth 6, learning rate 0.03 and random seed 13, fitted without tuning. Its 1,136 features summarize the record: seven statistics of each channel plus the static covariates. Probe P0 attached a logistic model or a multilayer perceptron to the frozen 64-dimensional state of seeds 29 and 23, the two checkpoints with the lowest validation loss. It also added the same state to the comparator’s features. Probe P1 was designed after P0 had read the tuning labels. It trained the encoder on the four endpoints, starting from the pretrained weights or from random initialization. A learned landmark embedding was added to the state, and a two-layer head read the endpoints. P1 also added the fine-tuned state of the pretrained encoder to the comparator’s features, and we call the resulting model the hybrid. Encoders were trained with AdamW at constant learning rates (weight decay 10−4 , batch size 256 stays, gradient clipping at 1, at most 15 epochs). Training stopped early on a 10% inner split of persons, with a patience of two epochs. Fine-tuning from the pretrained weights used learning rates of 10−4 for the encoder and 10−3 for the head, and training from random initialization used 5 × 10−4 for all weights. The comparison between the two encoders therefore confounds initialization with learning rate. Each model trained in about 30 seconds on one graphics processing unit (GPU). The pretrained encoder had been trained on every tuning-set patient, which favors the frozen and fine-tuned states. AUROC contrasts used 2,000 whole-person bootstrap replicates over 14,122 patients shared by all arms. No family-wise correction was applied to these development probes, whose gates are given in Table 2. The same recipes were refitted within MIMIC-IV for the registered read.

Registered MIMIC-IV read The two prediction results, synchronization and the endpoint probes, were read once on the person-disjoint test split of MIMIC-IV under a protocol committed and tagged before the read. The protocol fixed the analyses, the decision rules and the sentences that report each outcome, and every outcome is reported (Supplementary Note 7). The test manifest listed the stays that the read could use. It contained every MIMIC-IV stay of the test split with a harmonized time series (14,211 stays of 9,805 patients), with no filter on length of stay, context or outcome. It was built from identity columns alone and had to match the stay count, the permitted range of the patient count and the digests fixed in the protocol. This test split had been read before. The protocol discloses 34 earlier entries, 27 of them scientific reads (Supplementary Note 7 reproduces the list). These include zero-shot transfers of earlier versions of the eICU-CRD-trained twin, finetunes of the same pretrained model and outcomes that overlap ours: the blood-pressure rule that defines our hypotension endpoint, an earlier acute kidney injury definition, hypoxemia, heart rate above 130 per minute, SpO2 below 90%, and future heart and respiratory rate. We knew their results when designing the read. The quantiles used to clip and fill the harmonized records had also been computed over all splits, including the test covariates but no labels.

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Family A: synchronization with frozen weights. The five eICU-CRD checkpoints were applied without refitting, and no weight, intercept, calibration map or threshold was fitted. Inputs, including the continuous dose channels, were standardized with MIMIC-IV training statistics, so the crossing thresholds were the MIMIC-IV training means. The domain embedding was switched off, and no channel was remapped. In particular, MIMIC-IV’s hourly pulse oximetry was not copied into the arterial-saturation input, which in eICU-CRD carried the dense monitor saturation. Several inputs of the frozen model are recorded at different frequencies in MIMIC-IV. In the training data, arterial oxygen saturation is observed in 1.2% of MIMIC-IV hours against 49.2% of eICU-CRD hours. Heart rate and non-invasive mean arterial pressure are observed less often in MIMIC-IV (MIMIC-IV to eICU-CRD ratios 0.79 and 0.62), and most laboratory values more often (ratios 1.4 to 4.4). The SaO2 target was excluded before the read, because in MIMIC-IV that channel holds sparse blood-gas values and at most 10 training stays were evaluable in any setting. This left 9 settings, for heart rate, respiratory rate and SpO2 at (L, H) = (12, 6), (18, 12) and (30, 24). The MIMIC-IV SpO2 settings use hourly bedside pulse oximetry, unlike the nurse-charted pulse oximetry of the eICU-CRD SpO2 target. Each MIMIC-IV SpO2 setting can therefore be set beside either eICU-CRD saturation setting. Supplementary Note 7 shows both pairings, and neither enters a decision. The arms (S2, S1 and the patient’s last measured value), the labels and the hour-30 roster rule were those of the eICU-CRD analysis. The unit exit was the recorded unit length of stay rounded up to whole hours. Leaving the unit, alive or dead, counts as negative in the labels of families A and B, so these labels do not depend on the hospital-death flag. In both databases, the harmonized records can contain hours after the recorded unit exit. An endpoint-defining channel was measured after the recorded unit exit in 75.2% of MIMIC-IV training stays, against 58.5% of the eICU-CRD fitting set. Labels stop at the unit exit, but a stay can still enter a roster when rows recorded after the exit carry its record to hour 30. The primary contrast (A1) was S2 minus S1, and the registered secondary contrast (A2) was S2 minus the last measured value. Each estimate was a five-seed mean AUROC difference. Family A resampled whole patients with 10,000 replicates, used two-sided add-one P values and applied Holm correction over the 9 settings, separately for A1 and A2. The eICU-CRD analysis, by contrast, had resampled stays with 2,000 replicates, used P values without the add-one rule and corrected over 12 settings. Three registered sensitivities were run on MIMIC-IV. One used the stay-level inference of the eICU-CRD analysis, one dropped the hour-30 roster condition and one set the thresholds at the eICU-CRD training means. A setting survived when its Holm-adjusted P value was below 0.05 and its estimate was positive, with most replicates on the side of the estimate. It was significantly negative when its estimate was negative, its unadjusted P value was below 0.05 and most replicates lay below zero. The registered transfer criterion was met when at least 5 of 9 settings survived and none was significantly negative before correction. The outcome was classed as contradicted when any setting was significantly negative after Holm correction, and A2 received the same categories. Inference conditions on the five frozen checkpoints. The control inputs and calibrated metrics were not computed on MIMIC-IV. The disclosed secondary outcomes used the same checkpoints at hour 48 (Forecast evaluation). Family B: endpoint probes refitted within MIMIC-IV. The arms of the endpoint probes were 29

refitted within the MIMIC-IV training split (66,007 stays) with the recipes as run on eICU-CRD and with early stopping on 10% of training patients. There were four arms: the boosting comparator (one fit, tree seed 13), two encoders and the hybrid. One encoder was trained from random initialization and the other fine-tuned from the pretrained twin, each with five seeds (13, 17, 23, 29 and 31). The hybrid combined the boosting features with the fine-tuned state. The endpoints were hypotension and acute kidney injury (primary) and hypoxemia (secondary), defined as on eICU-CRD and scored at landmarks of 12, 24 and 48 hours with a 24-hour horizon. Death in the unit was not scored because the harmonized records lack its time, and the training loss therefore pooled three endpoints. The hypotension endpoint is the hypotension rule that earlier reads had scored on this split, apart from its at-risk rule, clock and readout. Acute kidney injury and hypoxemia as defined here had not been scored there, although an earlier kidney-injury definition and an earlier hypoxemia definition had been. Checks of physical units and invalid values in the endpoint channels ran on the training and test splits before any family-B label was written. The primary analysis was a one-sided test of the AUROC difference between the hybrid and boosting, pooled over landmarks. It used add-one P values from 10,000 whole-person bootstrap replicates over the 9,623 test patients whose record reached hour 12, with Holm correction over the two primary endpoints. For the endpoint probes on MIMIC-IV, intervals are two-sided 95% percentile intervals and the endpoint test is one-sided, so a gain significant at the second Holm step can have a lower interval bound at or below zero. Hypoxemia carried an unadjusted onesided P value. The difference between the encoder trained from random initialization and boosting was reported with intervals, without a test or a non-inferiority margin. Two seed analyses were registered as sensitivities and enter no test. When boosting was refitted under five random tree seeds, its AUROC spanned a range of 0.0008 for hypotension and 0.0004 for acute kidney injury. When each hybrid and boosting used the same tree seed (paired seeds), the hybrid’s gain over boosting was +0.0010 (−0.0012 to +0.0032) for hypotension and +0.0034 (+0.0006 to +0.0062) for acute kidney injury. For hypoxemia, a registered secondary endpoint, the hybrid’s change was +0.0009 (−0.0034 to +0.0054; unadjusted one-sided P = 0.35). The encoder trained from random initialization differed from boosting by −0.0109 (−0.0145 to −0.0072) for hypotension, −0.0294 (−0.0367 to −0.0226) for acute kidney injury and −0.0197 (−0.0275 to −0.0118) for hypoxemia; no non-inferiority margin was registered. The encoder fine-tuned from the pretrained twin was below the one trained from random initialization for hypotension and acute kidney injury, with intervals below zero, and differed from it by less than 0.0001 for hypoxemia (Supplementary Note 7). Separation of predictions from labels. The analysis code enforced the separation. The prediction stages saw the test records only through a view that refused every outcome field and showed hours 1 to 30 (family A) or 1 to 48 (family B and the secondary outcomes). Outside this view, the data loader still held the previously read hospital-death and length-of-stay columns in memory. The prediction processes also held later hours that fall inside label windows, and these hours were unused by construction. A fail-closed test, run on the real models with synthetic inputs, verified that every prediction issued at a landmark stayed bit-identical when all later hours were replaced at random. Predictions were frozen and hashed before any label was built, and the scorer opened the labels once. A verification stage recomputed every value with the same scoring code, which shows that the scoring is deterministic. A separate coding-agent pass with 30

read-only access re-applied the decision rules to the shipped contrasts (Use of large language models and coding agents). Supplementary Note 7 gives the power analysis written before the read, the deviations listed in the protocol and those that arose after it.

Historical rollout The historical benchmark of earlier work on this model (Fig. 4a) had five replicates, each drawn afresh from its data-generating model. Each replicate held 60 cells, one for each combination of outcome, treatment and horizon. Each cell held 19 to 30 treatment decisions, of which 1 to 10 were treated (Supplementary Note 1). Supplementary Fig. S2 gives stress analyses of the benchmark. The action-conditioned rollout scored on it had a clock defect. It wrote each transition sample into the row of the action that conditioned it, although the transition adapter had been trained to predict physiology one hour after that action. Generated physiology therefore led its time index by at least one hour, and by more where the adapter carried persistence through generated physiology. In deterministic toy tests (Supplementary Note 3), the defect inflated the mean response over the benchmark’s 12-hour window by factors of 1.00 to 1.38. Its effect on the historical readouts was not estimated. We report those readouts as run, and no positive finding of this paper rests on them.

Semi-synthetic generator and realism audit For each eICU-CRD stay, the generator takes the first 336 hourly rows of 12 outcome channels, standardized with the cohort normalization statistics (Fig. 3b; implementation details and constants in Supplementary Note 4 and Supplementary Tables S9 and S10). The channels are invasive mean arterial pressure, SpO2 , lactate, heart rate, respiratory rate, urine output, creatinine, pH, white cell count, platelets, bilirubin and temperature. A latent severity u and a fourdimensional mediator m evolve hourly, ut+1 = 0.85 ut + 0.10 m̄t + 0.05 āt + ξtu ,

mt+1 = Am mt + b ut + B ⊤ at + ξtm ,

(1)

with noise standard deviation 0.25 and states clipped at ±8. The states are driven by the indicators at of recorded doses of five treatments: phenylephrine, NaCl, norepinephrine, dextrose and inspired oxygen fraction. The generator therefore responds to whether a dose was recorded and ignores its size. The recorded doses are written back at single precision. The generated outcome real + 0.6 δ , with is ytk = ytk tk

δtk = 0.02 wk⊤ [ut , mt ] + εtk + 0.8 ηi (W at−2 )k ,

(2)

which combines a small latent background with a direct treatment route lagged by 2 hours. The sparse, sign-aligned weights W are drawn once, the structural change δ is clipped at 6 standard deviations, and y is clipped at 8 and to physiological bounds. The per-stay responsiveness ηi is drawn from 1 + N (0, 0.52 ), independently of the record, and clipped to [1/3, 3]. Individual responses therefore differ between stays, but the generated dataset carries no covariate-dependent

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effect heterogeneity. The generator’s hidden confounder is simulated without effect (λ = 0). For 674 test-split stays long enough for the 48-hour horizon, counterfactual outcomes were rolled out for 48 hours from the factual latent state at 20 random decision hours. Each treatment was set to 0 or 1 at the decision hour, later treatments were drawn from the generator’s propensity policy, and no common random numbers were shared across arms. The counterfactual truth is the minimum and maximum of the recorded outcome over 6, 12, 24 and 48 hours. Generated with the two corrections described next, this dataset passed the locked realism audit. The individual-effect analyses do not use this dataset. They use the washout panels (Washout panels and recoverability map). These sustain the intervention for 48 hours after a washout, share random numbers across counterfactual arms and define each patient’s effect against a placebo frame. Two parameter-free corrections were added to the generator as options that are off by default. The measurement hold records a new value only in hours in which the standardized real value differs from the previous hour’s value. In other hours it repeats the previous recorded value, as the forward-filled real record does. The hold applies to factual and counterfactual paths alike, so the counterfactual truth is the effect on the recorded value. The missingness correction restores the real missingness of the treatment channels and is permitted only when factual treatments replay the recorded doses. All other channels keep their real values and missingness. The locked realism audit compares 400 matched patients on 161 channels with a persongrouped, five-fold ridge logistic discriminator. Its 644 features are per-patient means, standard deviations, missingness and lag-1 autocorrelations. The audit requires an AUC of at most 0.75 together with standardized mean difference, missingness and label checks. The passing dataset contains all 200,825 eICU-CRD stays, generated without error. The two corrections were designed after diagnosing earlier audit failures. The audit was read for each generator version shown in Fig. 4b. The audit AUC bounds the distance between the real and generated summaries from below and never from above (Proposition 3). On the 3,000-patient probe with both corrections, a flexible classifier separated the real and generated summaries more than the audit did (out-of-fold AUC 0.735, against the audit’s 0.641; Supplementary Note 5). This classifier was histogram gradient boosting in one configuration, chosen after the audit. The same audit was applied to the three washout panels of the registered cell, under a protocol registered before it was run. It was run after all the panels’ other analyses, except the in-support control and the g-computation comparator, whose registered runs followed it (Supplementary Note 2). Each audited record combined the panel’s generated outcome values with the real record for every other channel. The generated values were set to missing wherever the real record is missing, and both were cut to hours 24 to 163 of the stay. Its treatment channels are therefore the real ones, while the outcome values respond to simulated dosing, a mismatch that the audit’s per-channel summaries cannot detect (Proposition 3(b), Supplementary Note 5). The summary-difference checks cannot detect a difference confined to the 12 written channels, so the discriminator is the operative check. Every audit, control and interval is in Supplementary Note 9 and Supplementary Table S29. The panels passed (audit AUC 0.714 to 0.727), but the audit did not flag measurement noise of 0.15 reference standard deviations added to the real records. For comparison, the semi-synthetic dataset that passed the audit gave audit AUC 0.581 on the same windows of the discovery and fixed-schedule stays and 0.591 on those of the replication stays. 32

Washout panels and recoverability map Each washout panel contains 400 eICU-CRD windows (Fig. 3c). The windows cover hours 24 to 163 of stays with at least 164 hourly rows, selected by a seeded permutation without regard to the eICU-CRD split. The real window supplies the 12 outcome channels, with missing values filled by fixed reference means. The panels use the generator’s code with a separate frozen parameterization. Its background scale is 1, and its mediator persistence is 0.82 (spectral radius of the latent map 0.93, half-life 10 hours). It has no per-stay multiplier on the ordinary treatment routes and uses common random numbers across counterfactual arms. Outcomes are standardized to fixed reference values (85 and 15 beats per minute for heart rate). The panels’ generated outcome values passed the realism audit described above, which on their windows did not flag measurement noise of 0.15 reference standard deviations added to the real records. The treatment columns are replaced by simulated dosing. In hours 6 to 53, one treatment follows a natural dosing pattern. It is taken from a factual rollout of the generator’s propensity policy on the same window and topped up to at least eight dosed hours. Hours 54 to 89 are a washout in which this treatment is not given. From hour 90, it is held for 48 hours at 1 in one counterfactual arm and at 0 in the other. The other four treatments follow the propensity policy in both arms. The injected memory-route effect on outcome k is stk = c tanh

X

 Lkj ζij qtj ,

qtj =

j

etj , 1 + etj

etj = 0.8 et−1,j + at−2,j .

(3)

Here c is the amplitude cap in reference standard deviations, and Lkj is the normalized loading of treatment j on outcome k (−1 for phenylephrine on heart rate). The patient’s trait for treatment j is ζij = min{max{0.6 lj ξi + 0.15 ϵij , −2/3}, 2}, with a shared standard normal factor ξi , treatment loadings lj drawn once from U (0.7, 1.3) and independent standard normal ϵij . The shift is applied after the ordinary clips, is attenuated near a bound and does not feed back into the latent state. The target τi starts from the do(1) minus do(0) difference in the responding outcome (heart rate in the registered cell), averaged over the 12 hours after the decision, in the active frame. From it we subtract the same average in a placebo frame with c = 0 and all random draws shared. This subtraction cancels the ordinary treatment routes. The target uses this outcome at every hour, without the measurement hold. Synthetic records, real windows observed every hour and real windows at their real cadence were all read with the same target. At real cadence, the generated record repeats its previous value in every hour in which the real value did not change. Its measurement times therefore come from the real record and do not depend on the injected effect. The discovery panel consists of the 400 stays used for the recoverability map. A fixed-schedule control on the same stays doses every patient in hours 6 to 24. The replication panel is a second sample of 400 stays, 395 of them new, so its pre-decision records and natural dosing differ from those of the discovery panel. Because it keeps the person-indexed generator draws, including the traits, its true effects are nearly identical to those of the discovery panel (correlation 0.9998). Readers are ridge regressions, fitted in five person-grouped cross-validation folds with the true per-patient effect as the label. They differ in the features they take from the pre-decision

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record. The history-blind reader uses the raw record. The occupancy-contrast reader uses the inner product of each channel’s record with the patient’s detrended occupancy profile, computed as q in the injected effect but with the reader’s lag and decay. The assimilating reader normalizes this inner product by the profile’s energy and adds exposure summaries and channel means. Like the other readers, it is a ridge regression and does not use the twin or its synchronization. The cadence-aware reader sums, over consecutive measurement times of each channel, the change in the recorded value times the change in the occupancy profile. Hours without a new value therefore contribute nothing. The occupancy-based readers use the generator’s exposure family. They choose its lag (0 to 3 hours) and decay (five values), together with the ridge penalty, by an inner four-fold cross-validation on the training patients. The grid contains the generator’s values. The oracle reference uses each patient’s true trait. Recovery is 1 − RMSE/SD(τ ), where RMSE is the root mean squared error of the reader’s prediction. Its intervals come from 2,000 whole-person bootstrap replicates. The figures show recovery, and the text reports the equivalent √ R2 = 1 − (1 − recovery)2 . A reader clears the registered bar of R2 = 0.2, which is 1 − 0.8 = 0.106 on the recovery scale, when its bootstrap probability of falling below the bar is under 0.05. It separates when it clears the bar and, if the history-blind reader also clears it, exceeds that reader with a bootstrap interval of the difference above zero. The second condition stops a reader from being credited with information that the history-blind reader already extracts from the raw record. Under a schedule shared by all patients, readers that apply one affine map to every record, such as the occupancy-contrast and assimilating readers, cannot exceed the history-blind reader in population R2 (Proposition 5). The cadence-aware reader is not such a reader, because it still sees each patient’s own measurement times. The proposition therefore does not explain why it did not separate on the fixed schedule. The cadence-aware reader was also compared with six learned controls and with two constant predictions, zero and the training mean. This comparison followed a development protocol written before the run and was not registered (provenance in Supplementary Table S1). The learned controls were ridge models fitted in the same way. Their features were the same actions together with summaries of the admission hours, of the six hours before the decision or of the full predecision record. Each comparison was made in the discovery, replication and fixed-schedule panels on two metrics, and Holm correction was applied over the resulting 48 contrasts. The first metric was the precision in estimation of heterogeneous effects (PEHE), the root mean squared error of the predicted individual effects. The second was centered PEHE (defined under CATE learners and oracle-structure references). After Holm correction, the cadence-aware reader won 32 of the 36 contrasts with the learned controls and all 12 with the two constant predictions. It lost none, and the four unresolved contrasts were all on the fixed schedule. The registered amplitude, selected after an exploratory sweep (Registration and provenance), was 0.75 standard deviations for phenylephrine acting on heart rate. On the panels’ reference scale of 15 beats per minute, it caps each patient’s shift at 11.25 beats per minute. The median shift is 3.8 beats per minute at sustained steady-state exposure and 4.5 at full occupancy, which the kinetics never reach. The realized target has a standard deviation of 3.5 beats per minute. Shifts of this size lie in the range of the heart-rate differences between norepinephrine and phenylephrine in randomized comparisons during spinal anesthesia for cesarean delivery 46,47 . 34

Those differences were measured in patients with intact baroreflexes. In septic shock, however, a randomized intensive care trial found no such difference 48 .

CATE learners and oracle-structure references Static learners were S-, T-, X-, DR- and R-learners with histogram gradient boosting or ridge base models 59–61 and causal forests with and without kernel-informed features 62,63 . They also included TARNet 64 , DragonNet 65 and random-slope distributed-lag models with and without the known kernel 66 . Sequence models were the Causal Transformer 22 , RMSN 20 and CRN 21 , run from their reference implementations. The g-computation sequence model was G-Net 23 , as implemented with the Causal Transformer code; it simulates every recorded variable forward under the queried treatments and averages 100 Monte Carlo trajectories with the same random draws in both arms. Besides the positive control, it had to recover a known average effect under timevarying confounding before it was scored. G-Transformer 24 was not run: no released code was found in the public sources searched on 8 October 2026 (Supplementary Note 9). Hyperparameter grids and a synthetic positive control with an effect standard deviation near 1 were fixed in the protocols. A learner that failed its positive control was excluded. The primary metrics were centered PEHE and the correlation with the true effect. Centered PEHE is the standard deviation of the per-person error, and it equals SD(τ ) for a zero prediction. Each learner was compared with the oracle-structure learner on both metrics in three panels. The comparisons used 10,000 whole-person bootstrap replicates and Holm correction within each family (78 static, 18 sequence and 6 g-computation comparisons); statements that combine the three families carry a family-wise error bound of 0.15 (Supplementary Note 1). A comparison was declared a win or a loss only if it survived correction and was otherwise unresolved, which never implies equivalence. G-Net was also compared with predicting zero (Holm over 2 comparisons, in the discovery and replication panels); a gain counted only if it survived correction in both panels, the model ranked patients in both, and the gain was at least 2% of the standard deviation of the true effects. The decision query is sustained dosing from hour 90, after a 36-hour washout in which no patient was dosed. No patient received the query regime, so its propensity is zero, and every estimator must extrapolate to it through assumptions (Proposition 6). The g-computation model and the control inside the support were added after the results at hour 90 were known, at the request of an internal review, under a protocol registered before they were run; the true effects at hour 90 had been read by earlier analyses (Supplementary Note 9), and the candidate hours (30 and 42) were set after label-free diagnostics of the Causal Transformer’s predictions at hour 42 had been seen. A control inside the treatment support was registered at hours 30 and 42, each to be scored if it met overlap criteria on the recorded treatments (at least 20 patients per panel receiving each regime; at least half of the query patients had at least 10 treated and 10 untreated training windows within three hours with the same number of recent doses); neither met them. The oracle-structure learner is a particle-filter response model (8,192 particles). It fits response weights and population-level nuisance parameters from factual data, but it knows the generator’s trait prior, bounded response map and measurement convention. The structureinformed linear response model weights the same particles with a linear memory increment in

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place of the bounded map. It was scored as a descriptive reference. Label-free diagnostics were descriptive and could not gate or select. One of them, a person-grouped four-fold domain classifier, separated the query from the training units, the factual 12-hour treatment windows (AUROC above 0.999). At most three treated training units per fold fell in the query’s recency cell (dosed earlier, no dose in the preceding 12 hours). An earlier label-free diagnostic at hour 42, which did not meet the overlap criteria, without true effects, had shown the Causal Transformer’s predicted effects varying as little across patients as at hour 90 (per fold, standard deviation 0.03 to 0.09 at hour 42 and 0.03 to 0.09 at hour 90).

ICare-RH and its placebo gate ICare-RH is a hierarchical random-slope response model. Its design drew on the structure of the oracle-structure learner and was fixed on synthetic data before the model was fitted to any real panel. The model describes each channel k at its measurement times as yik (t) = x⊤ t βk + γk bi ϕ(ei (t)) + aik + oik (t) + εik (t). Here et = λet−1 + At−ℓ is a learned exposure kernel with lag ℓ, and ϕ(e) = e/(κ + e) is its saturation. The patient’s response slope is bi ∼ N (m(si ), 1), whose prior mean m may depend on conditioning features si , and γk is its loading on channel k. The term aik is a random intercept, oik a mean-reverting Ornstein–Uhlenbeck baseline evaluated at measurement times, and x⊤ t βk a 12-lag common route with a learned exponential tail. The slope is integrated out analytically, and the remaining parameters are fitted by type-II marginal likelihood, with no constant taken from the generator. The individual effect is predicted from the posterior mean of bi given the patient’s own records before the query. In the registered arm the prior mean is a constant, so the arm uses no ICare-DT state. ICare-RH passed all four of its synthetic gates. They were scored on 400 simulated patients in five folds, from a generator matched to the registered cell (true effect standard deviation 0.227, query at hour 90). Its correlation with the true effect was 0.786 against a required minimum of 0.75, and its centered PEHE was 0.141 against a maximum of 0.15. On a null scenario, its predicted effects had a spread of 0.021 against a maximum of 0.05. It recovered the exposure kernel in four of five folds, the minimum required. Fifteen variants inside and outside the model family were scored descriptively (Fig. 7b). Outside its model family, ICare-RH beat predicting zero in seven of eight variants (correlation 0.64 to 0.84) and failed only when the response gain drifted (correlation 0.49; Supplementary Table S5). On real panels, the registered placebo gate applied label-free checks before any target was read. Its lag clause required a response lag within 1 to 3 hours in at least three of five folds of each natural-dosing panel. This clause also counted the folds in which a slope-variance test gave P < 0.05, but the count did not gate. Before any real panel was read, the test had already rejected in 199 of 200 folds on 40 synthetic null panels with baseline drift or level shifts. This rate is consistent with unmodeled baseline variance along the exposure direction (Proposition 7). The gate’s spread clauses concerned the zero-effect copies of every panel. On these copies, the spread of predicted effects had to be at most 0.05 standard deviations and at most 25% of the spread on active records. A final clause was to be read only after these checks. It required the 95% whole-person bootstrap interval of centered PEHE minus that of predicting zero to lie wholly

36

below zero in both natural-dosing panels.

Theoretical analysis of the evaluation Seven propositions state what each evaluation design can and cannot identify. They were derived after the results, and no eICU-CRD analysis was chosen, registered or stopped because of them. Three of them largely restate standard results. Proposition 1 rests on the Neyman– Pearson lemma and the refinement of proper scores. Proposition 3 rests on bounds of an AUC by distances between distributions, and Proposition 6 on the decomposition of mean squared error by correlation and spread. Propositions 5 and 7 build on standard results of generalized least squares and of the profile likelihood of a one-way variance component. Supplementary Note 5 gives the statements, assumptions, proofs and numerical checks. It also names the standard part of each proposition with its source and states where the idealized models and the record disagree. Throughout, moments are population moments, and AUROC gives half credit to ties. Proposition 1. Let Ft be the record through hour t and pG the probability of the label on the roster given information G (the ideal forecast). By standard results, pG has a receiver operating characteristic curve on or above that of every score computed from G 67,68 , and refining G cannot worsen its Brier score or log-loss 69,70 . For a fixed label, the ideal forecasts pFt form a martingale in t, the property that martingale-consistent self-supervised learning imposes on learned predictions 71 . Because L = 6 + H and roster stays are at risk at hour L, the ideal forecast S1∗ given the hour-6 arm’s input and conditioning event equals pF6 and differs from the ideal synchronized forecast S2∗ only in information. Hours from a donor whose record is conditionally independent of the patient’s label and later hours, given the first six hours, add no information. For AUROC and proper scores, the learned contrast S2 − S1 equals the ideal gain minus the shortfall of S2 from S2∗ plus that of S1 from S1∗ . It is therefore at most the ideal gain exactly when the hour-6 read falls short by no more than the synchronized read. The proposition uses only one property of the twin: its hour-t state is a function of the record through hour t. Proposition 2. Let V and V ′ be the patient’s and the donor’s last measured target values at hour L. In a latent-threshold Gaussian model, where each AUROC maps to a latent correlation ρ, consider a forecaster that reads hours 7 to L only through V , with index U6 + βV V (U6 affine in the first six hours, βV > 0). Under exact matching, its latent difference-in-differences is (wV − 1)(ρown − ρdonor ), where wV := βV SD(V )/SD(U6 + βV V ) is unobserved. For the ideal forecaster, two inequalities on the four latent correlations hold in every such model. Quadruples that satisfy them are attainable under the current-state hypothesis HCS (label independent of hours 7 to L given the first six hours and V ), and those that satisfy them strictly are attainable also without it. The four AUROCs therefore cannot reject HCS , or confirm it when both inequalities are strict. A violation would falsify the idealized model rather than HCS . Within that model, the label-free correlation of V and V ′ identifies HCS . Proposition 3. For a classifier two-sample test 44 , |AUC − 21 | is at most TV − TV2 /2, and hence at most the total-variation distance TV between the real and generated laws of the 644 summaries. The first bound is sharp, so the audit bounds that distance from below and never from above. The population audit (L2 -penalized logistic regression with an unpenalized intercept,

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fixed standardization) has zero slope exactly when the mean summaries agree, and its AUC is then 21 whatever other differences exist. A mean difference makes the slope nonzero and, under an equal-covariance Gaussian idealization, makes the AUC exceed 12 , although in general it need not. Proposition 4. With attenuation inactive, the washout target is τi = c Ti and the held record is exactly Zi0 + c Ui , where the zero-effect record Zi0 , the shift per unit amplitude Ui , Ti and the design (doses and measurement times) do not depend on the amplitude c. For a reader affine in the record at fixed kinetics, under an orthogonality condition, the population R2 gain over the reader’s design-only features satisfies 1/gain ≥ C0 + C1 /c2 , with equality when one statistic carries the signal. Here 1/C0 is the ceiling as c → ∞, C1 sets the gain at small amplitudes, and a change of sampling multiplies the amplitude needed for a small gain by (C1new /C1old )1/2 72 . For one responding channel in the linear regime with a random-walk baseline, the held record’s P Fisher information about the trait is proportional to J := (∆q)2 /g, summed over consecutive measurement times d < d′ with ∆q := q(d′ ) − q(d), g := d′ − d and q the generator’s occupancy P profile 73 . For hourly records the corresponding quantity is Q◦ := t (q(t) − q(t − 1))2 , and the retention χ := J/Q◦ lies in [0, 1]. Proposition 5. Among linear statistics of one channel that are invariant to a nuisance mean, the generalized least-squares weight is efficient 74 . Under white noise with a free quadratic trend, it gives, on hourly records, the detrended level contrast of the occupancy-contrast reader 75 . UnP der a random-walk baseline, the gap-weighted sum ∆Z ∆q/g is efficient among level-invariant P linear statistics. The implemented cadence-aware statistic ∆Z ∆q omits the gap weights. It is therefore efficient on hourly records, where every gap is one hour, but in general not on held records. Under a schedule shared by all patients, the assimilating and occupancy-contrast readers apply one affine map to every record and cannot exceed the history-blind reader in population R2 . Proposition 6. For a predictor τ̂ with correlation r with τ , the standard decomposition 76 cPEHE2 /Var(τ ) = (1 − r2 ) + {SD(τ̂ )/SD(τ ) − r}2 implies that τ̂ has lower centered PEHE than predicting zero exactly when 0 < SD(τ̂ ) < 2r SD(τ ). No function of data X has squared correlation with τ above R∗2 (X) := Var E[τ | X]/Var(τ ), which the conditional mean attains 77 . A supervised reader’s held-out R2 therefore estimates a lower bound on this information content and never bounds estimators from above. No dose occurs at or after hour 54, so the query regime has propensity zero, and the factual law fixes neither the conditional mean effect nor any predictor’s accuracy; every estimator must extrapolate through assumptions. Suppose that the trait that sets the pre-decision shift also sets the query effect, that the design and zero-effect record are independent of the trait, and that the response structure, including the query map, and the zero-effect law are known. Then the conditional mean effect is a known function of the pre-decision record. In large samples and with equal exposure information, an empirical-Bayes random slope 78 whose working model understates the baseline variance along the exposure direction has a spread ratio above its correlation. Proposition 7. For ICare-RH at a fixed kernel, loading direction and baseline (Ornstein–

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Uhlenbeck working covariance at the measurement times 79 ), let Ai and Bi be the exposure P P information and the exposure projection of the baseline residual, µ̂ := i Bi / i Ai , and ω := P P 2 i Ai the overdispersion along the exposure direction. With the prior mean proi (Bi − µ̂Ai ) / P filed out, the negative marginal log-likelihood of the slope variance v := k γk2 (bi has unit prior P variance) is Pn (v) = Pn (0) + v2 ( i Ai )(1 − ω) + O(v 2 ), as for a one-way variance component 80 . For ω ̸= 1, a small positive v therefore improves the fit exactly when ω > 1. With equal information and no mean response along the exposure direction, the exact profile likelihood prefers in large samples, of two kernels, the one whose population ω is larger and exceeds 1. When ω stays above 1, the fixed-baseline likelihood ratio for v diverges instead of following its 50:50 χ̄2 reference 81 . A decay of 0.99, the parameter’s upper bound, keeps at least 43% of the end-of-dosing exposure at hour 89, so in the random-walk limit of the working covariance the exposure profile is not orthogonal to linear drift. If the baseline is correct along the exposure direction and a globally optimized fit captures a response of the form the model assumes, then under regularity conditions the two spread clauses of the placebo gate pass with probability tending to one. The lag clause and the comparison with predicting zero are not covered. Idealizations. The propositions use simplified models that do not describe real records. They assume the linear regime of the bounded map, one responding channel, baselines whose jump variance grows with the gap and measurement times independent of the baseline, whereas in the panels the baseline and the measurement times both come from the real record. Proposition 1 sets aside that the checkpoints were selected on the validation split, and Proposition 2 assumes exact matching. Proposition 3 fixes the standardization that the audit refits within folds. Propositions 4 and 5 analyze population least squares at fixed kinetics, whereas the readers use cross-validated ridge regression over a kernel grid. The retention values use simulated draw patterns, because the panels’ draw masks were not shipped. Propositions 6 and 7 assume equal exposure information. Proposition 7 describes exact minimizers at a fixed baseline, whereas the implementation uses local optimizers and refits the baseline.

Statistics and reproducibility Unless stated otherwise, intervals are two-sided 95% percentile bootstrap intervals 82 . The seedlevel intervals in Fig. 2a and the replicate-level intervals in Supplementary Fig. S2 are t intervals. Replicate draws were shared across the arms of each analysis, so contrasts are paired. Resampling was over whole patients, except for the registered eICU-CRD test-set synchronization contrast and the disclosed last-value comparison, which resampled stays (2,000 replicates). The synchronization rosters hold only 1.02 to 1.12 stays per patient. A person-clustered bootstrap of the registered synchronization family, run afterwards as a sensitivity analysis, therefore changed little. It gave identical point estimates, widened the intervals by a median of 0.4% and at most 2.5%, and left all 12 settings significant after Holm correction. Two-sided P values use the add-one rule where stated. Where the observed statistic never crossed zero in 2,000 replicates, we report P < 0.001 rather than zero, and P < 0.012 after Holm correction over 12 settings. This was the case in every setting of the registered synchronization contrast on the eICU-CRD test set. Multiplicity was controlled by the Holm procedure within

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families fixed in the protocols 58 , and no correction was applied across families. No family-wise correction was applied to the registered development probes of clinical endpoints on eICU-CRD, so their intervals are unadjusted. For the benchmark families, a contrast is a win or a loss only when it survives correction with its interval wholly on one side of zero, and it is otherwise unresolved. Every learned treatment-effect estimator was also compared with a constant prediction of zero effect. A win over a comparator that did not itself beat zero is labeled as such. Subgroup and decision-curve analyses of the joint forecasts were descriptive. They excluded subgroups with fewer than 50 evaluable rows or a single class, and their intervals resample patients. The inference of the MIMIC-IV read is described above (Registered MIMIC-IV read).

Registration and provenance Each registered analysis followed a protocol that fixed the analysis and any decision rule and was committed and pushed to version control before the run. Some protocols also fixed the sentences that report their outcomes. Where this paper rewords such a sentence for readability, every number, outcome and caveat is unchanged, and the registered wording is in the protocols’ result notes. Such registrations in version control are not external preregistrations 33,34 . The protocols of the endpoint probes, the placebo gate, the MIMIC-IV read, the realism audit of the washout panels and the in-support control with the g-computation comparator also stated known biases or limitations and a forecast of the outcome. Those of the placebo gate, the MIMIC-IV read, the realism audit of the washout panels and the in-support control with the g-computation comparator were also tagged in version control. Automatic checks in the job-submission scripts and analysis stages backed these protocols. The submission scripts refused to submit a job from code with uncommitted changes or from a commit not yet pushed to the remote repository. They also refused any job whose registry entry did not match or whose pinned code or inputs had changed. The fitting stages could not open evaluation labels. A scoring stage opened the labels after the predictions had been sealed (frozen and hashed) and left a marker that blocks a second read by the same analysis. Each receipt, the record that a run writes, carries hashes of the run’s code, inputs and outputs. The receipts of the endpoint probes and the placebo gate also carry the SHA-256 hash of their own canonical content. Table 2 lists the principal analyses with their status and outcome, including those whose decision rules were not met. Supplementary Table S1 gives the provenance of each reported analysis. Several analyses reported here reused splits that had been read before, or involved choices made after an earlier read. Much of the eICU-CRD test set had been scored by models trained with an earlier objective (Supplementary Note 6). The roster rule of the registered synchronization contrast was set after an earlier read of that test set, and the last-value comparison was run after the synchronization result. The probability clip of the calibrated log-loss was changed after the test read, so we interpret that metric by its direction only. The control inputs were tested on a validation split that had been read for checkpoint selection, calibration maps and an earlier reader. The endpoint probes P0 and P1 shared a tuning set, and P1 was designed after P0 had read its labels. The registered cell was chosen after an exploratory amplitude sweep on the 400 discovery stays, and its discovery run reproduces the exploratory run on those stays. The new evidence for

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this cell therefore lies in the replication panel, which carries nearly identical injected effects, and in the fixed-schedule control. ICare-RH’s design was fixed before it was fitted to a real panel, but the panels had already been read by the reader and learner analyses. The in-support control and the g-computation comparator were registered after the results at hour 90 were known, and the control’s candidate hours were set after a label-free diagnostic at hour 42 had been seen. The MIMIC-IV test split had been read before the registered read (Registered MIMIC-IV read). The numerical checks of the theory, run after the results, read stored eICU-CRD test-set predictions and labels. Supplementary Note 2 lists these reads and choices for eICU-CRD, Supplementary Note 7 those for MIMIC-IV and Supplementary Note 9 those for the in-support control and the g-computation comparator. Earlier reads of evaluation splits that we could reconstruct are disclosed in a test-read registry, which states its coverage limits. The registry holds 46 plan rows, two of them for the MIMIC-IV read, two for the realism audit of the washout panels and one for the in-support control and the g-computation comparator, and 62 disclosure rows. Each plan row records a test-read plan or an analysis run under one, and each disclosure row records an earlier read of an evaluation split. A search ledger of 2,586 rows records the campaigns, runs and configurations of the project up to 26 September 2026. Of its 606 campaign and run rows, 280 were registered before their outcome and 189 were exploratory. A compute ledger records at least 17,683 central processing unit thread-hours and 362 GPU-hours (NVIDIA L40S) between 1 August and 26 September 2026. Both ledgers closed before eight analyses were run, so these analyses are reported here but not recorded in the ledgers (Supplementary Note 2). They are the endpoint probes, the control inputs, ICare-RH (its synthetic gates, 15 variants and placebo gate), the post hoc checks of Supplementary Note 5, the MIMIC-IV read, the realism audit of the washout panels, the in-support control and the g-computation comparator.

Software and compute Analyses used Python 3.10, PyTorch 2.5.1, NumPy 1.26.4, pandas 2.3.3, SciPy 1.15.3, scikitlearn 1.7.2, CatBoost 1.2.10, LightGBM 4.6.0 and XGBoost 3.2.0. Causal forests used R 4.3.3 with grf 2.6.1, and sequence baselines used their reference implementations under PyTorch 2.10.0 and PyTorch Lightning 1.4.5. Jobs ran on one shared node with an AMD EPYC 9555P processor and three NVIDIA L40S GPUs under Slurm.

Data availability eICU-CRD, MIMIC-IV and MIMIC-III are available to credentialed users through PhysioNet under their data use agreements. The data use agreements do not allow the authors to redistribute patient-level derived data, which credentialed users can regenerate with the preparation and split scripts that we will release (Code availability). Source data are provided with this paper and contain the aggregate tables behind every figure and behind Supplementary Tables S2–S8 and S20–S27, including all aggregate outputs of the registered MIMIC-IV read. Tables 1 and 2 and the other supplementary tables take their values from the reports and aggregate tables named in their captions, in Supplementary Table S1 and in the Supplementary Notes. 41

Code availability The data-preparation and split scripts, the model, generator, washout-panel, benchmark, scoring and figure code, the registered protocols and receipts, the test-read registry, the search and compute ledgers and the aggregate source tables will be released in a public repository of sanitized files that contains no patient-level data. They are available to editors and referees on request during review.

Use of large language models and coding agents Coding agents (Claude Code, Anthropic) were used to assist with the experimental implementation and writing polishing. The propositions of Supplementary Note 5 were derived after the results. Y.L. developed them, and coding agents assisted with their derivations, proofs, and numerical checks. Every registered gate followed a protocol committed before its run. Every registered gate result was recomputed from its shipped tables by a separate agent pass with its own context. For the MIMIC-IV read, that pass used the same model family as the agent that ran the read, so it was independent in context only. Scientific decisions were made by the authors, who take full responsibility for the content. No language model is an author, and no model output was used as data.

Ethics This study used de-identified research databases that are available to credentialed researchers. The institutional review board of Beth Israel Deaconess Medical Center approved the creation and sharing of MIMIC-IV with a waiver of informed consent. The institutional review boards of Beth Israel Deaconess Medical Center and the Massachusetts Institute of Technology approved those of MIMIC-III, also with a waiver of informed consent. eICU-CRD was released after certification of de-identification under the safe-harbor standard of the US Health Insurance Portability and Accountability Act, and its use is exempt from institutional review 10 . All analyses complied with the credentialing and data use requirements of each database.

Acknowledgements This study was partially funded by the JST Moonshot R&D Program (grant JPMJMS2034-18). The funder had no role in the design, analysis or writing. We thank the MIT Laboratory for Computational Physiology for making the MIMIC and eICU databases available and the contributors of the BlendedICU project for a standardized data pipeline.

Author contributions Y.L. conceived the study and designed the methodology. Y.L. and Y.Q. implemented the models, conducted experiments and analyzed the results. Y.L. developed the theoretical analysis, with 42

coding agents assisting in derivations, proofs and numerical checks (Methods). Y.K. contributed to data analysis, visualization design and manuscript revision. D.P.-G. provided critical suggestions on the methodology and the interpretation of results. Y.L. wrote the manuscript, with drafting assistance from coding agents described in Methods. All authors contributed to the manuscript revision and approved the final version.

Competing interests The authors declare no competing interests.

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48

Tables Table 1: Cohorts. Stays and patients in the three harmonized databases, in the person-disjoint eICU-CRD splits and in the semi-synthetic data built on eICU-CRD records. All analyses use eICU-CRD except the registered MIMIC-IV read of the two prediction results and its disclosed secondary outcomes. Earlier transfer analyses on MIMIC-III were not registered and are disclosed in Supplementary Note 8. “Read before” marks splits that earlier analyses had already read (Supplementary Notes 2 and 7). In the realism audit, a discriminator tries to tell generated from real records. Its area under the receiver operating characteristic curve, the audit AUC, is 0.5 when it cannot tell them apart and must be at most 0.75 for the audit to pass (Methods). Cohort or split

Stays

Patients

Role in this study

200,825

139,362

Training split

140,473

97,553

Fitting set Tuning set Validation split

119,330 21,143 30,213

82,946 14,607 20,904

Test set

30,099

20,886

200,825

139,362

400 per panel

–

MIMIC-IV

94,458

65,366

MIMIC-III CareVue

28,394

23,692

Model training and all analyses except the MIMIC-IV read Model training; clinical-endpoint probes (Supplementary Fig. S3) Fitting of the endpoint probes Scoring of the endpoint probes Checkpoint selection; control inputs and gradient-boosted comparison (read before) Hour-48 forecasts, synchronization and last-value comparison (read before) Generator applied to every eICU-CRD stay; passed the locked realism audit (audit AUC 0.572); counterfactual truth for 674 test-split stays Individual-effect analyses on real eICU-CRD windows with simulated dosing and injected responses (discovery, replication and fixed-schedule panels) Registered read of the prediction results, with hospital death and leaving the unit as disclosed secondary outcomes (person-disjoint test split of 14,211 stays; read before, including earlier zero-shot transfers and overlapping outcomes) CareVue-era stays of patients not in MIMIC-IV; used only in earlier transfer analyses, disclosed and not registered (Supplementary Note 8)

Experimental cohort

323,677

–

eICU-CRD (208 hospitals)

Semi-synthetic dataset

Washout panels

49

Three harmonized databases

Table 2: Analyses and outcomes. The principal analyses with their status, data, decision rule and outcome, including those whose decision rule was not met. The status in parentheses says whether and how each analysis was fixed before its run. Registered analyses followed a protocol committed to version control before their run (Methods). Prospective-protocol analyses followed a development protocol written before their run but were not registered. Disclosed, exploratory and post hoc analyses were not registered. The realism audit of the generated dataset applied a locked threshold. Supplementary Table S1 gives the provenance of each analysis. Earlier reads of the same data and post hoc choices are listed in Supplementary Note 2 for eICU-CRD, in Supplementary Note 7 for MIMIC-IV and in Supplementary Note 9 for the in-support control and the g-computation comparator. “Holm over n” denotes Holm correction within a family of n tests. Differences without a stated unit are in AUROC. Table 2 continues in three further parts. AUROC, area under the receiver operating characteristic curve. Analysis (status)

Data

Decision rule

Outcome

eICU-CRD test set, hour 48

All 17 settings and five seeds reported; nothing fitted on test data

MIMIC-IV, death and unit-exit forecasts (disclosed secondaries) Synchronization (registered)

MIMIC-IV test set, hour 48

Frozen eICU-CRD weights; hospital death and leaving the unit; all 6 settings reported; no criterion

eICU-CRD test set

Last-value comparison (disclosed) Gradientboosted comparator (prospective protocol)

eICU-CRD test set

Control inputs (registered)

eICU-CRD validation split

MIMIC-IV, synchronization (registered)

MIMIC-IV test set

MIMIC-IV, last-value comparison (registered secondary)

MIMIC-IV test set

Synchronized vs hour-6 forecast in stays with the target measured every hour through the label window (592 to 6,648 stays per setting), Holm over 12 settings Synchronized vs the patient’s last measured value on the same stays, Holm over 12 settings Twin forecasting heads (seed-29 checkpoint) vs multivariate CatBoost on the same rows, Holm over 120 contrasts; CatBoost with the twin state added vs CatBoost alone, Holm over 36 contrasts Synchronized vs input held at hour 6 (assimilation) and vs other-patient input (necessary condition for personalization), Holm over 36 primary contrasts; each condition requires ≥6/12 settings after correction and none worse at unadjusted P < 0.05 Synchronized vs hour-6 forecast, frozen eICU-CRD weights, Holm over 9 settings; transfer if at least 5 of 9 with none significantly worse before correction Synchronized vs patient’s last value, Holm over 9

AUROC 0.646 to 0.820 (threshold crossings, 16 settings), 0.798 (death in the unit within 48 h); mean death forecast far below the observed rate, in part by construction AUROC 0.687 (hospital death, unit exit within 48 h), 0.726 (hospital death), 0.765 (leaving the unit within 48 h); calibration in the text 12/12 settings, +0.073 to +0.153 AUROC

Forecasting Hour-48 forecasts (registered; death forecast descriptive)

eICU-CRD validation split

50

11/12 settings, median +0.044 AUROC (range +0.010 to +0.085) Heads: median −0.012 AUROC, worse in 3/12 settings after correction; state added: median +0.004 AUROC, no AUROC gain after correction Both conditions met in 12/12 settings (+0.080 to +0.146 and +0.111 to +0.185 AUROC); held input below the hour-6 forecast in 12/12 (significant in 5); information beyond the patient’s last value not shown Met: 8/9 settings, +0.025 to +0.114

Not met: worse in 2/9 settings after correction, better in 1/9

Table 2: Analyses and outcomes (continued). Boosting is a fixed CatBoost recipe on 1,136 summary features of the record, and the hybrid adds the fine-tuned twin state to these features. Differences from boosting are in AUROC. On eICU-CRD, the endpoint probes (Supplementary Fig. S3) were scored on a tuning set that the pretrained encoder had seen in training, and their intervals are unadjusted. AUROC, area under the receiver operating characteristic curve. Analysis (status)

Data

Clinical-endpoint probes Endpoint probe eICU-CRD P0, frozen state tuning set (registered)

Endpoint probe P1, encoders trained on the endpoints (registered)

eICU-CRD tuning set

MIMIC-IV, endpoint probes (registered)

MIMIC-IV test set

Decision rule

Outcome

Frozen state ≥ boosting on ≥2/4 endpoints, or within 0.02 of it on ≥3/4, or a gain with interval above 0 from adding the state to boosting on ≥2/4 Better encoder within 0.01 of boosting on ≥3/4 endpoints, or hybrid gain ≥ +0.005 with interval above 0 on ≥2/4

Not met: state 0.020 to 0.086 below boosting; gain from adding it to boosting ≤ +0.0014, no interval above 0

Hybrid vs boosting on hypotension and acute kidney injury, refitted within MIMIC-IV, one-sided, Holm over 2

51

Not met (near threshold): hybrid +0.0042 (hypotension) and +0.0075 (acute kidney injury), intervals above 0; better encoder (from random initialization) 0.004 to 0.021 below boosting Acute kidney injury only: +0.0035; hypotension +0.0010

Table 2: Analyses and outcomes (continued). Readers are ridge regressions fitted with the true perpatient effect as the label; they are not deployable. AUC, area under the receiver operating characteristic curve; SD, standard deviation. Analysis (status)

Data

Decision rule

Semi-synthetic data and individual treatment effects Realism audit SemiDiscriminator AUC ≤ 0.75 (locked) synthetic and all summary checks dataset Washout-panel Three Locked audit per panel: realism audit washout AUC ≤ 0.75 and all (registered, after panels summary checks all but two of the panels’ other analyses) Recoverability 400 A reader clears the bar map eICU-CRD (R2 = 0.2) and, if the (exploratory) stays, three history-blind reader also treatmentclears it, beats that reader response pairs

Registered cell, cadence-aware reader (registered after exploration)

Three washout panels

Clears the bar (R2 = 0.2) and, if the history-blind reader also clears it, beats that reader

52

Outcome Pass: AUC 0.572

Pass in 3/3: AUC 0.714 to 0.727, but the audit did not flag added measurement noise of 0.15 reference SD (weak evidence); other sensitivity controls not detected or not run At real cadence, of the mapped readers only the cadence-aware reader cleared the bar, and so separated: on heart rate (value changes in 57% of hours) for both drugs at amplitudes of 0.75 and 1 SD (R2 = 0.44 at 1 SD). At real cadence, none cleared it on creatinine (value changes in 4% of hours) Separated in the discovery panel (R2 = 0.33) and in a replication panel of 400 stays, 395 of them new, carrying nearly identical injected effects (R2 = 0.31); did not separate on the fixed schedule (R2 = 0.24, interval reaching below the bar)

Table 2: Analyses and outcomes (continued). The oracle-structure learner uses factual data only but is given the generator’s response structure; it is not deployable. A learner beats zero in a panel when the 95% bootstrap interval of its centered PEHE minus that of predicting zero lies wholly below zero. Comparisons that do not survive Holm correction are counted as unresolved. CATE, conditional average treatment effect; CRN, counterfactual recurrent network; PEHE, precision in estimation of heterogeneous effects; RMSN, recurrent marginal structural network. Analysis (status)

Data

Decision rule

Semi-synthetic data and individual treatment effects (continued) Static CATE Registered 13 learners vs the learners cell (decision oracle-structure learner, (registered) query outside Holm over 78 comparisons the support) Sequence CATE Registered Causal Transformer, RMSN models cell (decision and CRN vs the (registered) query outside oracle-structure learner, the support) Holm over 18 comparisons G-computation Registered Holm over 6 comparisons vs model cell oracle-structure learner; (registered) Holm over 2 vs predicting zero In-support query Two panels, Each learner vs predicting control hours 30 and zero, Holm over 64; vs (registered) 42 oracle-structure learner, Holm over 128 Label-free lag and Three ICare-RH placebo-spread rules, washout placebo gate checked before any target panels and (registered) was read their zero-effect copies Stored tables No criterion; run after the Theory checks of previously results (post hoc, read splits; descriptive) simulation

53

Outcome Oracle-structure learner: 77 wins, 0 losses, 1 unresolved; no learner beats zero Oracle-structure learner: 12 wins, 0 losses, 6 unresolved (the comparisons with the excluded CRN); no model beats zero Oracle-structure learner: 6 wins, 0 losses, 0 unresolved; G-Net (Causal Transformer code) does not beat zero No hour met the overlap criteria

Not met at the label-free stage; no target read, so ICare-RH was not scored on real records

Reported in Supplementary Note 5, including a flexible classifier on the realism probe and an hour-6 read of eICU-CRD test-set predictions and labels

Supplementary Information ICare-DT: a digital twin for continuous forecasting of intensive care patients Yanke Li, Yahang Qi, Yumi Kim and Diego Paez-Granados

Contents Supplementary Figures

2

Supplementary Tables

5

Supplementary Note 1: Registered wording

13

Supplementary Note 2: Prior reads, selection and deviations

14

Supplementary Note 3: Rollout clock defect

15

Supplementary Note 4: Semi-synthetic data generation

15

Supplementary Note 5: Theory and proofs

19

Supplementary Note 6: Cohort flow, forecasts and clinical-endpoint probes on eICU-CRD

90

Supplementary Note 7: The registered MIMIC-IV read

93

Supplementary Note 8: Earlier transfer analyses on MIMIC-III

116

Supplementary Note 9: Registered analyses of the washout panels added before submission

119

1

Supplementary Figures a

b ΔCalibrated NLL, adapted minus base

ΔAUROC, adapted minus base

AKI 6 h AKI 12 h AKI 24 h AKI 48 h Hypotension 6 h Hypotension 12 h Hypotension 24 h Hypotension 48 h Hypoxemia 6 h Hypoxemia 12 h Hypoxemia 24 h Hypoxemia 48 h Hyperlactatemia 6 h Hyperlactatemia 12 h Hyperlactatemia 24 h Hyperlactatemia 48 h Sepsis proxy 6 h Sepsis proxy 12 h Sepsis proxy 24 h Sepsis proxy 48 h −0.08

−0.04

0.00

0.04

ΔAUROC (95% CI) One head per event

−0.008

0.000

0.008

0.016

ΔCalibrated NLL (95% CI)

Joint configuration

Supplementary Fig. S1: Per-patient online adaptation of the forecasting heads did not improve on frozen weights in an earlier model version. Each marker shows the adapted model minus the base (frozen) model for one of five events (acute kidney injury, hypotension, hypoxemia, hyperlactatemia and a sepsis proxy) at a horizon of 6, 12, 24 or 48 hours on the eICU-CRD test split (1,895 patients; three seeds). The heads were adapted either as one isolated head per event or in a joint configuration. a, AUROC difference. b, Cross-fitted calibrated log-loss difference. A win required both metrics to improve after Holm correction over 20 settings, and no setting met this criterion. One setting in each configuration was significantly worse after correction, hypoxemia at 6 hours with isolated heads (AUROC −0.017) and hypotension at 48 hours in the joint configuration (AUROC −0.050). Filled markers survive correction. These descriptive re-reads of a completed campaign, which used an earlier readout, motivated the frozen-weight design of the twin.

2

a

Hidden-confounding stress 0.12

c

Overlap stress (λ = 0)

Predicted heterogeneity

0.12

Twin rollout Constant-effect floor

0.0177

Var(true effect) 0.10

0.10

PEHE

b

0.08

Bias2

0.0013

Var(predicted)

0.0000

−2 Cov

0.0000

0.08

0.06

0.06

0.04

0.04 0

¼ ½

1

Confounding strength λ

2

Moderate

Strong

Treatment overlap

Weak

0.00

0.01

0.02

Mean PEHE2 contribution

Supplementary Fig. S2: Stress analyses of the historical rollout benchmark. a, Individual-effect error (PEHE) of the twin rollout and of a constant-effect floor, which assigns the true average effect to every patient, by strength of hidden confounding (λ = 0, 0.25, 0.5, 1 and 2) at strong overlap. b, The same at λ = 0 for strong, moderate and weak overlap (treatment-policy logit scale γ = 0.5, 2 and 4). In a and b, values are means over five data-generating replicates, each averaged over three model seeds, with 95% t intervals across replicates. The rollout’s mean error was above that of the floor at every confounding strength and overlap. c, Decomposition PEHE2 = Var(τ̂ ) + Var(τ ) − 2 Cov(τ̂ , τ ) + bias2 of the historical model at λ = 0, averaged over the 60 outcome, treatment and horizon cells, five replicates and three model seeds. The model predicted almost no heterogeneity. The variance of its predicted individual effects was about 0.15% of that of the true effects, and their covariance with the true effects was near zero. The rollout ran with the clock defect described in Supplementary Note 3 and is reported as run.

3

a

ICU death

Hypotension

Kidney injury

Hypoxemia

Last value Frozen twin state Trained, pretrained Trained, scratch Gradient boosting Ensemble Boosting + state 0.7

0.8

0.9

0.76

AUROC

b

0.80

c

Difference from boosting

ICU death

Hypotension

Hypotension

Kidney injury

Kidney injury

Hypoxemia

Hypoxemia −0.04

0.00

42% 42%

76% 66%

0.72

0.80

0.88

AUROC

d

Log-loss vs boosting

Hypotension

Kidney injury

77% 51% 50

Hypoxemia 100

% of deficit closed

Trained, pretrained Ensemble Boosting + state

0.84

ICU death

94% 87%

0

ΔAUROC (95% CI)

Frozen state Boosting + frozen Trained, scratch

0.78

AUROC

Gap closed by training

ICU death

−0.08

0.72

AUROC

scratch pretrained

0.000

0.005

ΔLog-loss Trained, scratch Boosting + state

Supplementary Fig. S3: A fine-tuned twin state approaches gradient boosting and adds complementary signal on eICU-CRD. Registered development probes P0 and P1 on the tuning set of the eICU-CRD training split (Supplementary Note 6). The pretrained encoder had seen every stay of the tuning set’s 14,607 patients, and the 14,122 patients with at least one landmark row were scored. a, AUROC with 95% whole-person bootstrap intervals for death in the unit, hypotension (mean arterial pressure below 65 mmHg in two consecutive measured hours), acute kidney injury (KDIGO creatinine criterion) and hypoxemia (SpO2 below 90% in two consecutive measured hours) within 24 hours of landmarks at 12, 24 and 48 hours. The reference arms are logistic regression on the last measured value of the endpoint variable and the boosting comparator, a fixed CatBoost recipe on 1,136 summary features of the record (dotted line). The frozen twin state is read by a multilayer perceptron, and the encoder trained on the endpoints is either fine-tuned from the pretrained twin (pretrained) or trained from random initialization (scratch). Boosting and the fine-tuned encoder are also combined, in a mean-logit ensemble and in the hybrid (Boosting + state), which adds the fine-tuned state of the pretrained encoder to the boosting features. b, Differences from gradient boosting, with hollow markers for the frozen state. c, Share of the frozen-state deficit (frozen state minus boosting) closed by training the encoder on the endpoints. d, Log-loss difference from gradient boosting for the encoder trained from random initialization and for the hybrid (point estimates; negative is better). Two model seeds were averaged. Intervals are unadjusted. Neither probe met its registered gate, and P1 missed it narrowly (Table 2 of the main text). The registered MIMIC-IV read of this comparison is reported in the main text and Supplementary Note 7.

4

Supplementary Tables

5

Supplementary Table S1: Provenance of the reported analyses. Each row traces an analysis to its protocol and result reports and to the scheduler jobs whose receipts record the run. Statuses, given in parentheses, are registered (protocol committed to version control before the run), prospective protocol (development protocol written before the run and not registered), disclosed (run and reported without registration), exploratory, descriptive or locked audit. Protocol and result reports are in reports/revision_v3/, and aggregate tables are in source_tables/revision_v3/. Analysis

Protocol and result

Jobs

Joint model training

Corrected competing-event model, five seeds; validation-only checkpoint selection

Joint forecasts, test set (registered) Subgroups and decision curves, test set (descriptive) Further single-seed reads under the same plan (registered) Synchronization, test set (registered) Last-value comparison (disclosed) Isolation arms, validation (registered) Gradient-boosted comparator and the twin state added to it, vital signs, validation (prospective protocol) Online adaptation of the forecasting heads (descriptive re-read) Endpoint probe P0 (registered) Endpoint probe P1 (registered) Realism audit (locked audit) Washout-panel realism audit (registered) Recoverability map (exploratory) Registered washout cell (registered after exploration) Matched learned comparators (prospective protocol) Static CATE learners (registered) Sequence CATE models (registered) G-computation sequence model G-Net (registered)

PHASE1_TEST_ANALYSIS_PLAN (locked before the read) and all-results receipt; Supplementary Note 6 JOINT_FACTUAL_STRATIFIED_20260918; ethnicity re-read; Supplementary Tables S7 and S8 Earlier synchronization contrast (superseded) and frozen-state probe versus LightGBM; Supplementary Note 6

(before protocol registration began); selection 5255 5276, 5293, 5312 (verification) 10545; 10697

In-support query control (registered)

WASHOUT_Q5Q6_PROTOCOL_20261008 / RESULT_20261022; Supplementary Table S30

Oracle-structure learner F (prospective protocol) ICare-RH synthetic gates and variants (synthetic data, before any real-panel fit) ICare-RH placebo gate G7 (registered) Historical benchmark re-emission (disclosure test) Theory checks (post hoc, descriptive) Hour-6 read of the test panel (post hoc, disclosed) MIMIC-IV read: synchronization, endpoint probes and disclosed secondary outcomes (registered) Earlier MIMIC-III reads (disclosed)

Washout factual-response result

5264, 5275; 5265

State-synchronization v2 lock and release review; re-run without the hour-30 condition Landmark last-value baseline

10707, 10708; 10755, 10937 10754

STATE_SYNC_ISOLATION_PROTOCOL_20260927 / RESULT_20260928

13006, 13007

LANDMARK_TRAJECTORY_DEVELOPMENT_PROTOCOL_20260920 and LANDMARK_STATE_FUSION_PROTOCOL_20260920

11179–11181; 11208–11210

Rescore of the completed targeted-head campaign; Supplementary Fig. S1

2956–2973

ROUTE_B_P0_PROBE_PROTOCOL_20260927 / RESULT_20260928; Supplementary Note 6 ROUTE_B_P1_PROBE_PROTOCOL_20260928 / RESULT_20260929; Supplementary Note 6 Hold probe and realism gate reports WASHOUT_PANEL_REALISM_AUDIT_PROTOCOL_20261008 / RESULT_20261014; Supplementary Table S29 Twin identifiability map; amplitude calibration

12997, 13001–13003 13029–13032 10568–10581 13464–13469 10615–10689

Registered phenylephrine cell

10690–10694

Washout matched comparators

11227–11230

WASHOUT_STANDARD_CATE_PROTOCOL_20260923 and amendment

12770–12774

WASHOUT_SEQUENCE_CATE_PROTOCOL_20260924 and amendments

12939–12947

WASHOUT_Q5Q6_PROTOCOL_20261008 / RESULT_20261022; Supplementary Note 9

13479–13486, 13541–13546, 13772–13781 13479–13486, 13541–13546, 13772–13781 11287, 11288

WASHOUT_RESPONSE_HYBRID_IMPLEMENTATION_20260927; Supplementary Table S5

(local runs; no cluster job)

WASHOUT_RESPONSE_HYBRID_PROBE_PROTOCOL_20260929 / G7 result

13044 (failed before any read), 13046 (recomputed from shipped tables) (local re-reads and simulation) 10707, 10708, 10755, 10937 (re-read) 13123 (manifest); 13163–13173

A3_FOUR_ARM_REEMISSION_20260926 Supplementary Note 5; scripts and outputs in reports/revision_v3/theory_checks/ Supplementary Note 5, Section 5.1; per-stay predictions and labels of the synchronization runs MIMIC_IV_READ_PROTOCOL_20260930 / RESULT_20261002 and release review; Supplementary Note 7

Test-read registry rows TRB-M4-01, -05 to -16 and -18 to -25; Supplementary Note 8

6

As in the registry

Supplementary Table S2: Recoverability map. Recovery of the injected individual memory-route effect (12-hour target) by readers fitted with the true effect, for three treatment-response pairs on the same 400 real eICU-CRD records (Fig. 5a– c of the main text). On hourly records, the occupancy-contrast reader cleared the bar in all three pairs at 0.5 SD and above, whereas at real cadence only the cadence-aware reader cleared it, on heart rate at 0.75 and 1 SD. Recovery is 1−RMSE/SD(τ ). The main text reports R2 = 1−(1−recovery)2 , and the registered bar of 0.106 corresponds to R2 = 0.2. Size: amplitude cap in reference standard deviations (SD). Occupancy: occupancy-contrast reader. Hourly: every hour observed; real: real measurement cadence. Oracle: oracle reference that knows each patient’s trait. Dashes: not run. ∗ Clears the bar (bootstrap probability of falling below it under 0.05). The table has no hourly column for the cadence-aware reader. On hourly records it recovered 0.453, 0.392 and 0.342 at 1 SD for phenylephrine, norepinephrine and NaCl (jobs 10679, 10683 and 10675), and 0.369 for phenylephrine at 0.75 SD (job 10691). For phenylephrine at 1 SD, its hourly fits selected lag 2 and decay 0.95 in all five folds, and its fits at real cadence selected lag 2 and decay 0.8 (job 10678). Cell

Size (SD)

Occupancy, hourly

History-blind, real

Occupancy, real

Cadence-aware, real

Oracle

NaCl → creatinine NaCl → creatinine NaCl → creatinine NaCl → creatinine

0.25 0.5 0.75 1

0.025 0.147∗ 0.253∗ 0.345∗

-0.026 -0.014 0.006 0.030

0.004 0.009 0.019 0.032

-0.011 -0.008 – 0.078

0.930∗ 0.926∗ 0.920∗ 0.907∗

Phenylephrine → heart rate Phenylephrine → heart rate Phenylephrine → heart rate Phenylephrine → heart rate

0.25 0.5 0.75 1

0.077 0.212∗ 0.310∗ 0.385∗

-0.013 0.006 0.036 0.073

-0.009 0.030 0.064 0.115

0.019 0.095 0.181∗ 0.254∗

0.802∗ 0.802∗ 0.802∗ 0.802∗

Norepinephrine → heart rate Norepinephrine → heart rate Norepinephrine → heart rate Norepinephrine → heart rate

0.25 0.5 0.75 1

0.089 0.220∗ – 0.393∗

-0.026 0.000 0.038 0.081

0.004 0.035 0.070 0.101

-0.001 0.085 0.183∗ 0.254∗

0.675∗ 0.675∗ 0.675∗ 0.675∗

7

Supplementary Table S3: Registered cell across the five registered runs. The registered cell is phenylephrine acting on heart rate at 0.75 SD with the 12-hour memory-route target, scored on 400 stays per run with 2,000 whole-person bootstrap replicates (Fig. 5d of the main text). The oracle reference cleared the bar in every run, and every reader cleared it on synthetic and hourly records. At real cadence only the cadence-aware reader did so, in both the discovery and the replication panel, and no reader did on the fixed schedule. The replication run used a second sample of 400 stays, 395 of them new, carrying nearly identical injected effects (Supplementary Note 4). Recovered: 1 − RMSE/SD(τ ). The main text reports R2 = 1 − (1 − recovery)2 , and the bar of 0.106 corresponds to R2 = 0.2. Gap: difference from the historyblind reader. Clears bar: whether the bootstrap probability of falling below 0.106 was under 0.05. Readers are defined in Methods, and the assimilating reader, despite its name, does not use the twin. Dosing-window means are channel means over four windows set by the first and last dose (the window-mean reader of Supplementary Note 5, Definition 6). Run

Reader

Recovered [95% CI]

Gap over history-blind [95% CI]

Clears bar

Synthetic tapes Synthetic tapes Synthetic tapes Synthetic tapes Synthetic tapes Synthetic tapes

History-blind Dosing-window means Assimilating Occupancy contrast Cadence-aware Oracle (true trait)

0.520 [0.486, 0.551] 0.592 [0.560, 0.620] 0.667 [0.640, 0.690] 0.600 [0.571, 0.629] 0.461 [0.421, 0.500] 0.802 [0.780, 0.823]

– +0.072 [+0.043, +0.101] +0.147 [+0.113, +0.182] +0.079 [+0.039, +0.123] -0.059 [-0.106, -0.009] –

yes yes yes yes yes yes

Real, hourly observed Real, hourly observed Real, hourly observed Real, hourly observed Real, hourly observed Real, hourly observed

History-blind Dosing-window means Assimilating Occupancy contrast Cadence-aware Oracle (true trait)

0.219 [0.159, 0.274] 0.252 [0.190, 0.311] 0.358 [0.308, 0.404] 0.310 [0.261, 0.354] 0.369 [0.328, 0.406] 0.802 [0.780, 0.823]

– +0.033 [-0.015, +0.076] +0.139 [+0.082, +0.200] +0.091 [+0.027, +0.157] +0.150 [+0.082, +0.220] –

yes yes yes yes yes yes

Real cadence, discovery Real cadence, discovery Real cadence, discovery Real cadence, discovery Real cadence, discovery Real cadence, discovery

History-blind Dosing-window means Assimilating Occupancy contrast Cadence-aware Oracle (true trait)

0.036 [0.005, 0.064] 0.066 [0.019, 0.111] 0.060 [0.025, 0.090] 0.064 [0.031, 0.094] 0.181 [0.137, 0.221] 0.802 [0.780, 0.823]

– +0.030 [-0.000, +0.060] +0.024 [-0.011, +0.057] +0.028 [-0.013, +0.068] +0.144 [+0.094, +0.193] –

no no no no yes yes

Real cadence, replication Real cadence, replication Real cadence, replication Real cadence, replication Real cadence, replication Real cadence, replication

History-blind Dosing-window means Assimilating Occupancy contrast Cadence-aware Oracle (true trait)

0.056 [0.018, 0.092] 0.117 [0.065, 0.166] 0.052 [0.009, 0.092] 0.051 [0.015, 0.082] 0.170 [0.113, 0.220] 0.803 [0.780, 0.824]

– +0.061 [+0.029, +0.091] -0.004 [-0.043, +0.034] -0.005 [-0.051, +0.037] +0.114 [+0.054, +0.167] –

no no no no yes yes

Real cadence, fixed schedule Real cadence, fixed schedule Real cadence, fixed schedule Real cadence, fixed schedule Real cadence, fixed schedule Real cadence, fixed schedule

History-blind Dosing-window means Assimilating Occupancy contrast Cadence-aware Oracle (true trait)

0.089 [0.034, 0.141] 0.110 [0.062, 0.158] 0.112 [0.068, 0.153] 0.110 [0.067, 0.148] 0.129 [0.062, 0.186] 0.805 [0.782, 0.825]

– +0.021 [-0.016, +0.060] +0.023 [-0.019, +0.065] +0.021 [-0.024, +0.066] +0.040 [-0.020, +0.095] –

no no no no no yes

8

Supplementary Table S4: Every treatment-effect estimator on the registered cell. Centered PEHE (lower is better; predicting zero equals the SD of the true effect) and correlation with the true individual effect, by panel. Discovery cPEHE r

Estimator

Replication cPEHE r

Fixed schedule cPEHE r

S-learner (boosting) T-learner (boosting) X-learner (boosting) DR-learner (boosting) DR-learner (ridge) R-learner (boosting) R-learner (ridge) Causal forest TARNet DragonNet Random-slope lag model Random-slope lag model (kernel-informed) Causal forest (kernel-informed) Causal Transformer RMSN G-Net (Causal Transformer code) CRN (failed its synthetic gate) Linear response model (structure-informed) Oracle-structure learner (factual data) Population prior only Predict zero

0.232 0.353 0.328 0.291 0.523 0.530 0.345 0.248 0.482 0.467 0.546 0.572 0.240 0.239 0.245 0.241 0.242 0.189 0.189 0.232 0.232

0.08 0.01 -0.05 0.03 0.03 0.01 0.05 -0.09 0.02 0.04 0.26 0.31 -0.03 0.03 -0.01 0.05 -0.01 0.58 0.58 -0.00 –

0.231 0.348 0.260 0.276 0.592 0.380 0.358 0.241 0.395 0.404 0.574 0.603 0.247 0.235 0.245 0.238 0.240 0.179 0.179 0.232 0.232

0.10 0.12 0.11 0.02 0.06 0.05 0.06 0.09 0.02 0.01 0.25 0.28 0.02 0.04 -0.01 0.10 -0.01 0.64 0.65 -0.02 –

0.232 0.344 0.251 0.295 0.385 0.878 0.308 0.593 0.349 0.367 0.412 0.795 2.213 0.238 0.246 0.248 0.242 0.193 0.194 0.232 0.232

0.03 -0.00 -0.03 -0.01 0.03 -0.07 0.01 -0.04 0.05 0.09 0.20 0.21 0.04 0.03 0.04 0.04 0.03 0.56 0.55 0.04 –

SD of true effect

0.232

–

0.232

–

0.232

–

Supplementary Table S5: ICare-RH recovers individual effects in simulation. ICare-RH was fitted to factual data only, with no constant taken from the generator, on 400 simulated patients in five folds per scenario. Its registered arm, shown here, uses no ICare-DT state. The registered scenario is matched to the registered cell, and the other 15 scenarios, inside and outside the model family, were scored descriptively (Fig. 7b of the main text). ICare-RH beat predicting zero in every scenario except a drifting response gain, where the interval of the difference included zero. In the registered scenario, its per-patient 80% and 95% intervals covered 0.77 and 0.94 of the true effects. SD of truth: standard deviation of the true effects. Difference: centered PEHE minus that of predicting zero, with 95% bootstrap interval. Scenario

Family

Registered scenario New seed Slower washout (decay 0.6) Decay 0.73 Lag 3 h Fixed schedule Linear route shape Earlier query (hour 30) AR(2) baseline with drift Level shifts Heavy-tailed noise (t, 3 df) Clipped trait prior Linear response route Hill response (n = 2) Two-compartment kinetics Drifting gain

in family in family in family in family in family in family in family in family out of family out of family out of family out of family out of family out of family out of family out of family

SD of truth

Correlation

cPEHE

Difference from zero

0.227 0.212 0.209 0.221 0.204 0.227 0.227 0.147 0.227 0.227 0.227 0.199 0.252 0.257 0.157 0.195

0.786 0.788 0.789 0.788 0.796 0.790 0.793 0.640 0.813 0.766 0.770 0.745 0.817 0.841 0.645 0.493

0.141 0.131 0.128 0.136 0.124 0.139 0.138 0.113 0.136 0.146 0.146 0.133 0.146 0.139 0.121 0.183

-0.086 [-0.100, -0.073] -0.082 [-0.093, -0.070] -0.081 [-0.092, -0.069] -0.084 [-0.096, -0.072] -0.080 [-0.092, -0.069] -0.088 [-0.100, -0.075] -0.089 [-0.102, -0.076] -0.034 [-0.043, -0.025] -0.090 [-0.100, -0.079] -0.081 [-0.094, -0.068] -0.081 [-0.094, -0.067] -0.066 [-0.077, -0.055] -0.106 [-0.123, -0.090] -0.118 [-0.131, -0.103] -0.037 [-0.043, -0.030] -0.012 [-0.026, +0.002]

9

Supplementary Table S6: Synchronization contrasts by setting (AUROC differences). The synchronized forecast outperformed the model’s hour-6 forecast, the information-constant arm and the donor arm in all 12 settings, and the patient’s last measured value in 11 (Fig. 2c,d of the main text). The four “vs” columns give its AUROC gain over each comparator. On the test set, the comparators are the model’s hour-6 forecast (registered) and the patient’s last measured value (disclosed). On the validation split, the registered comparators are the information-constant arm and the donor arm, which replaces the patient’s hours from hour 7 to the landmark with those of the nearest other patient in the frozen hour-6 state. The last column, prespecified as descriptive, is the difference-in-differences between the advantage over the donor arm and the advantage of the patient’s own last measured value over the donor’s, on the validation split. It was significantly positive in no setting and significantly negative in two, so this contrast did not show information beyond the patient’s last measured value. n: test-set stays. ∗ Significant after Holm correction, over the 12 settings for the two test-set columns and the difference-in-differences, and over the 36 primary contrasts of the isolation arms for the information-constant and donor columns. Setting Heart rate, 12 h, next 6 h Heart rate, 18 h, next 12 h Heart rate, 30 h, next 24 h Resp. rate, 12 h, next 6 h Resp. rate, 18 h, next 12 h Resp. rate, 30 h, next 24 h SpO2 , 12 h, next 6 h SpO2 , 18 h, next 12 h SpO2 , 30 h, next 24 h SaO2 , 12 h, next 6 h SaO2 , 18 h, next 12 h SaO2 , 30 h, next 24 h

n

vs hour-6 forecast

vs last value

vs information constant

vs donor

Beyond last value

6648 4668 2141 6117 3998 1763 2487 1381 592 5699 3571 1637

∗

∗

∗

∗

-0.011∗ -0.003 +0.010 +0.005 +0.000 +0.007 -0.006 -0.004 -0.001 -0.015∗ -0.017 +0.012

+0.116 +0.107∗ +0.121∗ +0.102∗ +0.083∗ +0.108∗ +0.078∗ +0.101∗ +0.153∗ +0.073∗ +0.083∗ +0.136∗

+0.010 +0.027∗ +0.042∗ +0.048∗ +0.046∗ +0.049∗ +0.062∗ +0.060∗ +0.085∗ +0.023∗ +0.025∗ +0.017

+0.118 +0.133∗ +0.118∗ +0.106∗ +0.146∗ +0.131∗ +0.090∗ +0.080∗ +0.142∗ +0.088∗ +0.100∗ +0.124∗

+0.170 +0.185∗ +0.160∗ +0.131∗ +0.154∗ +0.142∗ +0.128∗ +0.111∗ +0.131∗ +0.121∗ +0.136∗ +0.150∗

Supplementary Table S7: Discrimination of the joint forecasts by subgroup on the eICU-CRD test set. Descriptive analysis (job 10545). Median AUROCs over settings ranged from 0.690 to 0.770 across the age, sex and unit-type subgroups. The settings are the crossings of four vital signs at four horizons and the 48-hour death forecast, all issued at hour 48 of the stay (Supplementary Note 6), and each setting’s AUROC in a subgroup is the median over five seeds. The columns give the number of settings with at least 50 evaluable rows and both classes, the median over these settings and their range. The last column gives the AUROC of the death forecast for death in the unit within 48 hours, for seed 13, with a 95% person-cluster bootstrap interval. The lowest single setting, with few stays, was 0.460 (SpO2 at 6 hours, 65 to 74 years, 64 stays). Discrimination of death was higher below 65 years (0.857 to 0.890) than at 65 years and older (0.771 to 0.803). Race, ethnicity and insurance are absent from the harmonized records, and the model did not use ethnicity. A descriptive re-read that joined ethnicity from the source eICU-CRD patient table (job 10697) gave median AUROCs over settings of 0.709 for White (17 settings), 0.712 for Black (17 settings) and 0.583 for Hispanic patients. Hispanic patients were evaluable in only five settings (SaO2 at four horizons and the death forecast; range 0.487 to 0.736), in which the medians were 0.709 for White and 0.700 for Black patients. Death AUROCs for seed 13 were 0.812 [0.782, 0.841] for White, 0.833 [0.745, 0.907] for Black and 0.761 [0.473, 0.974] for Hispanic patients. The other groups had too few evaluable stays for most settings, and eICU-CRD does not record insurance. Stratum

Value

Settings

Median AUROC

Range over settings

ICU death, 48 h

Age (years) Age (years) Age (years) Age (years) Age (years) Age (years)

<45 45–54 55–64 65–74 75–84 85+

15 13 16 16 16 12

0.758 0.736 0.717 0.700 0.702 0.770

0.608 to 0.860 0.597 to 0.875 0.656 to 0.890 0.460 to 0.821 0.587 to 0.844 0.609 to 0.910

0.857 [0.777, 0.922] 0.871 [0.754, 0.956] 0.890 [0.836, 0.934] 0.802 [0.745, 0.857] 0.803 [0.747, 0.857] 0.771 [0.674, 0.853]

Sex Sex

Female Male

17 17

0.690 0.703

0.621 to 0.819 0.621 to 0.830

0.824 [0.781, 0.863] 0.813 [0.771, 0.850]

Unit type Unit type Unit type Unit type Unit type

Cardiac Medical Medical-Surgical Neurological Surgical

13 12 17 17 15

0.699 0.744 0.706 0.735 0.715

0.583 to 0.838 0.577 to 0.854 0.547 to 0.850 0.663 to 0.798 0.526 to 0.871

0.814 [0.745, 0.880] 0.812 [0.719, 0.897] 0.809 [0.774, 0.846] 0.798 [0.687, 0.906] 0.878 [0.807, 0.939]

10

Supplementary Table S8: Decision curves of the joint forecasts on the eICU-CRD test set. Descriptive decision-curve analysis (job 10545) at the 50 alert thresholds t from 0.01 to 0.50, with net benefit TP/N − FP/N · t/(1 − t) taken as the median over five seeds. The table gives the lowest and highest thresholds at which the forecast’s net benefit exceeded both that of alerting on every stay and that of alerting on none. Count is the number of such thresholds, which need not be contiguous. Up to 24 hours ahead, the crossing forecasts of heart rate, respiratory rate and SaO2 had net benefit at most thresholds. For crossings at long horizons, where prevalence is high, alerting on every stay is nearly optimal, and fewer thresholds show net benefit. The mean death forecast was far below the observed rate (0.0023 against 0.026), in part by construction, because each target-specific head counts a death only if it precedes that target’s crossing. At thresholds from 0.03 to 0.15, the death forecast flagged at most 0.72 stays per 100 in any seed, so its net benefit in this range rests on very few alerts (Supplementary Note 6). Forecast

Horizon (h)

Prevalence

48 6 12 24 48 6 12 24 48 6 12 24 48 6 12 24 48

0.026 0.066 0.118 0.245 0.444 0.098 0.198 0.356 0.584 0.173 0.342 0.584 0.815 0.149 0.262 0.476 0.686

ICU death Heart rate Heart rate Heart rate Heart rate Respiratory rate Respiratory rate Respiratory rate Respiratory rate SpO2 SpO2 SpO2 SpO2 SaO2 SaO2 SaO2 SaO2

Thresholds with net benefit

Count

0.03 to 0.15 0.01 to 0.48 0.01 to 0.50 0.02 to 0.50 0.04 to 0.50 0.01 to 0.49 0.02 to 0.50 0.04 to 0.50 0.07 to 0.12 0.04 to 0.45 0.08 to 0.49 0.18 to 0.42 0.44 to 0.47 0.02 to 0.50 0.06 to 0.50 0.11 to 0.50 0.21 to 0.22

13 of 50 48 of 50 49 of 50 49 of 50 20 of 50 49 of 50 48 of 50 46 of 50 6 of 50 27 of 50 39 of 50 11 of 50 4 of 50 49 of 50 45 of 50 39 of 50 2 of 50

Supplementary Table S9: Direct-route weights W of the generator. Each entry is the change of the standardized outcome (row) per treated hour of a treatment (column), before the route scale of 0.8, the per-stay responsiveness ηi and the multiplier of 0.6 are applied. The weights are sparse, with 28 of the 60 entries at zero, and each nonzero weight moves its outcome in the direction that the generator treats as favorable (Supplementary Note 4). They were drawn once by the global random stream (seed 13) and are shared by the generated dataset and the washout panels, whose memory-route loadings are Lkj = Wkj / maxk′ |Wk′ j |. Values were recomputed from the shipped generator configuration (npj_ai_generator_constants_v1). Outcome Mean arterial pressure SpO2 Lactate Heart rate Respiratory rate Urine output Creatinine pH White cell count Platelets Bilirubin Temperature

Phenylephrine

NaCl

Norepinephrine

Dextrose

FiO2

0 0 0 -0.230 0 0 0 +0.173 -0.050 -0.184 0 0

+0.415 +0.765 -0.291 0 -0.261 +0.437 -0.812 0 0 -0.143 -0.326 0

+0.269 0 0 -0.228 -0.270 +0.111 0 0 -0.140 -0.304 -0.110 0

+0.255 +0.122 -0.175 0 -0.145 +0.636 0 0 0 -0.312 -0.048 0

0 +0.119 -0.466 -0.121 0 +0.019 -0.158 +0.286 0 0 0 0

11

Supplementary Table S10: The two constructions that use the generator. The generated dataset, which passed the realism audit, and the washout panels share the latent structural model, the background weights wk and the direct-route weights W (Supplementary Note 4). Generated dataset

Washout panels

Records

All 200,825 eICU-CRD stays, first 336 hours

Channels

12 outcome channels written; all 161 channels kept Cohort normalization statistics

400 eICU-CRD windows per panel, hours 24 to 163 12 outcome channels read

Standardization Missing cells Treatments Background scale Mediator persistence Responsiveness ηi Traits and memory route Hidden confounder Counterfactuals Truth or target Measurement hold Realism audit

Mean-filled for simulation; real missingness restored Recorded doses (indicator drives the model) 0.02 1.0 (spectral radius 1.03, not mean-reverting) 1 + N (0, 0.52 ), clipped to [1/3, 3] Off Simulated, no effect (λ = 0) One hour at 20 decision hours; independent draws across arms Minimum and maximum of the recorded outcome over 6 to 48 hours On Passed (AUC 0.572)

12

Fixed reference values (heart rate 85 and 15 beats per minute) Filled with reference means; not restored Simulated binary dosing of one treatment 1 0.82 (spectral radius 0.93, half-life 10 hours) 1 Patient traits; bounded memory route with cap c No effect (λ = 0) Sustained for 48 hours from hour 90; common random numbers 12-hour mean memory-route contrast on the unheld path On (real cadence) or off (hourly) Passed (AUC 0.714 to 0.727); low sensitivity on these windows

Supplementary Note 1: Registered wording Several registered analyses fixed in advance the sentences that may be used to report them. We quote them here verbatim. Key to internal terms: F is the oracle-structure learner; the decision-90 query is the treatment decision at hour 90; rs_dlm_mech and cf_grf_mech are the kernel-informed random-slope model and causal forest; UNRESOLVED marks a comparison that did not survive Holm correction; a “twin” in the map sentence is a supervised reader. Static CATE learners (registered headline). “no standard static learner extracted usable heterogeneity at the decision-90 query; learner failure at this signal level; positivity not separated. F not outperformed by static standard CATE learners on this cell; UNRESOLVED never means equivalence. F’s remaining structural knowledge (exact trait prior and shrinkage, bounded map, natural non-target policy prior), together with its Huber likelihood, adds accuracy beyond kernel-informed standard learners; F beat a comparator that did not beat zero (rs_dlm_mech, cf_grf_mech)” Sequence CATE models (registered headline). “no Causal Transformer or RMSN extracted usable heterogeneity at the decision-90 query; sequence-model failure at this signal level; positivity not separated (CRN failed the synthetic gate). F not outperformed by Causal Transformer and RMSN on this cell; UNRESOLVED never means equivalence”. The only permitted combined statement about the decision query is the conjunction of the three headlines (static, sequence and g-computation), with a family-wise error bound of 0.15; the control inside the support is reported separately, with a family-wise error bound of 0.10 when its two families are cited together. G-computation sequence model (registered headline). “F not outperformed by G-Net (Causal Transformer code) on this cell; UNRESOLVED never means equivalence” In-support query control (registered sentence). “Neither registered hour inside the dosing period (30 or 42) met the registered overlap criteria on the recorded treatments (Methods), so no control inside the support was scored, and the benchmark cannot distinguish a failure of the learners from the lack of treatment variation at the query.” Historical benchmark. “On a semi-synthetic benchmark with a known causal model (five data-generating replicates of 60 outcome, treatment and horizon cells, each cell with 19 to 30 decisions of which 1 to 10 were treated), the model’s average-treatment-effect estimates had lower absolute error than a naive treated-versusuntreated comparison (0.021 versus 0.213) but higher error than always predicting no effect (0.021 versus 0.017, worse in all five replicates), because 71% of the true cell effects were exactly zero; this benchmark therefore does not show that the model recovers treatment effects.” Identifiability map. “on real eICU records at their real cadence, for a treatment whose response carrier the record samples hourly and at an amplitude calibrated from clinical effect sizes, a twin that reads the record at its own draw times recovers 17 to 18 percent of the individual memory-route effect against a bar of 10.6, beats a history-blind ridge by 0.11 to 0.14 with lower bounds above +0.05, replicates on patients the first sample did not use, and has no advantage on a fixed schedule, where there is no patient-specific history to align to.” The readers are fitted with simulator effect labels; the claim concerns information content and is not a claim about a deployable factual-only estimator. The replication in this sentence uses a second sample of 400 stays, 5 of them shared with the first, and reuses the person-indexed generator draws, including the traits. Its true effects are therefore nearly identical to those of the first sample (correlation 0.9998), so it tests recovery from different records of nearly the same effects (Supplementary Notes 2 and 4). Personalization on validation. The registered sentence states that the protocol’s personalization condition was met on validation (12 of 12 cells after Holm, none significantly negative) and must carry three clauses: the matched patient’s hours were spliced onto the patient’s own first six hours, so a forecast degraded by the splice would also produce the advantage; the advantage “was not shown to exceed that of the patient’s own last measured target value over the donor’s under the protocol’s count rule (0 of 12 cells after Holm within that family)”; and in one cell (SpO2 , 18 to 12 hours) the number of measured hours separates the patient’s record from the donor’s. The splice-free comparison reported in the main text addresses the first clause.

13

Supplementary Note 2: Prior reads, selection and deviations This note records which data had been read before each analysis and which analysis choices were made after earlier results were known. Because of these reads and choices, the main text treats all of its results as development evidence. Supplementary Note 7 gives the corresponding detail for the MIMIC-IV read. Prior reads of evaluation splits. The eICU-CRD test set had been read in earlier analyses of the synchronization contrast, and the per-cell rosters were designed after such a read; the validation split had also been read before the isolation arms. The MIMIC-IV person-disjoint test split (14,211 stays) was scored in earlier campaigns for mortality, length of stay, legacy onset events and threshold events, before its registered read (Supplementary Note 7). The MIMIC-III test split (4,244 stays) was read in 21 earlier campaigns (Supplementary Note 8). All of these reads are disclosed in a test-read registry (46 plan rows and 62 disclosure rows). Selection of the registered cell. The new evidence for the registered cell comes from the replication panel and the fixed-schedule control, because the cell was chosen on the discovery stays. We registered the amplitude of 0.75 SD after an exploratory amplitude sweep on the 400 discovery stays, in which 0.75 SD was the smallest amplitude at which the cadence-aware reader separated. That reader had itself been added after the readers then in use did not separate at real cadence. The registered discovery run is bit-identical to the exploratory threshold run on the same stays. The replication panel, a second sample of 400 stays of which 395 are new, reuses the person-indexed generator draws of the discovery run, including the traits. It therefore carries nearly identical injected effects (correlation 0.9998; Supplementary Note 4) but differs from the discovery panel in real background physiology, cadence and natural dosing. Norepinephrine on heart rate was run as a sensitivity cell and was not registered. Because the amplitude and the reader were fixed after outcome-bearing runs, the search ledger classes the registered-cell runs as exploratory. Synchronization analyses. These analyses reused splits that had been read before, and some of their choices were made after an earlier read. The roster rule of the registered synchronization contrast was set after an earlier read of the eICU-CRD test set. The probability clip of the calibrated log-loss was changed after the test read, so we interpret the log-loss by its direction only. The last-value comparison was run after the synchronization result and is reported as a disclosed, unregistered analysis. The isolation arms ran on a validation split that had already been read for checkpoint selection, calibration maps and earlier landmark readouts. Theory and post hoc checks. The propositions of Supplementary Note 5 and their numerical checks were derived and run after the results, and no eICU-CRD analysis was chosen, registered or stopped because of them. The checks made no new prediction of the twin or of any benchmarked model, and they read washoutpanel targets only as the per-patient targets that the registered benchmark had already scored. Two groups of checks were new analyses of previously read data. The first read the shipped per-stay predictions, last measured values and labels of the synchronization runs on the eICU-CRD test set. It compared the hour-6 arm with the model’s hour-6 forecast of the earlier window and computed tie rates and prevalences of the last measured values. As an unreported diagnostic, it also fitted cross-fitted logistic combinations of the existing predictions on those labels. These reads are recorded as one disclosure row of the test-read registry (TRB-EI18). The second group fitted classifiers to the summaries of the realism probe, including a flexible classifier in one configuration chosen after the audit. Washout panels. The individual-effect analyses use washout panels of 400 real eICU-CRD windows whose outcome channels are real and whose treatment columns are replaced by simulated dosing; missing cells are filled with fixed reference means before the measurement hold is applied. The panels were realism-audited under a registered protocol (reports/revision_v3/WASHOUT_PANEL_REALISM_AUDIT_PROTOCOL_20261011_ V2.md, tag washout-realism-audit-protocol-v2) after their other analyses had been run, including the exploratory construction campaign from which the registered cell was chosen. The registered runs of the in-support control and the g-computation comparator followed the audit. Every audit of that protocol is in Supplementary Table S29, and two earlier realism reads of an earlier washout design on synthetic tapes are disclosed there. The panels’ stays were drawn without regard to the eICU-CRD split, so 117 of the 795 distinct stays belong to the test split. The realism audit of the generated dataset does not cover the panels; that dataset carries no covariate-dependent effect heterogeneity and is not the substrate of the individual-effect results (Supplementary Note 4).

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ICare-RH’s placebo gate (G7). ICare-RH’s design was fixed before it was fitted to any real panel, although the reader and learner analyses had already read the washout panels. The first dispatch of G7 stopped when its native test suite failed, before the analysis driver ran, and nothing was read. One test had required a recorded comparison of two scorers to be bitwise identical, which the cluster’s linear algebra library did not reproduce. A second frozen version, logged in the protocol, changed this test to a relative tolerance of 10−12 , and the gate then ran once. Clinical-endpoint probes. The two probes, P0 and P1, used the same tuning set, which was therefore read twice, and P1 was designed after P0 had read its labels. The pretrained encoder had also been trained on every tuning-set patient, so any further recipe would have to be selected on an inner split and read once on the validation split. The gradient-boosting comparator was a fixed recipe, and the planned set of tuned comparators was not built after P0 stopped. Supplementary Note 6 and Supplementary Fig. S3 report the results of both probes. Search and compute. A search ledger of 2,586 rows records every campaign, run and configuration of the project: of 606 campaign and run rows, 28 were positive, 110 negative, 17 null, 24 mixed, 245 descriptive, 72 failed, 23 not run, 69 withdrawn and 18 unrecorded; 280 were registered before their outcome and 189 were exploratory. The compute ledger records at least 17,683 allocated CPU-thread-hours and 362 GPUhours between 1 August and 26 September 2026; earlier months are not metered. Both ledgers stop on 26 September 2026 and do not contain the endpoint probes, the isolation arms, the placebo gate, the post hoc checks of Supplementary Note 5, the registered MIMIC-IV read or the registered analyses of the washout panels added before submission; the protocols of the probes and of G7 estimated small costs (P0 about 1 to 1.5 GPU-hours and 30 to 80 CPU core-hours; G7 about 6 CPU-hours), and the compute of the analyses of the washout panels is given in Supplementary Note 9.

Supplementary Note 3: Rollout clock defect A deterministic test suite (21 tests) found a clock defect in the action-conditioned rollouts of an earlier model version. The historical rollout benchmark (Fig. 4a of the main text and Supplementary Fig. S2) ran with this defect and is reported as run. No positive finding of the paper rests on its readouts, and the individual-effect analyses on the washout panels do not use these rollouts. All three rollouts (the joint competing-event rollout, the transition-adapter rollout and a dynamics-head rollout) wrote each generated transition into the row of the action that conditioned it. The transition adapter had been trained to map the state through hour t − 1 and the action at t to physiology at t + 1, so every generated row held a forecast for a later hour than its label. The error is a pure one-row shift where persistence is carried only by an exposure computed from the actions. Where persistence is carried through generated physiology, the error compounds, and a single dose is counted twice before decaying. Over the 12-hour window of the registered washout target, with sustained dosing, toy models inflated the mean response by factors of 1.00 to 1.38 (1.20 for an observed channel with autocorrelation 0.8), whereas a correctly clocked rollout read over the window one hour earlier would under-read it. The effect of the defect on the historical readouts was not estimated.

Supplementary Note 4: Semi-synthetic data generation This note gives the equations and constants of the semi-synthetic generator (Fig. 3 of the main text) for the two constructions that use it. Part A describes the generated dataset, which has treatment responses written onto every eICU-CRD stay and passed the realism audit. Part B describes the washout panels, which carry every individual-effect result. The two constructions share the latent structural model, the background weights and the direct-route weights, but they differ in several parameters and in the standardization, treatment policy, random-number regime and target (Supplementary Table S10). The generated dataset was produced at the code version whose hash is recorded in its shipped configuration. The washout panels ran a later version of the same module, whose added options are off by default and inactive in the generated dataset. We recomputed the constants below from the shipped configuration and the generator code without reading patient data (paper_npj_ai/make_generator_constants.py) and read the panel facts from the shipped receipts of the registered cell.

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A. The generated dataset Inputs and standardization. Each stay’s hourly record is cut to its first 336 rows. The generator writes 12 outcome channels (invasive mean arterial pressure, SpO2 by pulse oximetry, lactate, heart rate, respiratory rate, urine output, creatinine, pH, white cell count, platelets, bilirubin and temperature) and reads five treatment channels (phenylephrine, NaCl, norepinephrine, dextrose and inspired oxygen fraction). All other channels are copied from the real record. Missing cells are filled with the channel mean before simulation, and the unfilled record is kept to restore missingness afterward. Outcome k is standardized with the cohort normalization 0 statistics, ytk = (xtk − µk )/σk , and clipped to ±6 and to physiological bounds expressed in the same units (for example 20 to 250 beats per minute for heart rate and 20 to 200 mmHg for mean arterial pressure). A mean-filled cell therefore has y 0 = 0. Random numbers and fixed constants. Each stay has its own random stream, seeded from 13 and a hash of the stay identifier, so its draws do not depend on the other stays. A global stream seeded with 13 draws the background weights wk (templates per channel family with small jitter) and the direct-route matrix W once. Each entry of W is drawn from N (0, 0.352 ), set to zero with probability 0.35 and sign-aligned so that treatment moves each outcome in the direction that the generator treats as favorable (upwards for mean arterial pressure, SpO2 , urine output and pH, downwards otherwise). The realized W has 28 zero entries of 60, and temperature has no direct route (Supplementary Table S9). The mediator transition Am has diagonal entries 0.85 to 0.88 and off-diagonal entries 0.02 to 0.05, and severity enters the mediator through bu = (0.15, 0.12, 0.10, 0.12). The treatment-to-mediator rows are fixed for phenylephrine, NaCl and norepinephrine and drawn per stay from N (0, 0.082 ) for dextrose and inspired oxygen fraction. Each stay also draws the propensity weights and intercepts that drive the counterfactual policy, from N (0, 0.62 ) with a positive first column and from N (−2, 0.32 ), respectively. Its responsiveness ηi = min{max{1 + N (0, 0.52 ), 1/3}, 3} is drawn independently of its record, and about 9% of stays are expected at the lower clip. Latent dynamics. The severity u and the mediator m start from the mean standardized outcome of the first hour (clipped to ±2) and evolve as ut+1 = 0.85 ut + 0.10 m̄t + 0.05 āt + ξtu ,

mt+1 = Am mt + bu ut + B ⊤ at + ξtm ,

with ξ ∼ N (0, 0.252 ) and clipping at ±8. The generated dataset uses the legacy mediator persistence (the multiplier on Am ) of 1.0, under which the joint linear map of (u, m) has spectral radius 1.03 (doubling time 25 hours). Its latent state is therefore not mean-reverting and is bounded only by the clip, but its influence on the outcomes is scaled by 0.02. A hidden confounder vt+1 = 0.9 vt + N (0, 0.252 ) is simulated but enters neither the policy nor the outcomes (λ = 0). Patient traits are off, so the bounded memory route of the washout panels is absent. Factual treatments. The factual path replays the record. Treatment j is on in hour t (atj = 1) when it has a recorded dose above 10−9 in that hour, and only this indicator enters the dynamics. The recorded dose is written back at single precision, and a rounded dose not exceeding 10−9 is set to 0. Outcome.

The structural change and the generated outcome are  δtk = clip±6 0.02 (wk⊤ [ut , mt ] + εtk ) + 0.8 ηi (W at−2 )k ,

 0 ytk = clip ytk + 0.6 δtk ,

with ε ∼ N (0, 0.252 ) and y clipped to ±8 and to the physiological bounds, then back-transformed as µk + σk ytk and clipped to the same bounds in raw units. Per treated hour the direct route moves a standardized outcome by 0.6 × 0.8 ηi Wkj , for example by −0.39 ηi standard deviations on creatinine per hour of NaCl and −0.11 ηi on heart rate per hour of phenylephrine. Measurement hold. The measurement hold records a new value only in hours in which the real value changed. Because BlendedICU forward-fills, an unchanged value is taken to mark an hour without a new measurement. Hour t > 0 of channel k therefore counts as not re-measured when its standardized real value equals that of the previous hour exactly, and in such hours the recorded value repeats the previous recorded value. Only the recorded value is held. The latent state, the noise draws and the unheld outcome are computed every hour exactly as without the hold, so the random-number stream is unchanged. The first hour always receives a fresh value, and runs of missing (mean-filled) cells are held and masked afterward. With the hold off, the generator reproduces earlier datasets bit for bit.

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Missingness correction. The missingness correction restores the real missingness of the treatment channels. After simulation, every cell that is missing in the real record is set to missing in the generated record, and the correction extends this rule from the non-treatment channels to the five treatment channels. The 144 channels that the generator does not write are therefore the real record, the 12 outcome channels carry semisynthetic values where the real channel is observed, and the treatment channels equal the real ones. The generator permits the correction only when factual treatments replay the recorded doses. Under a simulated policy the treatment values are generated, and a missing value would assert an absence that the real record never made. The generator’s own synthetic measurement process is inactive. Counterfactuals. Counterfactuals were requested for the first 2,000 test-split stays. Of these, 674 had records longer than 51 hours, as the 48-hour horizon requires, and 1,326 were too short. For each eligible stay, 20 decision hours were drawn uniformly with replacement, and a 48-hour rollout started from the factual latent state for each decision hour, treatment and arm, giving 200 rollouts per stay. At the decision hour the chosen treatment was set to the arm’s value (0 or 1), and the others were drawn from the propensity policy pt = σ(clip±8 {2Wp [ut , mt ] + b}). Afterward all five treatments followed this policy rather than the recorded doses. The two arms used different random draws. The direct route used factual actions for lagged hours before the decision and counterfactual actions after it, and the hold used the real record’s change times. The stored truth is the minimum and maximum of the recorded counterfactual outcome over 6, 12, 24 and 48 hours. Under the hold, this truth is the effect on the recorded value, so the do(1) minus do(0) contrast of a channel is zero until the channel’s first real re-measurement after the decision. Scale. The audit found all 200,825 eICU-CRD stays in the generated dataset. The generator’s final log counts 198,825 stays without a counterfactual request, 674 with counterfactual truth and 1,326 requested but too short for it, which sum to 200,825. The index file has 200,826 lines including its header.

B. The washout panels Records. Each washout panel holds 400 real eICU-CRD windows. Candidate stays are the eICU-CRD stays present in both the time-series index and the label table, with no split filter. A seeded permutation selects the first 400 stays with at least 164 hourly rows, and the window is hours 24 to 163 of each stay. The discovery and fixed-schedule panels share one seed, and the replication panel uses another. From each window the 12 outcome channels are read, missing cells are filled with fixed reference means, values are kept strictly inside the physiological bounds, and the treatment columns are set to zero. Synthetic records replace the real window with smooth random curves (an offset, a slope, a sinusoid and small noise per channel). No two consecutive hours of these curves are equal, so the hold has no effect on them. Parameterization. The panels use a frozen parameter set (Supplementary Table S10). The background scale is 1, and a mediator persistence of 0.82 (spectral radius 0.93, half-life 10 hours) makes the latent state mean-reverting. There is no per-stay multiplier (ηi = 1), and patient traits act through a bounded memory route. The lag is 2 hours, counterfactual arms share common random numbers, and dosing is simulated and binary. All other structural constants, including W and wk , equal those of the generated dataset. Every person-level draw (traits, propensity weights and intercepts, noise and top-up) uses a stream keyed to the person index rather than to the stay. Traits. Each person carries one trait per treatment, which weights that treatment’s contribution to the memory route. The trait of person i for treatment j is ζij = min{max{0.6 lj ξi + 0.15 ϵij , −2/3}, 2}, with one shared standard normal factor ξi , loadings l = (1.09, 1.11, 0.99, 1.14, 0.78) drawn once and independent standard normal ϵij . For phenylephrine in the discovery panel the trait has standard deviation 0.59, and 14% of persons sit at the lower clip. Dosing. Each factual record runs through a lead-in, a dosing period and a washout before the decision at row 90. Rows 0 to 5 are the lead-in. Natural dosing in rows 6 to 53 is harvested from a separate factual rollout on the same window in which all five treatments follow the propensity policy, and only the scored treatment’s column is kept. If fewer than eight rows are dosed, a contiguous undosed block, or failing that random undosed rows, tops the count up to eight. All other rows and treatments are zero in the factual record. In the discovery panel the scored treatment was given in 21.4 rows on average (standard deviation 14.7, range 8 to 48). Rows 54 to 89 are the washout, after which the exposure memory at the decision averages 0.001. The fixed-schedule control doses rows 6 to 24 for every person.

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Memory route. The memory route adds a bounded, patient-specific response that builds with recent doses and decays after them. For each treatment, the exposure memory is etj = 0.8 et−1,j + at−2,j and its occupancy qtj = etj /(1 + etj ), both computed without noise and without feedback into the latent state. The shift on P outcome k is stk = c tanh( j Lkj ζij qtj ) with Lkj = Wkj / maxk′ |Wk′ j |. On heart rate the loadings are −1 for phenylephrine, −0.75 for norepinephrine, −0.26 for inspired oxygen fraction and 0 for NaCl and dextrose. The shift is added after the ordinary clips and attenuated near a bound, so the outcome stays inside it. Under sustained dosing from an empty memory, occupancy by outcome hour 90 + k is 0, 0.50, 0.64, 0.71, 0.75 and 0.77 for k = 1 to 6 and 0.82 by k = 12. It approaches 5/6 and never reaches full occupancy. The registered cell uses a cap of c = 0.75 reference standard deviations, which is 11.25 beats per minute on heart rate. For the 400 discovery traits, the scored treatment’s own shift has a median of 3.8 beats per minute at steady-state occupancy (95th percentile 8.7) and 4.5 at full occupancy. The recoverability map also used caps of 0.25, 0.5 and 1. Intervention, frames and target. The target isolates the memory-route effect of the scored treatment by comparing an active frame with a placebo frame. At row 90 the scored treatment is set to 1 or 0 and held for 48 hours. The other four treatments follow the propensity policy in both arms, and arm 1 replays arm 0’s random stream. Each person is simulated in two frames on the same record and actions, an active frame with cap c and a placebo frame with c = 0. Because the memory route does not feed back into the state, every trace except the outcome is identical between the frames (no mismatches in any receipt). The placebo frame’s factual record serves as the zero-effect copy used by the ICare-RH placebo gate. The target is the 12-hour mean of the do(1) minus do(0) difference in the unheld heart rate in the active frame, minus the same quantity in the placebo frame, in units of 15 beats per minute. The ordinary treatment routes are identical in the two frames and cancel in the target. They comprise the direct route of 0.6×0.8×WHR,phenylephrine = −0.11 standard deviations per dosed hour, the mediated route and any divergence of the other treatments’ policy draws. Their contribution, the do(1) minus do(0) contrast of the placebo frame, has a standard deviation across persons of 1.48 beats per minute in the discovery panel. The target itself has a standard deviation of 3.48 beats per minute in every real panel, with or without the hold, and a mean of −0.18 beats per minute in the discovery and replication panels. Because the policy also doses the other treatments after the decision, their memory enters the tanh, and the target is not a function of the scored trait alone. A predictor that is exact when the other treatments are held at zero correlates 0.977 with the target, and the oracle reference, which knows only the scored trait, reaches a recovery of 0.80 rather than 1. The value 0.977 comes from a simulation that is quoted in the registered protocol but whose code was not committed. Cadence. Measurement cadence changes what the readers see but not the target. At real cadence the generated record repeats its previous value in every hour whose real value did not change. Heart rate repeats in 43% of the hours of the discovery panel and in 41% of the replication panel. With hourly observation every hour receives a fresh value. The hold acts only on the pre-decision record of the active frame, which is what the readers see, and the target is computed from the unheld outcome. Replication and fixed schedule. The replication panel tests recovery of nearly identical injected effects from different records. It keeps the person indices, and hence the traits and person-level streams, but selects different stays (5 of 400 shared with the discovery panel) and re-harvests natural dosing on them. Its true effects, the targets defined above, correlate 0.9998 with those of the discovery panel. The fixed-schedule panel uses the discovery stays and persons with rows 6 to 24 dosed.

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Supplementary Note 5: Theory and proofs This note sets out what each evaluation design in the paper can and cannot identify, so that the main results can be read at the strength their designs support. It gives the definitions, formal statements, assumptions, proofs and numerical checks of the seven propositions summarized in the main text’s Methods (“Theoretical analysis of the evaluation”). Sections 5.1 to 5.7 take one proposition each, in the order of the Results. They cover synchronization (Proposition 1), the difference-in-differences and the current-value hypotheses (Proposition 2), the realism audit (Proposition 3), recoverability and cadence (Proposition 4), the efficient reader (Proposition 5), treatment-effect learners and transport (Proposition 6), and ICare-RH’s placebo gate (Proposition 7). Section 5.8 states what the theory does not show. Status. The seven propositions were derived after the results. The theory was not used to choose, register or stop any eICU-CRD analysis, and no part of it was registered in advance, in version control or externally. Each proposition lists its assumptions and says where the implemented models, code or data depart from the idealization it uses. The numerical checks are post hoc and descriptive. They re-read shipped tables of previously read splits, reconstructed inputs from shipped files or used simulation, and none made a new prediction of the twin or of any benchmarked model. The checks of Section 5.4 re-read only the per-patient washout targets that the registered benchmark had already scored. Two groups of checks are new analyses of previously read data, and both are disclosed in Supplementary Note 2. The first group read the per-stay predictions, last measured values and labels of the previously read eICU-CRD test panels. With them it formed the new contrast of two existing hour-6 reads reported in Section 5.1 and computed the label prevalences and the tie rates of the last measured values (Section 5.2). As an unreported diagnostic, it also fitted crossfitted logistic combinations of the existing predictions on those labels. These reads are recorded as one disclosure row of the test-read registry. The second group fitted classifiers to the summaries of the realism probe, including a flexible classifier in one configuration chosen after the audit (Section 5.3). Where a check depends on an idealization, it says so. For example, checks that need the panels’ draw masks, which were not shipped, use simulated draw patterns. The check scripts and their outputs will be released with the code, in reports/revision_v3/theory_checks/. Standard and specific parts. Several parts of the propositions restate standard results for the quantities of this evaluation: the Neyman–Pearson lemma and the refinement decompositions of proper scores (Section 5.1), bounds of an AUC by distances between distributions and equal-covariance Gaussian discrimination (Section 5.3), generalized least squares and trend partialling (Section 5.5), the decomposition of mean squared error by correlation and spread (Section 5.6), and the profile likelihood of a one-way variance component (Section 5.7). Each section begins by naming its standard parts and their sources, so that the reader can see which parts are specific to this evaluation. Conventions. Var, Cov, corr and SD are population moments; shipped tables report their sample versions with population standard deviations. AUROC gives half credit to ties. 1{·} is the indicator and Φ the standard normal distribution function. For a positive definite matrix Ω, ⟨u, v⟩Ω := u⊤ Ω−1 v and cos2Ω (u, v) := ⟨u, v⟩2Ω /(⟨u, u⟩Ω ⟨v, v⟩Ω ). Assumption labels such as (A1) are local to the section in which they appear, and a symbol introduced inside a proof is local to that proof. Code references give file names and line numbers at commit 82587202 of the analysis repository; apart from the check scripts added in reports/revision_v3/theory_checks/ and the files added for the registered analyses run after that commit (the realism audit of the washout panels, the in-support control and the g-computation comparator), the only code files changed between that commit and the commit of the submitted manuscript are the files added or changed for the registered MIMIC-IV read (among them icare_dt/measurement_occurrence.py and icare_dt/terminal_semantics.py) and the scripts that build the paper’s figures, tables, source data and cover letter (icare_dt/figures/npj_ai_evaluation_figures.py and paper_npj_ai/make_*.py). The trait ζij , ICare-RH’s channel loadings γk and the lag ℓ of an exposure kernel are those of Methods. Symbols with more than one use. sections named.

Some symbols carry more than one meaning, each confined to the

• L: landmark hour (5.1, 5.2). Lkj : loading of treatment j on channel k (5.4 to 5.7); it always carries two subscripts, and Λk := Lkj ∗ . • R: the roster event, with roster law PR (5.1, 5.2). PR , PG , fR , fG : laws and summaries of real and generated records (5.3).

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• η: the latent index of the threshold model (5.2 only). The generator’s per-stay responsiveness ηi (Supplementary Note 4) does not occur in this note. • u: spread ratio of a predictor (5.6, 5.7). The generator’s latent severity ut (Supplementary Note 4) occurs only as u0 , its shipped per-patient value at row 0, from which the natural doses were reconstructed (5.4 to 5.6). • F6 : the information in the first six hours (5.1, 5.2); the letter W is not used for it. W is the generator’s direct-route matrix (Supplementary Note 4) and enters this note only through Lkj . • λ: decay of an exposure kernel (5.4 to 5.7); penalty of the audit’s logistic regression (5.3). • j: treatment index in Γik , ζij and Lkj ; index of consecutive draws in dj , ∆j and gj (5.4, 5.5, 5.7). The two uses never occur in one formula. • Π: residual projector on a quadratic trend (5.4, 5.5). Πζ : posterior law of the trait (5.6). Πpool : donor pool (5.1). • v: weight vector of a linear statistic (5.5); slope variance of a random-slope model (5.6, 5.7). • b: trait-free baseline bik (5.4, 5.5); response slope bi , b∗i (5.6, 5.7). • δ: mean difference of the audit summaries (5.3); shortfalls δ2 , δa of learned reads (5.1); drift variance vδ (5.7). • r: a record read by the forecaster, with parts r1:6 and r7:L (5.2); correlation of a predictor with τ (5.6, 5.7), as in the main text; residual vectors ri and rik always carry a patient subscript. • gj : gap between consecutive draws (5.4, 5.5, 5.7). The linear-drift direction is t (5.7). • U6 : affine index of the first six hours (5.2). Uik : shift of the held record per unit amplitude (5.4). • K: exposure kernel. dKS (s): Kolmogorov distance between the two laws of a score (5.3). K1, K2, K3: clauses of the placebo gate (below).

Notation and shared definitions Forecasting (Propositions 1 and 2). A stay is indexed by i and its hours by t = 1, 2, . . . . Xt is the hour-t model input (161 standardized values and the observation mask) and xs the static inputs. Ft := σ(xs , X1 , . . . , Xt ) is the information in the record through hour t. The frozen twin state zt is a function of (xs , X1 , . . . , Xt ), and hence Ft -measurable. The encoder is causally masked, but the propositions do not rely on the mask, because every read is computed from the record truncated at its origin (Section 5.1, assumption (A0)). This measurability is the only property of the implemented encoder that the propositions use. A setting is (L, H) ∈ {(12, 6), (18, 12), (30, 24)}, and in every setting L = 6 + H. For a target k, Tev is the hour of the first event of any class (threshold crossing, live discharge, death), Y := 1{first crossing in (L, L + H]}, R is the event that the stay is on the setting’s roster and π := P (Y = 1 | R). The arms are those of the main text and Methods: S2, the synchronized forecast FL (H); S1, the hour-6 forecast [F6 (2H) − F6 (H)]/S6 (H) of the same window; BR and BR2, S1 and S2 with a training-fitted intercept per setting on the logit scale; IC, IC-A and IC-O, the information-constant arm and its two sensitivity versions; SH, the donor arm (the patient’s hours 1 to 6 followed by the matched donor’s hours 7 to L), and SH-A, its availability-preserving version; own and donor, the patient’s and the donor’s last measured target values; and three donor-only scores (the donor’s own forecast, last value and number of measured hours). The ten hour-6 and donor arms are S1, BR, IC, IC-A, IC-O, SH, SH-A and the three donor-only scores. V is the direction-signed last measured value of target k at hour L and V ′ the matched donor’s value. Definition 1 (ideal forecast, information-set value, donor independence). For a σ-field G, pG := P (Y = 1 | G, R) is the ideal forecast given G. The implemented S1 conditions on no event of any class by hour L, because L = 6 + H and the ratio [F6 (2H) − F6 (H)]/S6 (H) divides by the all-cause event-free probability. Every read at origin L conditions on the patient being at risk at L, because the heads are supervised only at at-risk origins. The information-set value of an arm is pGa , where Ga is generated by the arm’s input and the event its read conditions on; S1∗ and S2∗ are the information-set values of S1 and S2, and Proposition 1(b) shows that S1∗ = pF6 . Donor independence (D) states that the donor record is conditionally independent of the patient’s label and hours 7 to L given the patient’s first six hours. The implementation supports it, since donors are matched on hour-6 encoder states only, never on labels and never from the same person.

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It remains an assumption, because the standardization constants are roster-wide and the checkpoints were selected on validation data (Proposition 1, (D)). Exact matching, under which the donor’s hours 7 to L have the conditional law of the patient’s given the first six hours, is an idealization of nearest-neighbor matching in the 64-dimensional hour-6 state. The twin is a Transformer trained on a discriminative competing-risks likelihood and is not a Bayes filter, so treating its forecast as an ideal forecast is also an idealization. Definition 2 (current-value hypotheses). A forecaster obeys the current-value restriction (CVR) if its forecast at origin L is a function of the first six hours and V only. The data obey the current-state hypothesis HCS if Y ⊥ X7:L | (F6 , V, R), and current-value sufficiency (CVS) if Y ⊥ FL | (V, R). Personalization beyond the current state is the contrast AUROC(pFL ) − AUROC(pF6 ∨σ(V ) ), which no registered arm estimates. In the latent-threshold Gaussian model (LT), a standardized latent index η and the record are jointly Gaussian, Y = 1{η > tπ } with tπ := Φ−1 (1 − π), and each arm is a strictly increasing function of a linear index. Then AUROCa = Aπ (ρa ), where ρa is the correlation of arm a’s index with η and Aπ is a strictly increasing bijection of [−1, 1] onto [0, 1] (Proposition 2, Lemma 1). Definition 3 (classifier two-sample test). For a score s of the 644 audit summaries f , AUC(s) := P (s(fG ) > s(fR )) + 21 P (s(fG ) = s(fR )) for independent generated and real summaries. The audit statistic is the pooled out-of-fold AUC of an L2 -penalized logistic regression (C = 0.1, unpenalized intercept) over five persongrouped folds of 400 matched pairs. Its population target is the AUC of the minimizer of the penalized population logistic risk (Proposition 3). Washout panels (Propositions 4 to 7). Rows t = 0, . . . , 89, one per hour, precede the decision row t0 = 90. The horizon is h = 12, and the post-decision step s = 0, . . . , 11 refers to row 91 + s. Outcome channels are k = 1, . . . , 12 (k ∗ is heart rate in the registered cell, in units of 15 beats per minute) and treatments j = 1, . . . , 5 (j ∗ is the scored treatment). Ai (t) ∈ {0, 1} is the dose of j ∗ . Under natural dosing (NATMIN), doses fall within rows 6 to 53 and are topped up to at least eight dosed rows. Under the fixed schedule (FIX19), rows 6 to 24 are dosed. Rows 54 to 89 are a washout. A kernel K = (ℓ, λ, κ) with lag ℓ, decay λ and saturation κ defines the memory eK (t) = λeK (t − 1) + A(t − ℓ), eK (0) = 0, and the exposure profile pK (t) := φκ (eK (t)) with φκ (e) := e/(κ + e). The generator kernel is K0 = (2, 0.8, 1) and qi := pK0 is patient i’s occupancy. ζij is the trait of patient i for treatment j and ζi := ζij ∗ ∈ [−2/3, 2], with mean µζ and variance σζ2 . Lkj is the loading of treatment j on channel k and Λk := Lkj ∗ (Λk∗ = −1 for phenylephrine on heart rate), and c is Pthe amplitude cap. The latent standardized outcome is yik (t) = bik (t) + c αik (t) tanh Γik (t) with Γik (t) := j Lkj ζij qij (t). Here bik collects everything that involves no trait, namely the generator background, the ordinary treatment routes and the source tape (the stay’s real record), while αik ∈ [0, 1] is the attenuation near a physiological bound. The draw set Dik consists of row 0 and the rows at which the source value changes, ordered d0 = 0 < d1 < · · · < dn . The last draw at or before t is dik (t), and the held record is Zik (t) := yik (dik (t)). The design is Si := (Ai , Di1 , . . . , Di,12 ). Over consecutive draws, ∆j f := f (dj+1 ) − f (dj ) and gj := dj+1 − dj . The P89 P (m) exposure energies are Q◦i := t=1 (qi (t) − qi (t − 1))2 (hourly) and Qik := j gjm (∆j qi )2 for m ∈ {−1, 0, 1} P (−1) (sampled), with Jik := Qik . The cadence-aware statistic is Cik := j ∆j Zik ∆j qi (with the companion P (0) feature Qik ), and the plain occupancy contrast is Oik := t (Πqi )(t)Zik (t), with Π the residual projector on a quadratic trend is rec := 1 − RMSE/SD(τ ) and R2 = 1 − (1 − rec)2 . The √ over rows 0 to 89. Recovery 2 bar rec = 1 − 0.8 = 0.106 corresponds to R = 0.2. For a predictor τ̂ , r := corr(τ̂ , τ ), u := SD(τ̂ )/ SD(τ ) (spread ratio) and cPEHE := SD(τ̂ − τ ). For a σ-field or random vector X, R∗2 (X) := Var E[τ | X]/ Var(τ ) is the information content of X about τ . Ph−1 Definition 4 (washout target). τi := h−1 s=0 [YisA (1̄)−YisA (0̄)−Yis0 (1̄)+Yis0 (0̄)], a frame-by-regime interaction on the latent (every-hour) heart-rate path at row 91 + s. Frame A has cap c and frame 0 has cap 0. Both are rolled out from the same factual trace with the same seed, so they share every state, action and noise path. Regime 1̄ gives j ∗ on rows 90 to 101 and regime 0̄ withholds it. The other four treatments follow the generator’s policy with common random numbers. The target is computed from the unheld path, whereas the readers see the held record. Definition 5 (design exogeneity). The design Si is independent of the traits. Natural doses are harvested from a policy run on the same real window, which depends on the stay’s tape and latent state but not on the trait. The draw sets are fixed by the source tape, and the trait enters only the injected shift, which does not feed back into the latent state, the doses or the draw sets. Independence of the doses from the trait was checked by reconstructing all 800 NATMIN dose columns of the discovery and replication panels from shipped inputs (dosed-row counts equal the shipped table for 400 of 400 patients in each panel). The design is not 21

independent of the baseline b, because the doses and the draw sets both depend on the stay’s tape. Any statement that treats them as independent is an idealization and says so. Definition 6 (linear readers and information content). A reader is a feature map that is affine in the held record for a fixed design. The history-blind, window-mean, plain, assimilating and cadence-aware readers are of this kind. The window-mean reader takes channel means over four windows set by the first and last dose, and the plain reader is the occupancy-contrast reader of the main text. The oracle reference, which uses the true trait, is not a reader in this sense. Design-only features depend on Si alone. The population R2 of a reader is that of population least squares on its features with the kinetics fixed. The shipped out-of-fold ridge regression with nested cross-validation estimates it with a finite-sample loss. Every predictor that is a function of the record has R2 ≤ R∗2 (record) (Proposition 4(c)(i), Proposition 6(b2)). A supervised reader’s out-of-fold R2 therefore estimates a lower bound on the information content of the record and does not bound from above what another estimator using the same record can reach. Definition 7 (retention and aliasing). Under a random-walk baseline, the retention χik := Jik /Q◦i ∈ [0, 1] is the share of the hourly record’s Fisher information about the trait that the held record of channel k keeps. An interval (dj , dj+1 ] is aliased when the exposure has the same value at its two draws although it moved between them. Under the generator’s binary dosing no interval is exactly aliased (Proposition 4(d)(iii)), but nearly aliased intervals occur, and χik measures their effect. Definition 8 (beating zero). The constant prediction 0 has centered PEHE SD(τ ), and a predictor beats predicting zero when cPEHE < SD(τ ). This is a population criterion. In the benchmark, a learner was counted as outperforming predicting zero only if the 95% bootstrap interval of its centered PEHE minus that of predicting zero lay wholly below zero. Definition 9 (transport). Under NATMIN and FIX19 no dose of j ∗ occurs at or after row 54, so the query regime, sustained dosing from row 90, has propensity zero for every patient. Treated training windows in the query’s recency cell of Methods (dosed earlier, no dose in the preceding 12 hours) are factual windows of other regimes, not instances of the query regime. Structural transport assumes T1 (invariance: the same ζi that sets the pre-decision shift sets the query effect), T2 (exogeneity: Definition 5) and T3 (known structure: kernel, bounded map, loadings, trait law and the law of the zero-effect record given the design are known or identified); see Proposition 6. Neither registered hour inside the dosing period (30 or 42) met the overlap criteria of Methods. Definition 10 (overdispersion along the exposure direction). For one channel (or a fixed loading direction), a kernel K and a panel copy, let Ai := ⟨pK , pK ⟩Ωi and Bi := ⟨pK , ri ⟩Ωi be the exposure information and the exposure projection of the baseline residual ri at the drawP rows under ICare-RH’s working covariance P P P Ωi . The overdispersion along the exposure direction is ω(K) := i (Bi − µ̂Ai )2 / i Ai with µ̂ := i Bi / i Ai . If the residual has conditional covariance Σi , then on a mean-zero copy E[Bi2 | Si ] = Ai ωi (K) with −1 −1 ⊤ −1 ωi (K) := p⊤ K Ωi Σi Ωi pK /pK Ωi pK , which equals 1 when the working covariance is correct along pK . ω0 denotes the population value on the zero-effect copy. Definition 11 (pseudo-true parameter). The pseudo-true parameter of a working model is the parameter value that maximizes the working model’s expected log-likelihood under the true law. Definition 12 (clauses of ICare-RH’s placebo gate). K1 is the lag clause. The analysis stops if, in either natural-dosing panel, the profile likelihood favors a lag outside 1 to 3 hours in at least three of five outer folds. The count of folds in which a slope-variance test has P < 0.05 was reported descriptively and did not gate. K2 is the pair of spread clauses on the zero-effect copies of every panel: the standard deviation σ0 of the predicted effects must be at most 0.05 (absolute clause) and at most 25% of the standard deviation σA on the active copy (ratio clause). K3, which was to be read only after K1 and K2 and was never reached, required the 95% whole-person bootstrap interval of centered PEHE minus that of predicting zero to lie wholly below zero in both natural-dosing panels. Proposition 7(e) concerns K2 only.

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5.1 Synchronization is information refinement Proposition 1 (Synchronization is information refinement, and the isolation arms add model error to it). Standard and specific parts. Part (a) collects standard facts: the ROC dominance of the ideal forecast is the Neyman–Pearson lemma applied within a sub-σ-field (Lemma 2 below), and the Brier and log-loss identities are the refinement decompositions of proper scoring rules (DeGroot and Fienberg; Gneiting and Raftery). Part (d) is an algebraic identity whose sign constraints come from (a). What is specific to this evaluation is the identification, in (b) and (c), of the information-set values of the implemented arms, including the exact Brier cost of a splice in (c)(iv). Setting. Fix a target k and a setting (L, H) ∈ {(12, 6), (18, 12), (30, 24)}, so that L = 6 + H. Let R be the event that a stay is on the roster of this target and setting, PR := P( · | R) with expectation ER , and Y ∈ {0, 1} the label (first crossing of target k in (L, L + H]), with π := PR (Y = 1) ∈ (0, 1). Let xs be the static inputs (including the domain indicator), Xt the hour-t model input (161 standardized values and the observation mask), ωt := (xs , X1 , . . . , Xt ), Ft := σ(ωt ) and X7:L := (X7 , . . . , XL ), so that FL = F6 ∨ σ(X7:L ). Tev is the hour of the first event of any class for target k (measured threshold crossing, discharge, death). For a σ-field G, pG := PR (Y = 1 | G) is the ideal forecast given G, and S1∗ := pF6 ∨σ({Tev >L}) ,

S2∗ := pFL .

Arms (as implemented, with trained weights, intercepts, orientation sign and matching rule held fixed). S2 = ψ(ωL ) is the frozen model’s onset cumulative incidence at bin H read from hours 1 to L; ψ(ω6 , x7:L ) denotes the same read on the input whose statics and hours 1 to 6 are ω6 and whose hours 7 to L are x7:L . S1 = [F6 (2H) − F6 (H)]/S6 (H), clipped to [0, 1] and read from ω6 alone. For a ∈ {IC, IC-A, IC-O}, a = ψ(ω6 , κa (ω6 )), where κIC repeats the hour-6 values in every channel with every mask entry 0; κIC-A repeats the hour-6 values and the hour-6 non-target mask, with target-channel occurrence 0; κIC-O equals κIC-A except that the occurrence ′ of target k at hour L is 1. ωL is the record (statics and hours 1 to L) of the donor that the seed’s matching rule ′ are its hours 7 to L; donors are roster assigns to the stay, a random element under PR (see (D)), and X7:L ′ ′ )), where κSH-A keeps a stays and therefore at risk at L. SH = ψ(ω6 , X7:L ) and SH-A = ψ(ω6 , κSH-A (X6 , X7:L non-target channel available, with its hour-6 value, wherever the recipient had it at hour 6 and the donor’s row ′ is unavailable. The three donor-only scores read the donor’s record alone: the donor’s own S2 = ψ(ωL ), the ′ donor’s direction-signed last target value V at L, and the donor’s number of measured target hours in 7, . . . ,
L times a fixed sign. BR = gb (S1) and BR2 = gb′ (S2) with gb (p) := expit logit(min{max{p, ϵ}, 1 − ϵ}) + b , ϵ = 10−6 , b, b′ fixed. Write A6 := {S1, BR, IC, IC-A, IC-O} (hour-6 arms), ASH := {SH, SH-A} (spliced arms) and Adon for the three donor-only scores. The information-set value of an arm a is pGa with Ga := σ(ιa ) ∨ σ(Ea ), ′ ) and where ιa is the input the arm reads (for BR and BR2 that of S1 and S2; for the donor-only scores ιa := ωL Ea is the event its read conditions on: {Tev > L} for S1 and BR through the ratio form; the recipient being at risk at L for the recipient’s reads at origin L (S2, BR2, the IC-type and spliced arms); the donor being at risk at L for the donor’s own S2; and the sure event for V ′ and the measured-hours score. Metrics. For a score s (a real random variable) and α ∈ [0, 1], ROCs (α) is the true-positive rate, at falsepositive rate exactly α, of the test that calls a stay positive when s exceeds a threshold, ties at the threshold being broken by independent randomization (Lemma 1 of the proof). AUROC(s) := P(s1 > s0 ) + 12 P(s1 = s0 ), with s1 , s0 independent draws from the laws of s given Y = 1 and Y = 0 under PR . For a forecast q ∈ [0, 1], Brier(q) := ER (Y − q)2 and LogLoss(q) :=
ER [−Y log q − (1 − Y ) log(1 − q)] ∈ [0, ∞]. For σ-fields F ⊆ G, MI(Y ; G | F) := ER KL Ber(pG ) ∥ Ber(pF ) in nats. (a) Dominance. Let F ⊆ G. For every G-measurable score s and every α ∈ [0, 1], ROCs (α) ≤ ROCpG (α); hence AUROC(s) ≤ AUROC(pG ). For every G-measurable forecast q,
Brier(q) − Brier(pG ) = ER (q − pG )2 , LogLoss(q) − LogLoss(pG ) = ER KL Ber(pG ) ∥ Ber(q) . In particular Brier(pF ) − Brier(pG ) = ER (pG − pF )2 ≥ 0 and LogLoss(pF ) − LogLoss(pG ) = MI(Y ; G | F) ∈ [0, log 2], and each of the two vanishes if and only if pF = pG almost surely. (b) Hour-6 arms; lead time is not a separate factor. Under the roster rule (A1), S1∗ = pF6 almost surely. The information-set value of each arm in A6 is S1∗ , and that of S2 and BR2 is S2∗ . Consequently LogLoss(S1∗ ) − LogLoss(S2∗ ) = MI(Y ; FL | F6 ) ≥ 0,

Brier(S1∗ ) − Brier(S2∗ ) = ER (S2∗ − S1∗ )2 ≥ 0,

ROCS1∗ ≤ ROCS2∗ pointwise and AUROC(S2∗ ) ≥ AUROC(S1∗ ). The two proper-score differences vanish if and only if Y ⊥ X7:L | F6 under PR , in which case S2∗ = S1∗ . The ideal synchronized and hour-6 forecasts of the same window, under the same at-risk conditioning, therefore differ only through their information sets. (c) Another patient’s hours. Assume donor independence (D). ′ (i) pF6 ∨σ(ωL′ ) = S1∗ almost surely. Every F6 ∨ σ(ωL )-measurable score s, in particular every arm in ASH ∪ Adon , satisfies ROCs ≤ ROCS1∗ pointwise and AUROC(s) ≤ AUROC(S1∗ ).

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′ (ii) Every F6 ∨ σ(ωL )-measurable forecast q satisfies Brier(q) = Brier(S1∗ ) + ER (q − S1∗ )2 .

(iii) The information-set value of SH and of SH-A is S1∗ . The information-set value of each donor-only score is ′ S1∗don := pσ(ωL′ ) = ER [S1∗ | ωL ], with AUROC(S1∗don ) ≤ AUROC(S1∗ ), Brier(S1∗don ) − Brier(S1∗ ) = ER (S1∗ − S1∗don )2 and ′ ′ LogLoss(S1∗don ) − LogLoss(S1∗ ) = MI(Y ; F6 ∨ σ(ωL ) | σ(ωL )); the two proper-score gaps vanish if and ∗ ′ only if S1 is almost surely equal to a function of ωL . (iv) If in addition exact matching (EM) holds, then for the implemented map ψ Brier(SH) − Brier(S1∗ ) = ER (S2 − S1∗ )2 , ′ and for the ideal map (ψ(ωL ) = S2∗ , spliced forecast SH∗ := ψ(ω6 , X7:L )) one has Brier(SH∗ ) − ∗ ∗ ∗ Brier(S1 ) = Brier(S1 ) − Brier(S2 ): on the Brier score the ideal donor contrast Brier(SH∗ ) − Brier(S2∗ ) is exactly twice the ideal information gain Brier(S1∗ ) − Brier(S2∗ ).

(d) Decomposition for the learned model. For every real-valued metric M of an arm’s scores on the roster (population or sample, per seed or averaged over seeds, and in each bootstrap replicate that weights all arms alike) and every arm a,     M (S2) − M (a) = M (S2) − M (S1) + M (S1) − M (a) . From here on M is the population AUROC, −Brier or −LogLoss under PR (−LogLoss only when the loglosses involved are finite; for the two proper scores only arms that are forecasts, which excludes V ′ and the measured-hours score). Let A := A6 without (D) and A := A6 ∪ ASH ∪ Adon under (D), and define I ∗ := M (S2∗ ) − M (S1∗ ),

δ2 := M (S2∗ ) − M (S2),

δa := M (S1∗ ) − M (a) (a ∈ A).

(d1) I ∗ ≥ 0, δ2 ≥ 0 and δa ≥ 0 for a ∈ A, and M (S2) − M (S1) = I ∗ − δ2 + δS1 ,

M (S1) − M (a) = δa − δS1 .

(d2) For a ∈ A6 and, under (D), a ∈ ASH , the arms S1 and a have the same information-set value S1∗ : the second bracket compares two uses of the same information and vanishes when every arm is replaced by its information-set value. For a donor-only score a (under (D)), δa = ε + δa′ ,

ε := M (S1∗ ) − M (S1∗don ) ≥ 0,

δa′ := M (S1∗don ) − M (a) ≥ 0;

ε is the loss from not reading the recipient’s own statics and hours 1 to 6, the second bracket equals ε when every arm is replaced by its information-set value, and for the two proper scores ε = 0 if and only if ′ . S1∗ is almost surely a function of ωL ′ )-measurable (a forecast, for the proper (d3) For every score b that is F6 -measurable or, under (D), F6 ∨ σ(ωL ∗ scores), M (b) ≤ M (S1 ); hence δS1 ≥ M (b) − M (S1) and M (S2) − M (S1∗ ) ≤ M (S2) − M (b). For every FL -measurable score (forecast) b, δ2 ≥ M (b) − M (S2). In particular, for every nonempty B ⊆ A,

M (S2) − M (S1∗ ) ≤ M (S2) − max M (a), a∈B

so among the tested arms the smallest contrast M (S2) − M (a) is the tightest upper bound they give on the advantage of the learned synchronized forecast over the ideal hour-6 forecast. It is not a bound on I ∗ . For sample metrics, the smallest estimated contrast is a selected estimate of that bound and is biased downwards. (d4) M (S2) − M (a) ≤ I ∗ if and only if δa ≤ δ2 . In particular the learned contrast M (S2) − M (S1) is at most the ideal information gain I ∗ exactly when the hour-6 read falls short of the ideal hour-6 forecast by no more than the synchronized read falls short of the ideal synchronized forecast. (d5) AUROC(BR) = AUROC(S1) and AUROC(BR2) = AUROC(S2) whenever gb (respectively gb′ ) is injective on a set carrying the values of the score (a set of probability one for the population metric, the set of observed values for the sample metric); gb is injective on a set A ⊆ [0, 1] if and only if A has at most one point in [0, ϵ] and at most one in [1 − ϵ, 1].

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Assumptions. (A0) Roster law and fixed maps. P(R) > 0 and 0 < π < 1. All statements concern the population law PR of ′ one stay drawn from the roster population, together with its donor record ωL as described in (D). Held ′ fixed: the five trained checkpoints; the TRAIN intercepts b, b of BR and BR2; the TRAIN orientation sign of the measured-hours score; and, for each seed, the matching rule of match_donors (the nearest stay of a different person in that seed’s standardized hour-6 encoder state, among candidates whose record reaches L, exact ties going to the smallest stay identifier) together with its standardization constants. Every learned arm is then a fixed measurable function of the input listed in the Setting. No property of the encoder mask is needed for this, because every read is computed from the truncated prefix alone (the producer passes only the first 6 or L rows, statics and domain to the model). Nothing treats the twin as a Bayes filter: learned arms are arbitrary measurable functions of their inputs, and only the ideal forecasts pG are conditional probabilities. Ties are randomized, matching the half-credit AUROC of the code. Two dependences are set aside. Each checkpoint is the epoch with the lowest validation log-loss (r2_joint_train_validate.py L148–L198; selection status VALIDATION_SELECTED_NO_TEST_ACCESS), so on the validation split the learned maps depend on the stays’ labels through that choice; and the standardization constants are roster-wide moments. Both affect the reading of the validation sample values, not the population statements. (A1) Roster rule (code fact). Every roster stay is at risk at origin L: target k is measured at hour L and not past its threshold there, no earlier measured value crossed the threshold, and L precedes the terminal (discharge or death) hour; in addition its label window is resolved. Hence R ⊆ {Tev > L} and R ⊆ {at risk at L}. (terminal_semantics.py L409–L414, L755–L835, L958–L975, L1073–L1079; revision_v3_state_sync_producer_v2.py L187–L191.) (A2) Input structure (code fact). The IC-type transforms replace hours 7 to L by fixed functions of hour 6 and constants and leave statics, domain and hours 1 to 6 unchanged; SH and SH-A replace them by functions of the donor’s hours 7 to L and the recipient’s hour 6; the donor-only scores ′ ; S1 is a function of the hour-6 prefix read, clipped to [0, 1]; BR, BR2 are gb ◦ are functions of ωL S1, gb′ ◦ S2. (revision_v3_state_sync_isolation_v1.py L526–L638, L983–L1045, L1118–L1150; revision_v3_state_sync_cells_v2.py L172–L194.) ′ ) case of (d3)). Under PR , (D) Donor independence (used in (c), for ASH ∪ Adon in (d), and for the F6 ∨ σ(ωL ′ ′ | F6 . the donor record ωL is conditionally independent of (Y, X7:L ) given F6 ; the proofs use only Y ⊥ ωL ′ By Lemma 4 of the proof, (D) holds whenever ωL = r(ω6 , Πpool ) for a fixed measurable map r and a donor pool Πpool that is independent of the recipient’s (ωL , Y ) under PR . The construction has this form if persons are independent, the seed’s encoder and the standardization constants are fixed (A0), and Πpool is the list of roster stays of persons other than the recipient’s, with their records: match_donors uses hour-6 encoder states, person identifiers and record lengths, no label enters it, and it never returns a stay of the recipient’s person; the pool is the evaluable roster, whose membership depends only on each candidate’s own record. Two departures make (D) an assumption that the construction supports rather than a code fact: the standardization constants are roster-wide means and standard deviations (standardize_states, L681–L690), to which the recipient’s person contributes with weight of order 1/|R| (|R| between 551 and 7,707), and the checkpoints that define the encoder state were selected with validation data (A0). ′ (EM) Exact matching (idealization; used only in (c)(iv)). Under PR , (ω6 , X7:L ) has the same law as (ω6 , X7:L ). The implementation takes the nearest different-person neighbor in the standardized 64-dimensional hour6 state from a finite pool; (EM) would follow from exact state matches together with sufficiency of that state for the law of hours 7 to L, neither of which holds exactly.

(RI) Targets of the learned reads (interpretive; used in no proof). Let R6 := R ∩ {target k measured at hour 6}, and call a head population-optimal when it maximizes the population training likelihood over measurable functions of the prefix. (RI) states that the population-optimal hour-L read equals S2∗ almost surely on R and the population-optimal hour-6 ratio read equals S1∗ almost surely on R6 . It holds, for example, if (i) a label Ỹ and a first-event time are defined for every stay at risk at the origin in question, for instance from the gap-free hourly path, with Ỹ = Y on R; (ii) training draws persons from the same population and censoring by measurement gaps is noninformative, so that the population-optimal head at origin t returns the conditional law of the event process given ωt among stays at risk at t; and (iii) Ỹ ⊥ 1R | FL among stays at risk at L, and Ỹ ⊥ 1R | F6 among stays at risk at 6 with Tev > L. It is an assumption because the roster requires the label window to be resolved. On R \ R6 origin 6 is not a supervised origin type, the population optimum does not determine the hour-6 read, and (RI) says nothing there. (RI) serves only to read δ2 , and the part of δS1 that arises on R6 , as model error of the heads rather than as a mismatch between their targets and S1∗ , S2∗ . 25

Proof. Conventions. All probabilities and expectations are under PR . For y ∈ {0, 1} and a bounded or nonnegative random variable Z, Ey Z := ER [Z1{Y = y}]/PR (Y = y). If Z is bounded, or [0, ∞]-valued, and G-measurable, the tower property (with monotone convergence in the second case and 0 · ∞ := 0) gives ER [Z1{Y = 1}] = ER [Z pG ],

ER [Z1{Y = 0}] = ER [Z(1 − pG )].

(1)

For σ-fields H ⊆ G the tower property also gives pH = ER [pG | H] almost surely. A test function is a [0, 1]-valued random variable φ, the probability of calling the stay positive; its false-positive rate is E0 φ and its true-positive rate E1 φ. Write h(p) := −p log p − (1 − p) log(1 − p) and KL(p∥q) := p log(p/q) + (1 − p) log((1 − p)/(1 − q)) for the Bernoulli divergence, with 0 log(0/x) := 0 for x ≥ 0 and a log(a/0) := ∞ for a > 0. With these conventions, for all p, q ∈ [0, 1], −p log q − (1 − p) log(1 − q) = h(p) + KL(p∥q) ∈ [0, ∞], (2) 0 ≤ h(p) ≤ log 2, and KL(p∥q) ≥ 0 with equality if and only if p = q (Gibbs’ inequality). Lemma 1 (randomized threshold tests). Let s be a score with conditional laws µy := PR (s ∈ · | Y = y), let U ∼ Unif(0, 1) be independent of (Y, s) (on a product extension of the probability space), and set Vs := µ0 ((s, ∞)) + U µ0 ({s}). For α ∈ (0, 1) let cα := inf{x ∈ R : µ0 ((x, ∞)) ≤ α}, and let γα := (α − µ0 ((cα , ∞)))/µ0 ({cα }) if µ0 ({cα }) > 0 and γα := 1 otherwise. Then: (i) cα ∈ R and µ0 ((cα , ∞)) ≤ α ≤ µ0 ([cα , ∞)), so γα ∈ [0, 1]; (ii) almost surely 1{Vs ≤ α} = 1{s > cα } + 1{s = cα }1{U ≤ γα }; (iii) given Y = 0, Vs is uniform on [0, 1]; (iv) φsα := 1{s > cα } + γα 1{s = cα } = P(Vs ≤ α | s, Y ) is a σ(s)-measurable test function with E0 φsα = α and E1 φsα = ROCs (α), where we define ROCs (α) := PR (Vs ≤ α | Y = 1) for α ∈ [0, 1]; R1 (v) 0 ROCs (α) dα = AUROC(s). By (ii) and (iv), ROCs (α) is the true-positive rate at exact false-positive rate α of the upper-threshold test of s with randomization at the threshold; as α runs through [µ0 ((c, ∞)), µ0 ([c, ∞))] for a tied value c, the rejected fraction of {s = c} grows linearly, which is the linear interpolation across ties that corresponds to half credit for ties. This is the ROC curve used in the statement. Proof of Lemma 1. (i) G0 (x) := µ0 ((x, ∞)) is nonincreasing and right-continuous with limits 1 at −∞ and 0 at +∞. As 0 < α < 1, the set {x : G0 (x) ≤ α} is nonempty and bounded below, so cα is finite; right-continuity gives G0 (cα ) ≤ α, and G0 (x) > α for x < cα gives µ0 ([cα , ∞)) = limx↑cα G0 (x) ≥ α. (ii) If s > cα , then Vs ≤ µ0 ([s, ∞)) ≤ µ0 ((cα , ∞)) ≤ α. If s < cα , then Vs ≥ G0 (s) > α. If s = cα , then Vs ≤ α if and only if U µ0 ({cα }) ≤ α − G0 (cα ); when µ0 ({cα }) > 0 this is U ≤ γα , and when µ0 ({cα }) = 0, (i) gives G0 (cα ) = α, so Vs = α and the event holds, as γα = 1 prescribes. (iii) By (ii) and the independence of U , PR (Vs ≤ α | Y = 0) = µ0 ((cα , ∞)) + γα µ0 ({cα }) = α for α ∈ (0, 1), in both cases of the definition of γα . Since 0 ≤ Vs ≤ µ0 ([s, ∞)) ≤ 1, Vs given Y = 0 is uniform on [0, 1]. (iv) Immediate from (ii), because U is independent of (s, Y ); then Ey φsα = PR (Vs ≤ α | Y = y), which equals α for y = 0 by (iii). (v) By Tonelli, R1 R1 P (V ≤ α | Y = 1) dα = E1 0 1{Vs ≤ α} dαR = 1 − E1 Vs . Given Y = 1, (s, U ) has law µ1 ⊗ Unif(0, 1), so 0 R s with s0 ∼ µ0 and s1 ∼ µ1 independent, E1 Vs = [µ0 ((x, ∞)) + 21 µ0 ({x})] µ1 (dx) = P(s0 > s1 ) + 12 P(s0 = s1 ). Hence the area is 1 − P(s0 > s1 ) − 12 P(s0 = s1 ) = P(s1 > s0 ) + 21 P(s1 = s0 ) = AUROC(s). □ Lemma 2 (Neyman–Pearson on a sub-σ-field). Let G be a σ-field, t ∈ [0, 1), and φ∗ a G-measurable test function with φ∗ = 1 on {pG > t}, φ∗ = 0 on {pG < t} and E0 φ∗ = α. Then E1 φ ≤ E1 φ∗ for every G-measurable test function φ with E0 φ ≤ α. Proof. Pointwise (φ∗ − φ)(pG − t) ≥ 0, and pG − t = (1 − t)pG − t(1 − pG ). Taking expectations and applying (1) to the bounded G-measurable Z = φ∗ − φ, 0 ≤ (1 − t) ER [(φ∗ − φ)pG ] − t ER [(φ∗ − φ)(1 − pG )] = (1 − t)π(E1 φ∗ − E1 φ) − t(1 − π)(E0 φ∗ − E0 φ). The last term is ≤ 0 because t ≥ 0 and E0 φ∗ = α ≥ E0 φ. Hence (1 − t)π(E1 φ∗ − E1 φ) ≥ 0, and (1 − t)π > 0. □ Lemma 3 (conditionally irrelevant information). Let G, H be σ-fields with Y ⊥ H | G under PR . Then pG∨H = pG almost surely. This applies in particular to H = σ(E) for an event E with PR (E) = 1, since a PR -trivial σ-field is independent of every random variable. Proof. For A ∈ G and B ∈ H, conditional independence gives ER [Y 1B | G] = pG PR (B | G), so ER [Y 1A∩B ] = ER [1A pG PR (B | G)] = ER [1A pG 1B ], the last step by the tower property since 1A pG is Gmeasurable. The sets A ∩ B form a π-system generating G ∨ H, and the class of C ∈ G ∨ H with ER [Y 1C ] = ER [pG 1C ] is a λ-system (it contains Ω and is closed under proper differences and increasing 26

unions, by linearity and monotone convergence). By Dynkin’s π–λ theorem it is all of G ∨ H. As pG is G ∨ Hmeasurable, it is a version of pG∨H . □ Lemma 4 (a donor drawn from an independent pool). Let Πpool be a random element of a measurable ′ space that is independent of (ωL , Y ) under PR , and let ωL = r(ω6 , Πpool ) for a measurable map r. Then ′ (Y, X7:L ) ⊥ ωL | F6 under PR ; that is, (D) holds. Proof. Write Ξ := (Y, X7:L ), so that (ω6 , Ξ) is a function of (ωL , Y ) and Πpool is independent of (ω6 , Ξ). Let f, g, k be bounded measurable functions and put Γ(u) := ER g(r(u, Πpool )), which is measurable in u by Fubini’s theorem. Because the joint law of ((ω6 , Ξ), Πpool ) is the product of its marginals, Fubini’s theorem gives       ER k(ω6 )f (Ξ)g(r(ω6 , Πpool )) = ER k(ω6 )f (Ξ)Γ(ω6 ) = ER k(ω6 )Γ(ω6 ) ER [f (Ξ) | F6 ] . ′ As k is arbitrary and F6 = σ(ω6 ), ER [f (Ξ)g(ωL ) | F6 ] = Γ(ω6 ) ER [f (Ξ) | F6 ] almost surely. With f ≡ 1 this ′ ′ ′ gives ER [g(ωL ) | F6 ] = Γ(ω6 ), hence ER [f (Ξ)g(ωL ) | F6 ] = ER [f (Ξ) | F6 ] ER [g(ωL ) | F6 ]. Taking f, g to be indicators gives the conditional independence. □

Proof of (a). ROC dominance. Let s be G-measurable and write p := pG , a G-measurable score with values in [0, 1]; let µp0 be its law given Y = 0. Case α ∈ (0, 1). By Lemma 1(iv), φsα is σ(s)-measurable, hence Gmeasurable, with E0 φsα = α and E1 φsα = ROCs (α). Applied to p, Lemma 1 gives φpα = 1{p > c} + γ1{p = c} with c := cα (p) and E0 φpα = α. We have c ∈ [0, 1): for x < 0, µp0 ((x, ∞)) = 1 > α, so c ≥ 0; and by (1), µp0 ({1}) = ER [(1 − p)1{p = 1}]/(1 − π) = 0, so µp0 ([1, ∞)) = 0 < α ≤ µp0 ([c, ∞)) (Lemma 1(i)), which forces c < 1. Thus φpα satisfies the hypotheses of Lemma 2 with t = c, and ROCs (α) = E1 φsα ≤ E1 φpα = ROCp (α). Case α = 1. V ≤ 1 always, so ROCs (1) = ROCp (1) = 1. Case α = 0. Since U > 0 almost surely, P(Vs ≤ 0 | s, Y ) = 1{µ0 ([s, ∞)) = 0} =: φs0 , a G-measurable test function with E0 φs0 = 0 by Lemma 1(iii). Any G-measurable test function φ with E0 φ = 0 has ER [φ(1 − p)] = 0 by (1), so φ = 0 almost surely on {p < 1} and E1 φ = ER [φp]/π ≤ ER [p1{p = 1}]/π = PR (p = 1 | Y = 1). For s = p, µp0 ([1, ∞)) = 0 shows {p = 1} ⊆ {µp0 ([p, ∞)) = 0}, hence ROCp (0) ≥ PR (p = 1 | Y = 1) ≥ ROCs (0). Integrating over α and using Lemma 1(v) gives AUROC(s) ≤ AUROC(pG ). Brier score. For a G-measurable forecast q, ER (Y −q)2 = ER (Y −pG )2 +ER (pG −q)2 +2ER [(Y −pG )(pG −q)], and the cross term vanishes by (1) with the bounded G-measurable Z = pG − q. Log-loss. The variables − log q and − log(1 − q) are G-measurable with values in [0, ∞], so by (1) and (2)   LogLoss(q) = ER −pG log q − (1 − pG ) log(1 − q) = ER h(pG ) + ER KL(pG ∥q). With q = pG this gives LogLoss(pG ) = ER h(pG ) ≤ log 2 < ∞, and subtracting gives the stated identity in [0, ∞]. Nested information sets. If F ⊆ G, then q = pF is G-measurable, so Brier(pF ) − Brier(pG ) = ER (pG − pF )2 and LogLoss(pF ) − LogLoss(pG ) = ER KL(pG ∥pF ) = MI(Y ; G | F). The log-loss identity applied to F with q = pF gives LogLoss(pF ) = ER h(pF ), so MI(Y ; G | F) = ER h(pF ) − ER h(pG ) ∈ [0, log 2]. Both differences are expectations of nonnegative quantities that vanish exactly where pG = pF (Gibbs’ inequality for the divergence), so each is zero if and only if pF = pG almost surely. This proves (a). Proof of (b). Step 1. By (A1), R ⊆ {Tev > L}, so PR (Tev > L) = 1 and Lemma 3 gives S1∗ = pF6 ∨σ({Tev >L}) = pF6 almost surely. Step 2 (information sets). For S1 and BR the input is ι = ω6 , so σ(ι) = F6 . For a ∈ {IC, IC-A, IC-O}, (A2) gives ιa = (ω6 , κa (ω6 )) with κa a fixed measurable map (the transforms listed in the Setting; the occurrence value 1 written by κIC-O is a constant). Then σ(ιa ) ⊆ F6 because ιa is a measurable function of ω6 , and F6 ⊆ σ(ιa ) because ω6 is a coordinate of ιa ; so σ(ιa ) = F6 . The conditioning event Ea is {Tev > L} for S1 and BR and the recipient being at risk at L for the reads at origin L; by (A1) both contain R, so PR (Ea ) = 1 and Lemma 3 gives pGa = pσ(ιa ) = pF6 = S1∗ . For S2 and BR2, σ(ι) = σ(ωL ) = FL and the same argument gives pG = S2∗ . Step 3. Apply (a) with F = F6 ⊆ G = FL and use S1∗ = pF6 : this gives the log-loss and Brier identities, the pointwise ROC dominance of S2∗ over the F6 -measurable (hence FL -measurable) score S1∗ , and the AUROC inequality. By (a) the two proper-score differences vanish if and only if pFL = pF6 almost surely, that is PR (Y = 1 | F6 , X7:L ) = PR (Y = 1 | F6 ), which for binary Y is Y ⊥ X7:L | F6 . In that case S2∗ = S1∗ . Both S1∗ and S2∗ are forecasts of the same event {Y = 1} under the same law PR , so every difference in any metric is a functional of the joint law of (Y, F6 , X7:L ) under PR alone; the forecast origin enters only through the information set. For the implemented S1 this matching of events is what the design L = 6 + H secures: from origin 6 the window (L, L + H] is the bin interval (H, 2H], and dividing by the all-cause event-free probability S6 (H) conditions on {Tev > L}. ′ ′ Proof of (c). Put GD := F6 ∨ σ(ωL ). (i) By (D), Y ⊥ σ(ωL ) | F6 under PR , so Lemma 3 and Step 1 ∗ ′ of (b) give pGD = pF6 = S1 . Each arm in ASH ∪ Adon is GD -measurable: SH = ψ(ω6 , X7:L ); SH-A = ′ ′ ψ(ω6 , κSH-A (X6 , X7:L )) with X6 a coordinate of ω6 ; the donor’s own forecast ψ(ωL ), its last value V ′ and its ′ oriented count of measured hours are functions of ωL (the orientation sign is fixed under (A0)). Part (i) is

27

then (a) with G = GD , whose ideal forecast is S1∗ . (ii) This is the Brier identity of (a) for G = GD . (iii) For a ∈ ASH the conditioning event (the recipient at risk at L) has PR -probability one by (A1), so Lemma 3 gives ′ ′ pGa = pσ(ιa ) . The input ιa is (ω6 , X7:L ) for SH and (ω6 , κSH-A (X6 , X7:L )) for SH-A. In both cases ω6 is a ′ coordinate of ιa , so F6 ⊆ σ(ιa ), and ιa is a measurable function of (ω6 , ωL ), so σ(ιa ) ⊆ GD . By the tower ∗ ∗ property and (i), pσ(ιa ) = ER [pGD | ιa ] = ER [S1 | ιa ] = S1 , the last equality because S1∗ = pF6 is F6 ′ measurable and F6 ⊆ σ(ιa ). For a donor-only score, ιa = ωL and Ea is either the sure event or the donor being at risk at L, which has PR -probability one because donors are roster stays. Lemma 3 gives pGa = pσ(ωL′ ) , ′ ′ and since σ(ωL ) ⊆ GD the tower property and (i) give pσ(ωL′ ) = ER [S1∗ | ωL ] = S1∗don . Now apply (a) with ′ ∗ ∗ F = σ(ωL ) ⊆ G = GD , for which pF = S1don and pG = S1 : this gives AUROC(S1∗don ) ≤ AUROC(S1∗ ) (S1∗don ′ is GD -measurable), the Brier gap ER (S1∗ − S1∗don )2 , the log-loss gap MI(Y ; GD | σ(ωL )), and that both gaps ∗ ∗ vanish if and only if S1 = S1don almost surely. The last condition holds if and only if S1∗ is almost surely ′ equal to a σ(ωL )-measurable variable f : if it holds, take f = S1∗don ; conversely, if S1∗ = f almost surely then ∗ ′ ′ S1don = ER [f | ωL ] = f = S1∗ almost surely. By Doob–Dynkin, σ(ωL )-measurable variables are the mea′ ∗ surable functions of ωL . (iv) Under (EM), write S1 = f1 (ω6 ) with f1 measurable (Doob–Dynkin). Because ′ (ω6 , X7:L ) and (ω6 , X7:L ) have the same law and (u, v) 7→ (ψ(u, v) − f1 (u))2 is measurable,
2
2 ′ ER (SH − S1∗ )2 = ER ψ(ω6 , X7:L ) − f1 (ω6 ) = ER ψ(ω6 , X7:L ) − f1 (ω6 ) = ER (S2 − S1∗ )2 , and (ii) with q = SH, a GD -measurable forecast, gives Brier(SH) − Brier(S1∗ ) = ER (S2 − S1∗ )2 . For the ideal map take ψ to be a version of pFL as a function of ωL ; two versions agree almost everywhere for the ′ law of ωL , which under (EM) is also the law of (ω6 , X7:L ), so SH∗ does not depend on the version. Then ER (S2∗ −S1∗ )2 = Brier(S1∗ )−Brier(S2∗ ) by (b), which gives Brier(SH∗ )−Brier(S1∗ ) = Brier(S1∗ )−Brier(S2∗ ) and hence Brier(SH∗ ) − Brier(S2∗ ) = 2[Brier(S1∗ ) − Brier(S2∗ )]. Proof of (d). Identity. Add and subtract M (S1). For seed averages the identity holds because averaging is linear; in a bootstrap replicate that gives every arm the same stay weights (the whole-person bootstrap of the isolation run shares its weights across arms and seeds) each M is computed on the same weighted sample, so the identity holds replicate by replicate. (The two-way bootstrap of donor contrasts uses different weights, so for those contrasts the identity concerns the point estimates.) The remaining claims concern the population metrics under PR ; for sample metrics the signs below can fail through sampling error. (d1). I ∗ ≥ 0 by (b). S2 = ψ(ωL ) is FL -measurable by (A0), so (a) with G = FL gives AUROC(S2) ≤ AUROC(S2∗ ), Brier(S2) ≥ Brier(S2∗ ) and LogLoss(S2) ≥ LogLoss(S2∗ ), that is δ2 ≥ 0 for each of the three metrics. Each a ∈ A6 is F6 -measurable by Step 2 of (b), and pF6 = S1∗ by Step 1, so (a) with G = F6 gives δa ≥ 0. Under (D) each a ∈ ASH ∪ Adon is GD -measurable with pGD = S1∗ by (c)(i), so (a) with G = GD gives δa ≥ 0. By definition M (S2) = M (S2∗ ) − δ2 and M (a) = M (S1∗ ) − δa , so M (S2) − M (S1) = I ∗ − δ2 + δS1 and M (S1) − M (a) = δa − δS1 ; more generally M (S2) − M (a) = I ∗ − δ2 + δa . (d2). The information-set value of S1 is S1∗ by (b). For a ∈ A6 it is S1∗ by (b), and for a ∈ ASH it is ∗ S1 by (c)(iii). Replacing S1 and a by their information-set values therefore turns the second bracket into M (S1∗ ) − M (S1∗ ) = 0. For a donor-only score a the information-set value is S1∗don by (c)(iii), so the same replacement turns the second bracket into M (S1∗ ) − M (S1∗don ) = ε, and ε ≥ 0 by (c)(iii). The split δa = ε + δa′ ′ ′ is algebra. As a is σ(ωL )-measurable (A2) and pσ(ωL′ ) = S1∗don , (a) with G = σ(ωL ) gives δa′ ≥ 0 (for the proper scores a is the donor’s own forecast). For the proper scores ε is the Brier or log-loss gap of (c)(iii), which ′ . vanishes if and only if S1∗ is almost surely a function of ωL (d3). If b is F6 -measurable, (a) with G = F6 gives M (b) ≤ M (pF6 ) = M (S1∗ ); if b is GD -measurable and (D) holds, (a) with G = GD gives the same through (c)(i). Hence δS1 = M (S1∗ ) − M (S1) ≥ M (b) − M (S1) and M (S2) − M (S1∗ ) ≤ M (S2) − M (b). If b is FL -measurable, (a) with G = FL gives M (b) ≤ M (S2∗ ), so δ2 ≥ M (b) − M (S2). Every a ∈ A is a score of the first kind by (d1), so for nonempty B ⊆ A one has maxa∈B M (a) ≤ M (S1∗ ), which is the displayed bound; the smallest of the bounds M (S2) − M (a), a ∈ B, is the one attained at the maximizing arm. (d4). By (d1), M (S2) − M (a) = I ∗ − δ2 + δa , which is ≤ I ∗ if and only if δa ≤ δ2 . With a = S1 this is the stated condition. (d5). Let g be nondecreasing and injective on a set A that carries the values of s (with probability one for the population metric; the observed values for the sample metric, whose AUROC is that of the empirical law). For s0 , s1 with values in A, {g(s1 ) > g(s0 )} = {s1 > s0 } and {g(s1 ) = g(s0 )} = {s1 = s0 }, so AUROC(g ◦ s) = AUROC(s). The map gb is the composition of p 7→ min{max{p, ϵ}, 1 − ϵ}, which is nondecreasing, constant on [0, ϵ] and on [1 − ϵ, 1] and strictly increasing on [ϵ, 1 − ϵ], with the strictly increasing maps logit, z 7→ z + b and expit. It is therefore nondecreasing, and injective on a set A ⊆ [0, 1] if and only if A has at most one point in [0, ϵ] and at most one in [1 − ϵ, 1]. The same holds for gb′ and S2. This completes the proof. □ Reading of part (d) for the implemented arms. The following facts about the code locate the shortfalls δa ; they are descriptions and are not used in the proof. The heads are supervised only at at-risk origins, where 28

the target is measured (A1). IC and IC-A present the synchronized map with target occurrence 0 at hour L, an origin type never supervised; IC also sets every non-target availability indicator to 0 on hours 7 to L, although that indicator records availability after forward filling and the isolation module itself treats a fall from 1 to 0 as atypical of real records. IC-O is read on the supervised origin type, but its input asserts that the target was measured at L with its hour-6 value and that nothing else was newly measured. The ideal synchronized map would return a version of pFL evaluated at that input; such inputs may lie outside the support of ωL , where the value is not determined by the law of the data, and in any case it need not equal S1∗ . For SH and SH-A, (c)(iv) shows that under (EM) a loss relative to S1∗ arises even for an ideal forecaster on spliced records that are distributed exactly as real records: it is the cost of treating another patient’s hours as the patient’s own, and it does not require the spliced input to be unusual. Nearest-neighbor matching and the junction at hour 7 can add a distributional component for the learned model. The hour-6 read S1 is taken for every roster stay, including stays whose target is not measured at hour 6, for which origin 6 is not a supervised type; this part of δS1 has no model-error reading under (RI). For the IC-type arms and SH, SH-A, S1 − a compares two uses of the same information set; for the donor-only scores it also contains ε, the loss from not reading the recipient’s own statics and hours 1 to 6. Finally, (d3) gives lower bounds on δS1 and δ2 from any further hour-6 or hour-L score, but none of the arms bounds either shortfall from above, so the arms do not decide whether δS1 ≤ δ2 . Numerical check. Table S11 re-reads the validation isolation run (job 13007). The identity of (d) holds with residual 0 in all 12 settings. S1 and BR, whose AUROCs are equal by construction, are the best of the ten hour-6 and donor arms (S1, BR, IC, IC-A, IC-O, SH, SH-A and the three donor-only scores) in 12 of 12 settings, by 0.001 to 0.034 over the next arm, so S2 − S1 (+0.066 to +0.126) is also the smallest of the upper bounds of (d3) that these arms give; |AUROC(BR) − AUROC(S1)| = |AUROC(BR2) − AUROC(S2)| = 0 in all 60 seed-by-setting cells. S1 − SH is positive in 12 of 12 settings and in 59 of 60 seed-by-setting cells; it belongs to no registered Holm family, so these are point estimates, and since (c) orders SH only against S1∗ , this ordering of two learned arms is consistent with (c) without being implied by it. The donor measured-hours score has AUROC 0.495 to 0.517; for the donor-only scores S1 − a contains the information term ε of (d2), which the arms do not separate from δa′ . On the cross-fitted (isotonic) scale IC is worse than S1 on the Brier score in 12 of 12 settings (Holm-significant in 5) and on log-loss in 10 of 12 (Holm-significant in 2, both in that direction); the shipped log-loss clips probabilities to [10−3 , 1 − 10−3 ], so the identity with MI in (a) does not apply to these values and only their signs are read. Post hoc and unregistered (a new analysis of the per-stay predictions and labels of the previously read test panel, disclosed in Supplementary Note 2 and the test-read registry; no new predictions were made): on the test panel of jobs 10707/10708, the model’s hour-6 cumulative incidence F6 (H), which is F6 -measurable and forecasts the earlier window (6, L], exceeds S1 in AUROC at arterial oxygen saturation (30, 24) by 0.021 (95% whole-person bootstrap interval 0.004 to 0.039) and at pulse-oximetry saturation (30, 24) by 0.050 (0.010 to 0.089); it exceeds S1 in one further setting, by 0.002, and falls below it in the other nine. Neither of the two larger excesses survives Holm correction over the 12 settings (adjusted one-sided P = 0.066 and 0.1265). By (d3) these are estimates of lower bounds on δS1 , equal to 16% and 33% of S2 − S1 in those two settings, so the condition δS1 ≤ δ2 of (d4) cannot be taken for granted there. S2 − max(S1, F6 (H)) is +0.073 to +0.121 and positive in 12 of 12 settings.

5.2 The difference-in-differences and current-value hypotheses Proposition 2 (The four AUROCs of the difference-in-differences do not identify the current-value hypothesis). Standard and specific parts. Lemmas 0 to 3 of the proof are elementary Gaussian facts. The statements (a) to (c) concern the difference-in-differences of this evaluation and are specific to it. Setting. Fix a target k and a setting (L, H). All laws are conditional on the roster event R, and π := P (Y = 1 | R) ∈ (0, 1). A record r = (r1:6 , r7:L ) is an input that the frozen forecaster M reads at origin L; r1:6 denotes its statics, domain indicator and hours 1 to 6, r7:L its hours 7 to L (values and masks), and v(r) its direction-signed target value at hour L. For a roster stay with record r let V := v(r). Its donor is a roster stay of another person whose target is measured at hour L, and V ′ is the donor’s value there. The spliced record rSH keeps r1:6 and takes hours 7 to L from the donor, so v(rSH ) = V ′ ; the own-value splice rOV is rSH with its hour-L target value set to V (both targets are measured at L, so the masks agree). The arms are S2 := M(r), SH := M(rSH ), OV := M(rOV ), LVown := V and LVdonor := V ′ , each scored against the recipient’s label Y by the AUROC with half credit for ties; OV is proposed here and was not run. The analysis protocol fixes in advance the following descriptive contrast, Holm-corrected within the 12 settings; by the protocol’s own rules it cannot change a decision outcome and acts only on the choice of reported wording: DiDAUROC := (AUROCS2 − AUROCSH ) − (AUROCown − AUROCdonor ). Latent-threshold Gaussian model (LT). There are random vectors w ∈ Rp (statics and hours 1 to 6) and x ∈ Rq (hours 7 to L) with Cov((w, x)) nonsingular, F6 = σ(w), FL = σ(w, x) and V = x1 , and a latent index η (a 29

Target SaO2 SaO2 SaO2 SpO2 SpO2 SpO2 Heart rate Heart rate Heart rate Respiratory rate Respiratory rate Respiratory rate

(L, H) (12, 6) (18, 12) (30, 24) (12, 6) (18, 12) (30, 24) (12, 6) (18, 12) (30, 24) (12, 6) (18, 12) (30, 24)

n 6206 3579 1650 2582 1395 551 7707 4799 2229 7127 4301 1899

S2 0.739 0.733 0.694 0.730 0.735 0.693 0.806 0.787 0.724 0.735 0.745 0.694

S1 0.661 0.641 0.580 0.651 0.669 0.576 0.694 0.660 0.616 0.650 0.635 0.590

S2−S1 0.077 0.092 0.114 0.079 0.066 0.117 0.112 0.126 0.108 0.085 0.110 0.104

S1−IC 0.010∗ 0.008 0.010 0.010 0.013 0.025 0.006∗ 0.006 0.010 0.021∗ 0.036∗ 0.027∗

S1−IC-A 0.013 0.011 0.019 0.028 0.038 0.033 0.005 0.001 0.009 0.019 0.034 0.029

S1−IC-O 0.012 0.008 0.015 0.019 0.023 0.024 0.005 0.001 0.009 0.019 0.034 0.029

S1−SH 0.044 0.045 0.036 0.049 0.045 0.014 0.058 0.058 0.052 0.046 0.044 0.039

S2−IC 0.088 0.100 0.124 0.090 0.080 0.142 0.118 0.133 0.118 0.106 0.146 0.131

Supplementary Table S11: Isolation arms on the validation split (Proposition 1). Job 13007 (source_tables/revision_v3/state_sync_isolation_13007_v1/score/isolation_metrics.csv); raw scale, AUROC averaged over the five per-seed values; n is the roster size. All contrasts are positive in all 12 settings. Entries are rounded independently, so a row need not add exactly; unrounded, S2 − IC = (S2 − S1) + (S1 − IC) with residual 0. S2 − S1 is Holm-significant in 12 of 12 settings (a replication family of the protocol, not independent of the test-set result; correction within its 12 settings); an asterisk marks the 5 settings in which IC − S1 is Holm-significantly negative in the primary family of 36 contrasts. S1 − SH is not a registered contrast and is reported as a point estimate. By Proposition 1(d), S2 − S1 is the learned contrast I ∗ − δ2 + δS1 and S1 − IC = δIC − δS1 compares two reads of the same information. These are sample values on a split used for checkpoint selection; the proposition concerns population quantities.

symbol of this section only), such that (w, x, η) is jointly Gaussian, Eη = 0, Var η = 1, Var(η | w, x) > 0 and Y = 1{η > tπ } with tπ := Φ−1 (1 − π). A record is identified with a point (w, x) of Rp+q , with r1:6 = w, r7:L = x and v(r) = x1 . Aπ (ρ) is the AUROC for Y of any strictly increasing function of a variable J such that (J, η) is bivariate Gaussian with Var J > 0 and corr(J, η) = ρ (Gaussianity of J alone is not enough); Aπ is a continuous strictly increasing bijection of [−1, 1] onto [0, 1] (Lemma 1). The latent correlation of arm a is ρa := A−1 π (AUROCa ); write ρown , ρdonor for the two last-value arms, ∆LV := ρown − ρdonor and DiDρ := (ρS2 − ρSH ) − ∆LV . Put mV := E[V | w], eV := V − mV , mη := E[η | w], θV := Cov(η, eV )/ Var(eV ) and ι2 := Var(eV )/ Var(V ). Let µ be the affine function on Rp+q with µ(w, x) = E[η | w, x] almost surely; it is unique 2 := Var(η | w, x). because Cov((w, x)) is nonsingular. Put ση|L Exact matching (EM). The donor’s hours 7 to L are represented by x′ ∈ Rq with V ′ = x′1 , and conditionally on (w, x, η) the law of x′ is the conditional law of x given w. The spliced record is then (w, x′ ), and the own-value record is (w, x′′ ) with x′′1 := x1 and x′′j := x′j for j ≥ 2. Hypotheses and forecasters. M obeys the current-value restriction (CVR) if M(r) = g(r1:6 , v(r)) for some function g and every record r in a domain that contains the natural, spliced and own-value records. In LT, M obeys linear CVR if moreover g(r1:6 , v) = Ψ(U6 + βV v) with Ψ strictly increasing, U6 an affine function of w and βV > 0; then I := U6 + βV V , ISH := U6 + βV V ′ and wV := βV SD(V )/ SD(I). The data obey the current-state hypothesis HCS if Y ⊥ x | (w, V ), and current-value sufficiency (CVS) if Y ⊥ (w, x) | V . The ideal forecaster is M∗ (r) := Φ{(µ(r) − tπ )/ση|L }; it is defined at every record and is a version of P (Y = 1 | FL , R). S1∗ := P (Y = 1 | w, R) is the ideal hour-6 forecast and CS∗ := P (Y = 1 | w, V, R) the ideal currentstate forecast, with latent correlations ρS1∗ and ρCS∗ . By Definition 2, AUROC(M∗ ) − AUROC(CS∗ ) is the personalization beyond the current state. Parts (a)(i) to (a)(iv) and (b) assume LT and ∆LV > 0. Part (a)(v), except its last sentence, and part (c) need neither. (a) The null value of the difference-in-differences. (i) Under EM, if M obeys linear CVR, then ρS2 − ρSH = wV ∆LV , so DiDρ = (wV − 1)∆LV . DiDρ is zero if and only if Var(U6 ) + 2βV Cov(U6 , V ) = 0, negative if and only if this quantity is positive, and positive if and only if it is negative, which requires Cov(U6 , V ) < 0. (ii) Under EM and HCS , the ideal forecaster obeys linear CVR at every record, with U6 = mη − θV mV and βV = θV > 0. Its DiDρ is zero if and only if Var(mη ) = θV2 Var(mV ), and negative if and only if Var(mη ) > θV2 Var(mV ). (iii) Any donor assignment. Without EM, if M obeys linear CVR, the indices I, ISH , V, V ′ have positive variances, and (ISH , η) and (V ′ , η) are bivariate Gaussian, then ρS2 − sSH ρSH = wV (ρown − sLV ρdonor ),

sSH :=

SD(ISH ) , SD(I)

sLV :=

SD(V ′ ) , SD(V )

so that DiDρ = (wV − 1)∆LV + (sSH − 1)ρSH − wV (sLV − 1)ρdonor . Under EM, sSH = sLV = 1. If instead, conditionally on (w, x, η), x′ has the conditional law of x given z(w) for a linear map z (exact matching on a linear 30

summary of the first six hours, a Gaussian stand-in for the frozen hour-6 state z6 on which donors are matched), then (ISH , η) and (V ′ , η) are bivariate Gaussian, sLV = 1, and s2SH = 1+2βV Cov(U6 , E[V | z(w)]−mV )/ Var(I), which equals 1 if and only if Cov(U6 , E[V | z(w)] − mV ) = 0. (iv) Scale. DiDAUROC = Aπ (ρS2 )−Aπ (ρSH )−Aπ (ρown )+Aπ (ρdonor ), and DiDρ = 0 does not imply DiDAUROC = 0: the proof exhibits an LT model with EM and HCS whose ideal forecaster has DiDρ = 0 and DiDAUROC = +0.0023 at π = 0.25. (v) Current-value sufficiency. Suppose CVS holds and Ψ0 (v) := P (Y = 1 | V = v, R) is strictly increasing. Then r 7→ Ψ0 (v(r)) is a version of the ideal forecaster, and for it AUROCS2 = AUROCown and AUROCSH = AUROCdonor exactly; hence ϕ(AUROCS2 ) − ϕ(AUROCSH ) − ϕ(AUROCown ) + ϕ(AUROCdonor ) = 0 for every function ϕ, for every donor assignment, and for the empirical AUROCs on every finite roster. Neither Gaussianity nor matching is used. The same holds for every forecaster r 7→ Ψ̃(v(r)) with Ψ̃ strictly increasing. In LT under CVS (where Var V > 0 because Cov((w, x)) is nonsingular), Ψ0 is strictly increasing if and only if ρown > 0, and under EM this follows from ∆LV > 0. (b) Non-identification. Assume EM and consider the ideal forecaster. Let Q = (ρS2 , ρSH , ρown , ρdonor ) with 0 < ρSH < ρS2 < 1 and ∆LV > 0. Call Q attainable by a class of LT models if some model of the class, with EM, gives Q as the latent correlations of the four arms of its ideal forecaster. (i) In every LT model with EM and Var µ > 0: ρSH ρS2 = Var(mη ) = ρ2S1∗ , hence ρSH ≤ ρS1∗ ≤ ρS2 , and ρS2 (ρS2 − ρSH ) = Var(µ − mη ). (ii) Q is attainable under HCS if and only if (R1)

∆LV ≤ [ρS2 (ρS2 − ρSH )]1/2

and

(R2)

h ρ2donor ≤ ρSH ρS2 1 −

i ∆2LV . ρS2 (ρS2 − ρSH )

(R2 implies R1.) (iii) R1 and R2 hold in every LT model with EM, whether or not HCS holds, and Q is attainable by a model that violates HCS if and only if R1 and R2 hold with strict inequality. (iv) Consequently every attainable Q is attainable under HCS , and every Q that satisfies R1 and R2 strictly is attainable both under HCS and under its failure. For known π the four AUROCs and Q determine each other, so no function of the four AUROCs, including a difference-in-differences on any scale, can reject HCS at an attainable quadruple, or confirm it at a quadruple that satisfies R1 and R2 strictly. The same holds when the AUROC of the ideal hour-6 forecast S1∗ is added, because ρS1∗ = (ρSH ρS2 )1/2 is a function of Q by (i). For the ideal forecaster in LT, CVR holds if and only if HCS holds, so the same is true of CVR; and because the two models that attain such a Q carry one forecaster that obeys CVR and one that does not, the conclusion persists for every class of forecasters that contains the ideal ones. Part (b) does not model the measuredhours arms, the second pre-specified difference-in-differences built from them, the donor’s own forecast, or the learned same-information arms, and says nothing about them. (v) Size of a departure, and a label-free completion. In every LT model with EM whose ideal forecaster has 2 2 and ι2max := 1 − ρ2donor /(ρSH ρS2 ), and := ρS2 (ρS2 − ρSH ), ι20 := ∆2LV /σnew a quadruple Q as above, put σnew 2 let ς := corr (µ − mη , eV ), the share of the new-hour latent variance Var(µ − mη ) carried by the current value (through its part eV not predicted by hours 1 to 6); 1 − ς is the share that lies outside it. Then: (1) ι2 = 1 − corr(V, V ′ ), a quantity computable without labels; (2) ι0 ≤ ι ≤ ιmax and ς = ι20 /ι2 ∈ [ι20 /ι2max , 1]; HCS holds if and only if ς = 1, if and only if corr(V, V ′ ) = 1 − ι20 ; 2 2 2 (3) ρ2CS∗ = ρSH ρS2 + ς σnew , hence ρ2S2 − ρ2CS∗ = (1 − ς)σnew ≤ (1 − ι20 /ι2max )σnew and

2 AUROC(M∗ ) − AUROC(CS∗ ) ≤ Aπ (ρS2 ) − Aπ (ρSH ρS2 + σnew ι20 /ι2max )1/2 ; (4) every ς ∈ [ι20 /ι2max , 1] occurs in an LT model with EM whose ideal forecaster has quadruple Q, so the bounds in (2) and (3) are attained. Within this idealized class the four AUROCs therefore bound the size of a departure from HCS without detecting one, and adding corr(V, V ′ ) identifies HCS . This identification uses EM: (5) if EM is replaced by exact matching on a linear summary z(w) as in (a)(iii), then corr(V, V ′ ) = Var(E[V | z(w)])/ Var(V ) ≤ 1 − ι2 , with equality if and only if E[V | z(w)] = mV almost surely, so 1 − corr(V, V ′ ) overstates ι2 ; the quadruple, and with it ι0 , is changed by the same layer, and the direction of the net error of the completed test is not determined here. (c) A label-free test of the current-value restriction. This part uses neither LT, EM nor labels, except for the AUROCs in (ii). (i) If M obeys CVR, then OV = S2 for every stay, and SH = S2 for every recipient-donor pair with V ′ = V . Conversely, for records r, r̃ let the own-value splice rOV (r̃) be the record whose first part is r1:6 and whose second part is r̃7:L with the hour-L target value replaced by v(r). If N is a set of records whose target is 31

measured at hour L that contains rOV (r̃) for all r, r̃ ∈ N , and M(r) = M(rOV (r̃)) for all r, r̃ ∈ N , then M obeys CVR on N . A single stay with OV ̸= S2, or a single pair with V ′ = V and SH ̸= S2, shows that M does not obey CVR. (ii) Quantitative form. For every forecaster, S2 − SH = (S2 − OV) + (OV − SH), where OV and SH are read from records that differ only in the hour-L target value, and S2 and OV from records that differ only in the other entries of hours 7 to L. The label-free ratio δOV := E[(S2−OV)2 ]/E[(S2−SH)2 ] (defined when the denominator is positive; it is not bounded by 1 in general) is zero under CVR. For every ε > 0, |AUROCS2 − AUROCOV | ≤ P (|S2 − OV| > ε | Y = 1) + P (|S2 − OV| > ε | Y = 0) + P (|S21 − S20 | ≤ 2ε), with S21 ∼ L(S2 | Y = 1) and S20 ∼ L(S2 | Y = 0) independent; the same bound holds for empirical AUROCs with the empirical class-conditional laws. A forecaster with |S2 − OV| ≤ ε for all but a small fraction of the stays of each class therefore has AUROCOV close to AUROCS2 when few pairs of positive and negative stays have S2 within 2ε of each other. Assumptions. (A0) Roster and splice as implemented. All laws are conditional on the roster event R of the setting, with π = P (Y = 1 | R) ∈ (0, 1). Every roster stay and every donor has its target measured at hour L: the roster admits only at-risk origins, which require a measurement at L, donors are drawn from the same roster, and the last-value score refuses a reading that is not taken at L (none of the 40,702 last-value inputs of the test-set last-value comparison, the only split for which per-stay last values are shipped, was taken at another hour). Hence V and V ′ exist and v(rSH ) = V ′ . The splice replaces hours 7 to L (values and masks) and leaves statics, domain and hours 1 to 6 unchanged. The arms are scored on the same recipients and labels, so π is common to them. (A1) Latent-threshold Gaussian model (LT), used in (a)(i) to (a)(iv), the last sentence of (a)(v), and (b): (w, x, η) jointly Gaussian with Cov((w, x)) nonsingular, Var η = 1, Var(η | w, x) > 0, Y = 1{η > tπ }, V = x1 , and each arm a strictly increasing function of a linear index. Nonsingularity makes µ, mV and mη unique affine functions defined at every record, which is what lets the ideal forecaster be evaluated on ownvalue records, whose values need not occur among natural records of a degenerate Gaussian model. LT is an idealization. On the roster V is truncated at the crossing threshold (an at-risk origin is not past the threshold at L and has no earlier observed crossing); some targets take few distinct values, so V has ties (independent-pair tie probability 0.29 to 0.34 for pulse-oximetry saturation, 0.09 to 0.12 for arterial saturation, at most 0.005 for heart and respiratory rate); the real record contains exact dependencies (masks, carried-forward values) that a nonsingular covariance excludes; the label is a first crossing in (L, L + H] that competes with discharge and death; and the learned forecaster (a Transformer with a competing-risks head) is not known to be an increasing function of a linear index. The current-value share wV is defined only for linear-index forecasters. (A2) Exact matching (EM), used in (a)(i), (a)(ii), the last clause of (a)(v) and (b): conditionally on (w, x, η) the donor’s x′ has the conditional law of x given w. EM combines donor independence with equality of conditional laws. Donor independence holds by construction provided stays of different persons are independent: donors are chosen from hour-6 encoder states only, never from labels, and never from the recipient’s person. Equality of conditional laws is idealized in two layers. First, the implementation takes the nearest neighbor, with reuse, where EM draws from the exact conditional law. Second, it matches on a summary of w, the standardized 64-dimensional frozen hour-6 state z6 , and exact matching on a summary equals EM only if the summary carries all of the information in w about x. Part (a)(iii) removes EM from the identity of (a)(i) and shows that exact matching on a linear summary keeps sLV = 1 but not sSH = 1; part (b) is proved under EM and under neither of the two implemented layers, and (b)(v)(5) shows that the summary layer alone already biases corr(V, V ′ ). Size, by simulation only (Gaussian worlds that reproduce four observed quadruples, with a 64-dimensional state whose geometry was calibrated to the shipped donor summaries): nearest-neighbor matching moved the ratio (ρS2 − ρSH )/∆LV away from wV by 0.006 to 0.025 (Monte Carlo standard errors 0.001 to 0.004) and lowered ρSH by 0.005 to 0.040 relative to EM. With the roster truncation and the ties of (A1) added, singly or together, the ratio differed from wV by −0.027 to +0.067 under nearest-neighbor matching and by −0.023 to +0.048 under exact matching within the truncated roster. In these worlds the null value of (a)(i) therefore holds only to about 0.02 on the latent scale, comparable to several observed latent differences-in-differences. (A3) CVR on records. In (a)(iii) and (c) the restriction must hold on the spliced and own-value records themselves, which need not lie in the support of natural records. Under EM spliced records have the law of natural records, so (a)(i) and the necessity parts of (b) need CVR only almost surely on natural records; the ideal forecaster of (a)(ii) and (b)(iv) obeys CVR at every record because µ is unique. 32

(A4) ∆LV > 0. On validation (job 13007) the latent ∆LV lies between 0.252 and 0.381, and the AUROC-scale difference between 0.115 and 0.188, in all 12 settings. (A5) In (a)(v): v 7→ P (Y = 1 | V = v, R) strictly increasing. In LT under CVS this is equivalent to ρown > 0 and follows from (A4) under EM. (A6) Ideal forecaster in (a)(ii), (a)(iv), (a)(v) and (b): the forecast P (Y = 1 | FL , R) on the roster. Treating the twin’s forecast as this object is an idealization that the record does not support. Under LT, EM and roster ignorability (Definition 1), a forecaster that is ideal at origins 6 and L has hour-6 latent correlation (ρSH ρS2 )1/2 by (b)(i); computed from the twin’s S2 and SH, this value exceeds the latent correlation of the twin’s own hour-6 arm S1 in 12 of 12 settings, by 0.003 to 0.071 (sampling uncertainty not propagated). So the twin is not ideal at both origins, or LT or EM fails; the record does not say which, although the nearest-neighbor layer, which lowered ρSH in the simulations cited under (A2), would push the computed value down rather than up. Part (b)(iv) states what survives for learned forecasters; the bounds of (b)(v) hold only for the ideal forecaster. The only implemented property used elsewhere is that the forecast at origin L is a function of the record read at L. Proof. Throughout, Var, Cov, corr and SD are population moments of the roster law, L(·) denotes a law and ⊥ denotes (conditional) independence. For a score S and independent draws S1 ∼ L(S | Y = 1) and S0 ∼ L(S | Y = 0), AUROC(S) := E hc(S1 , S0 ) with hc(a, b) := 1{a > b} + 21 1{a = b}; the empirical AUROC is the same functional of the empirical class-conditional laws. In LT, conditional expectations given w or (w, x) are taken in their affine versions, which are unique because Cov((w, x)) is nonsingular; the same nonsingularity gives Var V > 0 and Var(eV ) = Var(x1 | w) > 0. Lemma 0 (invariance). If Ψ is strictly increasing on the range of S, then AUROC(Ψ(S)) = AUROC(S), in the population and empirically. Proof. Pointwise, {Ψ(S1 ) > Ψ(S0 )} = {S1 > S0 } and {Ψ(S1 ) = Ψ(S0 )} = {S1 = S0 }. □ Lemma 1 (AUROC link). Let (J, η) be bivariate Gaussian with Var J > 0 and ρ := corr(J, η), and let Ψ be strictly increasing. Then AUROC(Ψ(J)) = Aπ (ρ), where Aπ (1) = 1, Aπ (−1) = 0 and, for |ρ| < 1, ! ρ (η1 − η0 ) , Aπ (ρ) = E Φ p 2(1 − ρ2 ) with η1 ∼ L(η | η > tπ ) and η0 ∼ L(η | η ≤ tπ ) independent. Aπ is continuous and strictly increasing on [−1, 1] with Aπ (0) = 21 , and a constant score has AUROC 12 = Aπ (0). Hence ρa = corr(Ja , η) for every arm whose score is a strictly increasing function of a variable Ja with (Ja , η) bivariate Gaussian and Var Ja > 0. Proof. By Lemma 0, and because an increasing affine map preserves the AUROC, we may take J = ρη + (1 − ρ2 )1/2 ε with ε ∼ N (0, 1) independent of η; this representation uses the joint Gaussianity of (J, η). Let (η1 , ε1 ) and (η0 , ε0 ) be independent, ηy ∼ L(η | Y = y), εy ∼ N (0, 1) independent of everything else; then Jy := ρηy +(1−ρ2 )1/2 εy has the law of J given Y = y. If |ρ| < 1, ε0 −ε1 ∼ N (0, 2) is independent of (η1 , ηp 0 ), so J1 −J0 has a density, P (J1 = J0 ) = 0, and conditioning on (η1 , η0 ) gives P (J1 > J0 ) = E Φ{ρ(η1 − η0 )/ 2(1 − ρ2 )}. If ρ = ±1, Jy = ±ηy and η1 > tπ ≥ η0 almost surely,pso the AUROC is 1 or 0. Since D := η1 − η0 > 0 almost surely, k 7→ Φ(kD) is strictly increasing, and ρ 7→ ρ/ 2(1 − ρ2 ) is a strictly increasing bijection of (−1, 1) onto R; so Aπ is strictly increasing on (−1, 1) with values in (0, 1), continuous there by dominated convergence, and tends to 1 and 0 as ρ → 1 and ρ → −1, again by dominated convergence. Aπ (0) = Φ(0) = 12 . A constant score has P (S1 = S0 ) = 1. □ Note. Gaussianity of J alone does not suffice. J := η (1{|η| < 1.5} − 1{|η| ≥ 1.5}) is standard normal by symmetry, and corr(J, η) = E[η 2 ; |η| < 1.5] − E[η 2 ; |η| ≥ 1.5] = −0.044; at π = 0.25 its AUROC is about 0.67 (two Monte Carlo runs of 2 × 106 draws gave 0.6671 and 0.6679), while A0.25 (−0.044) = 0.479. Lemma 2 (conditional independence in LT). Assume LT, and let G ⊆ FL be generated by finitely many affine functions of (w, x). Then Y ⊥ (w, x) | G if and only if µ = E[η | G] almost surely. In particular, CVS holds if and only if µ = E[η | V ], and HCS holds if and only if µ = E[η | w, V ] = mη + θV eV . Proof. Let MG := E[η | G] and D := µ − MG . In the jointly Gaussian family both conditional expectations are affine in (w, x), so D is Gaussian with mean zero and uncorrelated with the generators of G; being jointly 2 Gaussian with them, it is independent of G. Conditionally on (w, x), η ∼ N (µ, ση|L ), so P (Y = 1 | w, x) = Φ{(MG + D − tπ )/ση|L }. As Y is binary, Y ⊥ (w, x) | G holds if and only if P (Y = 1 | w, x) is almost surely equal to a G-measurable variable (which is then P (Y = 1 | G)). If D = 0 almost surely this holds. Conversely,

33

if P (Y = 1 | w, x) = Z almost surely with Z G-measurable, then Z ∈ (0, 1) and D = ση|L Φ−1 (Z) + tπ − MG almost surely, so D is G-measurable and independent of G, hence almost surely constant, and ED = 0 gives D = 0. For G = σ(w, V ) = σ(w, eV ): eV is Gaussian and uncorrelated with w, hence independent of w, and the projection of η on span{1, w, eV } is mη + {Cov(η, eV )/ Var eV }eV . □ Lemma 3 (splicing under exact matching). Assume LT and EM, and put ξ := x − E[x | w] and ξ ′ := x′ − E[x | w]. Then ξ ⊥ w; ξ ′ is independent of (w, x, η) and has the law of ξ; (w, x, η, x′ ) is jointly Gaussian; d

and (w, x′ ) = (w, x). For an affine J = c0 + c⊤ w + b⊤ x with Var J > 0 and its splice J ′ := c0 + c⊤ w + b⊤ x′ , Var J ′ = Var J,

Cov(J ′ , η) = Cov(E[J | w], η),

corr(J, η) − corr(J ′ , η) =

Cov(b⊤ ξ, η) . SD(J)

Proof. ξ is Gaussian and uncorrelated with w, hence independent of w, and Cov ξ is the constant conditional covariance of x given w. By EM, conditionally on (w, x, η), x′ ∼ N (E[x | w], Cov ξ), so ξ ′ ∼ N (0, Cov ξ) conditionally on (w, x, η); thus ξ ′ is independent of (w, x, η) with the law of ξ, and (w, x, η, ξ ′ ), hence (w, x, η, x′ ), d

is jointly Gaussian. Since ξ and ξ ′ are both independent of w with the same law, (w, E[x | w] + ξ ′ ) = (w, E[x | w] + ξ). Write J = E[J | w] + b⊤ ξ and J ′ = E[J | w] + b⊤ ξ ′ . Independence gives Var J ′ = Var E[J | w] + Var(b⊤ ξ) = Var J and Cov(J ′ , η) = Cov(E[J | w], η). Subtracting from Cov(J, η) = Cov(E[J | w], η) + Cov(b⊤ ξ, η) and dividing by SD(J) = SD(J ′ ) (with SD η = 1) gives the last identity. □ d

Proof of (a)(i). By Lemma 3, (w, x′ ) = (w, x), so a spliced record has the law of a natural record and an almost-sure version of CVR applies to it. Linear CVR gives S2 = Ψ(I) and SH = Ψ(ISH ), where ISH = U6 +βV x′1 is the splice of I = U6 + βV x1 ; the last-value scores are V and its splice V ′ . Lemma 3 with J = V and Lemma 1 give SD(V ′ ) = SD(V ) and Cov(eV , η) . (P2.1) ∆LV = corr(V, η) − corr(V ′ , η) = SD(V ) Since ∆LV > 0, Cov(eV , η) > 0. As I −E[I | w] = βV eV with eV ⊥ w, Var I = Var(U6 +βV mV )+βV2 Var eV > 0, and Lemma 3 with J = I, Lemma 1 and (P2.1) give ρS2 − ρSH =

βV Cov(eV , η) βV SD(V ) = ∆LV = wV ∆LV . SD(I) SD(I)

Hence DiDρ = (wV − 1)∆LV , which has the sign of wV − 1. Because βV > 0 and Var I = Var U6 + 2βV Cov(U6 , V ) + βV2 Var V , wV = βV SD(V )/ SD(I) is smaller than, equal to or larger than 1 according as Var U6 + 2βV Cov(U6 , V ) is positive, zero or negative. The last case requires Cov(U6 , V ) < 0 because Var U6 ≥ 0. □ Proof of (a)(ii). By (P2.1), θV = Cov(η, eV )/ Var eV > 0. By Lemma 2, HCS gives µ = mη + θV eV = (mη − θV mV ) + θV V almost surely. Both sides are affine functions of (w, x), and two affine functions that agree almost surely under a law with nonsingular covariance agree at every point of Rp+q ; so µ(w, x) = U6 (w) + θV x1 for every record, with U6 := mη − θV mV affine in w. Hence M∗ (r) = Ψ(U6 (r1:6 ) + θV v(r)) at every record, natural, spliced or own-value, with Ψ(υ) := Φ{(υ − tπ )/ση|L } strictly increasing: the ideal forecaster obeys linear CVR with βV = θV . Because eV ⊥ w, Cov(U6 , V ) = Cov(U6 , mV ), and Var U6 + 2θV Cov(U6 , mV ) = Var mη − 2θV Cov(mη , mV ) + θV2 Var mV + 2θV Cov(mη , mV ) − 2θV2 Var mV = Var mη − θV2 Var mV . Part (i) gives the conclusion. □ Proof of (a)(iii).

CVR holds on the spliced records, so I − ISH = βV (V − V ′ ) for every stay and Cov(I, η) − Cov(ISH , η) = βV {Cov(V, η) − Cov(V ′ , η)}.

Lemma 1 applies to each of the four pairs, and SD η = 1, so Cov(I, η) = ρS2 SD(I), Cov(ISH , η) = ρSH SD(ISH ), Cov(V, η) = ρown SD(V ) and Cov(V ′ , η) = ρdonor SD(V ′ ). Substituting and dividing by SD(I) gives the identity. Writing ρS2 − ρSH = (ρS2 − sSH ρSH ) + (sSH − 1)ρSH and wV (ρown − sLV ρdonor ) = wV ∆LV − wV (sLV − 1)ρdonor , and subtracting ∆LV , gives the expression for DiDρ . No property of the donor assignment beyond the Gaussianity of the two pairs is used. Under EM, Lemma 3 with J = I and J = V gives SD(ISH ) = SD(I) and SD(V ′ ) = SD(V ), which recovers (i). 34

Now let z be linear and, conditionally on (w, x, η), x′ ∼ L(x | z(w)). As (z(w), x) is jointly Gaussian, L(x | z(w)) = N (E[x | z(w)], Σz ) with E[x | z(w)] affine in w and Σz constant; so x′ = E[x | z(w)] + ξ ′′ with ξ ′′ ∼ N (0, Σz ) independent of (w, x, η), and (w, x, η, x′ ) is jointly Gaussian. Hence (ISH , η) and (V ′ , η) d

d

are bivariate Gaussian, (z(w), x′ ) = (z(w), x), and in particular V ′ = V , so sLV = 1 and Var V ′ = Var V > 0; also Var ISH ≥ βV2 Var ξ1′′ = βV2 Var(V | z(w)) ≥ βV2 Var(eV ) > 0, so the hypotheses of (iii) hold. Put mzV := E[V | z(w)]. As U6 is a function of w and E[V ′ | w] = mzV , Cov(U6 , V ′ ) = Cov(U6 , mzV ), while Cov(U6 , V ) = Cov(U6 , mV ). Therefore Var ISH − Var I = 2βV Cov(U6 , mzV − mV ), and dividing by Var I > 0 gives the formula for s2SH . □ Proof of (a)(iv). The display restates ρa = A−1 π (AUROCa ). For the example let w = (w1 , w2 ), εV and ε0 be independent standard normal, mV := 2−1/2 w1 , eV := 2−1/2 εV , V := mV + eV , x := V , m := 0.181/2 ( 21 w1 + √ 3 2 w2 ) and η := m+0.6 eV +0.8 ε0 , with donors drawn as in EM. Cov((w1 , w2 , V )) is nonsingular (its determinant is 21 ). Then Var V = Var η = 1, mη = m, and µ = m + 0.6 eV is σ(w, V )-measurable, so HCS holds (Lemma 2), θV = 0.6, and Var mη = 0.18 = θV2 Var mV ; by (ii), DiDρ = 0. Explicitly: Cov(µ, η) = Var µ = 0.36, so ρS2 = 0.6; Lemma 3 gives Cov(µ′ , η) = Var mη = 0.18 and ρSH = 0.3; Cov(mV , m) = 12 (0.5 · 0.18)1/2 = 0.15, so ρown = 0.15 + 0.6 · 0.5 = 0.45 and, by Lemma 3, ρdonor = 0.15. At π = 0.25, evaluating the integral of Lemma 1 by adaptive quadrature at 30-digit working precision gives Aπ (0.60) = 0.79164, Aπ (0.30) = 0.64402, Aπ (0.45) = 0.71715 and Aπ (0.15) = 0.57179 (an independent evaluation as a trivariate normal orthant probability agrees −5 within 3×10 So DiDAUROC = 0.14762−0.14536 = +0.0023 ̸= 0 (to more digits +0.002265). On the binormal √ ).−1 ′ scale d = 2 Φ (AUROC) the same model gives +0.070. □ Proof of (a)(v). CVS gives P (Y = 1 | FL , R) = P (Y = 1 | V, R) = Ψ0 (V ) almost surely, so r 7→ Ψ0 (v(r)) is a version of the ideal forecaster, and for it S2 = Ψ0 (V ) and SH = Ψ0 (V ′ ) whatever the donor. Lemma 0 gives AUROCS2 = AUROCown and AUROCSH = AUROCdonor , in the population and empirically, for any joint law and any donor assignment; the ϕ-difference-in-differences is then ϕ(AUROCown ) − ϕ(AUROCdonor ) − ϕ(AUROCown )+ϕ(AUROCdonor ) = 0. The same argument applies to r 7→ Ψ̃(v(r)). In LT under CVS, Var V > 0 by nonsingularity, Lemma 2 gives µ = E[η | V ] = θ0 (V − EV ) with θ0 := Cov(η, V )/ Var V , and Var(η | 2 V ) = 1 − Var µ = ση|L > 0, so Ψ0 (v) = Φ{(θ0 (v − EV ) − tπ )/ση|L }, which is strictly increasing if and only if θ0 > 0, that is (Lemma 1) if and only if ρown = corr(V, η) > 0. Under EM, since eV is affine in (w, x), Cov(η, eV ) = Cov(µ, eV ) = θ0 Var eV , and (P2.1) gives Cov(η, eV ) = ∆LV SD(V ) > 0, so θ0 > 0. □ Proof of (b)(i). Fix an LT model with EM and Var µ > 0. The ideal forecaster is Ψ(µ) with Ψ(υ) = Φ{(υ − tπ )/ση|L }, and its spliced read is Ψ(µ′ ) with µ′ the splice of µ. Put ν := µ − mη = µ − E[µ | w], a linear function 2 := Var ν. Because η − µ is uncorrelated with every of ξ and hence independent of w, and σ62 := Var mη , σnew 2 2 affine function of (w, x), Cov(µ, η) = Var µ = σ6 + σnew , and Lemma 1 gives ρS2 = SD(µ). Lemma 3 with J = µ gives Var µ′ = Var µ and Cov(µ′ , η) = Cov(mη , η) = σ62 , so ρSH = σ62 / SD(µ). Hence ρSH ρS2 = σ62 ,

2 ρS2 (ρS2 − ρSH ) = σnew .

(P2.2)

S1∗ = Φ{(mη − tπ )/(1 − σ62 )1/2 } is strictly increasing in mη (note σ62 ≤ Var µ < 1), and Cov(mη , η) = σ62 , so ρS1∗ = σ6 when σ6 > 0 (Lemma 1), and ρS1∗ = 0 = σ6 when mη is constant. Thus ρ2S1∗ = ρSH ρS2 ; as 0 ≤ ρSH = σ62 /ρS2 ≤ ρS2 , the geometric mean lies between them. □ Proof of (b)(ii) and (b)(iii): necessity. Let Q be the quadruple of an LT model with EM, with 0 < ρSH < ρS2 < 1 and ∆LV > 0. By (P2.2), σ6 = (ρSH ρS2 )1/2 > 0 and σnew = [ρS2 (ρS2 − ρSH )]1/2 > 0. With ι = SD(eV )/ SD(V ) and ι0 := ∆LV /σnew > 0, R1 reads ι0 ≤ 1 and R2 reads ρ2donor ≤ σ62 (1 − ι20 ). Since the right side of R2 is negative when ι0 > 1, R2 implies R1. Because eV is affine in (w, x), Cov(eV , η) = Cov(eV , µ), and because eV ⊥ w, Cov(eV , mη ) = 0. With (P2.1), ∆LV SD(V ) = Cov(eV , ν) ≤ SD(eV ) σnew ,

that is

∆LV ≤ ι σnew .

(P2.3)

As Var V = Var mV + Var eV , ι ≤ 1, which gives R1, and (P2.3) gives ι ≥ ι0 . By Lemma 3 with J = V , Cov(V ′ , η) = Cov(mV , η) = Cov(mV , mη ) (as mV is w-measurable) and SD(V ′ ) = SD(V ), so Lemma 1 and the Cauchy–Schwarz inequality give ρdonor =

Cov(mV , mη ) , SD(V )

|ρdonor | ≤

SD(mV ) σ6 = (1 − ι2 )1/2 σ6 ≤ (1 − ι20 )1/2 σ6 , SD(V )

which gives R2. This proves the first claim of (iii) and the “only if” part of (ii).

35

(P2.4)

HCS holds if and only if ν is almost surely a multiple of eV : under HCS , Lemma 2 gives ν = θV eV ; conversely, if ν = c eV then µ = mη + c eV is σ(w, V )-measurable, so µ = E[η | w, V ] and Lemma 2 gives HCS . Since ν and eV have mean zero, positive variances and Cov(eV , ν) > 0 by (P2.3), equality holds in the Cauchy–Schwarz step of (P2.3) if and only if ν is almost surely a multiple of eV . Hence ι = ι0 under HCS , while under a violation ι > ι0 , so that R1 holds strictly (ι0 < ι ≤ 1) and, by (P2.4) and σ6 > 0, R2 holds strictly. This proves the “only if” part of the second claim of (iii). Proof of (b)(ii) and (b)(iii): sufficiency. Let Q satisfy R1 and R2, define σ6 , σnew , ι0 as above, and let ι ∈ [ι0 , 1] satisfy ρ2donor ≤ σ62 (1 − ι2 ). Build the model M(ι) as follows. Let w1 , w2 , εV , ε2 , ε0 be independent standard normal, w := (w1 , w2 ), and set mV := (1 − ι2 )1/2 w1 , c1 := ρdonor (1−ι2 )−1/2 (ι < 1),

eV := ι εV ,

c1 := 0 (ι = 1),

V := mV + eV ,

c2 := (σ62 −c21 )1/2 ,

x := (V, ε2 ),

θ := ∆LV /ι2 ,

2 γ := (σnew −∆2LV /ι2 )1/2 ,

η := c1 w1 + c2 w2 + θ eV + γ ε2 + (1 − ρ2S2 )1/2 ε0 , with donors drawn as in EM. The constraint on ι makes c2 real (if ι = 1 it forces ρdonor = 0), ι ≥ ι0 > 0 makes θ finite and γ real, and γ = 0 if and only if ι = ι0 . The map from (w1 , w2 , εV , ε2 ) to (w1 , w2 , V, ε2 ) is linear with determinant ι > 0, so Cov((w, x)) is nonsingular. The vector (w, x, η) is jointly Gaussian with mean zero, 2 2 Var V = 1, and, using σ62 +σnew = ρ2S2 from (P2.2), Var η = σ62 +θ2 ι2 +γ 2 +1−ρ2S2 = σ62 +σnew +1−ρ2S2 = 1. Since eV = V − mV and ε2 = x2 are functions of (w, x) and ε0 is independent of (w, x), µ = c1 w1 + c2 w2 + θeV + γε2 , 2 2 ση|L = 1 − ρ2S2 > 0, mη = c1 w1 + c2 w2 with variance σ62 , and ν = θeV + γε2 with variance θ2 ι2 + γ 2 = σnew . By 2 2 1/2 2 = ρS2 and σ6 /ρS2 = ρSH . By (P2.1) with (b)(i), the model’s latent correlations of S2 and SH are (σ6 + σnew ) SD V = 1, its ∆LV is Cov(eV , η) = θι2 = ∆LV ; by (P2.4), its ρdonor is Cov(mV , mη ) = (1 − ι2 )1/2 c1 = ρdonor ; hence its ρown is ρdonor + ∆LV = ρown . So M(ι) attains Q, and its Var(eV )/ Var(V ) is ι2 . Finally, ε2 is independent of eV with unit variance, so ν = θeV + γε2 is a multiple of eV if and only if γ = 0; by the characterization above, HCS holds in M(ι) if and only if ι = ι0 . Under R1 and R2 the choice ι = ι0 is admissible, so Q is attainable under HCS ; this is the “if” part of (ii). Put ιmax := (1 − ρ2donor /σ62 )1/2 ≤ 1; strict R2 is equivalent to ι0 < ιmax . If R1 and R2 hold strictly, every ι ∈ (ι0 , ιmax ) is admissible, satisfies ι < 1 and gives γ > 0; so Q is attainable by a model that violates HCS . This is the “if” part of the second claim of (iii). □ Proof of (b)(iv). By (iii) every attainable Q satisfies R1 and R2, and by (ii) it is then attainable under HCS ; by (ii) and (iii) every Q that satisfies R1 and R2 strictly is attainable under HCS and under its failure. The threshold tπ is fixed by π, which is common to all models considered, and Aπ is a bijection (Lemma 1), so the four AUROCs and Q determine each other and any function of the four AUROCs is a function of Q; it therefore takes at every attainable Q a value that it also takes under some HCS model, and at every Q satisfying R1 and R2 strictly a value that it also takes under some model violating HCS . By (i) and Lemma 1, AUROC(S1∗ ) = Aπ ((ρSH ρS2 )1/2 ) in every such model, a function of Q, so the same argument applies to the five AUROCs. For the ideal forecaster, if HCS holds, the proof of (a)(ii) shows that it obeys linear CVR at every record. If HCS fails, µ is not almost surely σ(w, V )-measurable (otherwise µ = E[η | w, V ] and Lemma 2 gives HCS ), and Ψ is injective, so Ψ(µ) is not almost surely a function of (w, V ) and CVR fails even in its almost-sure form on natural records. Hence the two models M(ι0 ) and M(ι), ι ∈ (ι0 , ιmax ), that attain a strictly interior Q carry an ideal forecaster that obeys CVR and one that does not; any class of forecasters containing the ideal ones contains both. The last sentence of (iv) is a statement of scope. □ Proof of (b)(v). Fix an LT model with EM whose ideal forecaster has quadruple Q with 0 < ρSH < ρS2 < 1 2 and ∆LV > 0. By (P2.2), σnew as defined in (v) equals Var ν, and ρSH ρS2 = σ62 , so ιmax as defined in (v) is the ιmax of the sufficiency proof. (1) By Lemma 3, V ′ = mV + ξ1′ with ξ1′ independent of (w, x, η) and distributed as ξ1 = eV ; and eV ⊥ w. Hence Cov(V, V ′ ) = Cov(mV + eV , mV + ξ1′ ) = Var mV and Var V ′ = Var V > 0, so corr(V, V ′ ) = Var mV / Var V = 1 − ι2 . No label enters. (2) The necessity proof gives ι ≥ ι0 (from (P2.3)) and ρ2donor ≤ (1 − ι2 )σ62 (from (P2.4)), that is ι ≤ ιmax . The variables ν and eV are jointly Gaussian with mean zero, Cov(ν, eV ) = Cov(η, eV ) = ∆LV SD(V ) by (P2.3), 2 Var ν = σnew and Var eV = ι2 Var V , so ς=

i ι20 h ι20 ∆2LV Var V = ∈ , 1 , 2 σnew ι2 Var V ι2 ι2max

36

and ς = Var(E[ν | eV ])/ Var ν because E[ν | eV ] is the linear projection of ν on eV . By the characterization in the necessity proof, HCS holds if and only if ν is almost surely a multiple of eV , that is if and only if ς = 1; by ς = ι20 /ι2 and (1), this is ι2 = ι20 , that is corr(V, V ′ ) = 1 − ι20 . (3) By Lemma 2, µCS := E[η | w, V ] = mη + θV eV , a Gaussian variable with Var µCS = σ62 + 2 Cov(η, eV )2 / Var eV = σ62 + ∆2LV /ι2 = σ62 + ςσnew > 0, using eV ⊥ w. Since µCS is the projection of η on span{1, w, eV }, Cov(µCS , η) = Var µCS , and Var(η | w, V ) = 1 − Var µCS ≥ 1 − Var µ > 0. So CS∗ = Φ{(µCS − tπ )/ Var(η | w, V )1/2 } is strictly increasing in µCS , and Lemma 1 gives ρCS∗ = SD(µCS ) = 2 2 2 (ρSH ρS2 +ςσnew )1/2 . With ρ2S2 = σ62 +σnew , ρ2S2 −ρ2CS∗ = (1−ς)σnew , and (2) bounds it. AUROC(M∗ ) = Aπ (ρS2 ) ∗ 2 2 2 1/2 and AUROC(CS ) = Aπ (ρCS∗ ) ≥ Aπ ((ρSH ρS2 + σnew ι0 /ιmax ) ) because Aπ is increasing, which gives the AUROC bound. (4) The model is LT with EM, so by (iii) Q satisfies R1 and R2 and ι0 ≤ ιmax . For every ι ∈ [ι0 , ιmax ] the constraint ρ2donor ≤ σ62 (1 − ι2 ) holds, so M(ι) of the sufficiency proof is an LT model with EM that attains Q and has Var(eV )/ Var(V ) = ι2 , hence ς = ι20 /ι2 by (2). As ι runs over [ι0 , ιmax ], ς runs over [ι20 /ι2max , 1]; the endpoints ι = ι0 and ι = ιmax attain the bounds in (2) and (3). The sentence after (4) restates (2) to (4). (5) As in the proof of (a)(iii), x′ = E[x | z(w)] + ξ ′′ with ξ ′′ independent of (w, x, η), so V ′ = mzV + ξ1′′ with d

mzV := E[V | z(w)], and V ′ = V , so Var V ′ = Var V . Since z(w) is a function of w, mzV = E[mV | z(w)], and mV − mzV is uncorrelated with every affine function of z(w). Hence Cov(V, V ′ ) = Cov(mV + eV , mzV ) = Cov(mV , mzV ) = Var mzV , using eV ⊥ w, and corr(V, V ′ ) = Var(mzV )/ Var(V ). Finally Var mV = Var mzV + Var(mV − mzV ), so Var mzV ≤ Var mV = (1 − ι2 ) Var V , with equality if and only if mV = mzV almost surely. □ OV SH Proof of (c)(i). If M(r) = g(r1:6 , v(r)) on the domain, then, because r1:6 = r1:6 = r1:6 , v(rOV ) = V and SH ′ ′ v(r ) = V , we have OV = g(r1:6 , V ) = S2 and SH = g(r1:6 , V ), which equals S2 when V ′ = V . For the converse, let r, r̃ ∈ N with r1:6 = r̃1:6 and v(r) = v(r̃). The splice rOV (r̃) has first part r1:6 = r̃1:6 and second part r̃7:L with the hour-L target value replaced by v(r) = v(r̃), which leaves r̃7:L unchanged (the mask at L is 1 in both); so rOV (r̃) = r̃ and M(r) = M(rOV (r̃)) = M(r̃). Hence g(a, v) := M(r) for any r ∈ N with (r1:6 , v(r)) = (a, v) is well defined, and M(r) = g(r1:6 , v(r)) on N . No law, matching property or label enters. The last sentence of (c)(i) is the contrapositive of the first. □

Proof of (c)(ii). The decomposition is an identity, and OV and SH are read from records that differ only in the hour-L target value by the definition of rOV . Under CVR, S2 = OV for every stay by (i), so δOV = 0. For the bound, draw (S21 , OV1 ) ∼ L((S2, OV) | Y = 1) and, independently, (S20 , OV0 ) ∼ L((S2, OV) | Y = 0); the marginals give AUROCS2 = E hc(S21 , S20 ) and AUROCOV = E hc(OV1 , OV0 ). On the event Bε := {|S21 − OV1 | ≤ ε, |S20 − OV0 | ≤ ε, |S21 − S20 | > 2ε}, if S21 − S20 > 2ε then OV1 − OV0 ≥ (S21 − ε) − (S20 + ε) > 0 and both values of hc are 1; if S21 − S20 < −2ε then OV1 − OV0 < 0 and both are 0. So hc(S21 , S20 ) − hc(OV1 , OV0 ) vanishes on Bε and lies in [−1, 1] elsewhere, and |AUROCS2 − AUROCOV | ≤ P (Bεc ) ≤ P (|S21 − OV1 | > ε) + P (|S20 − OV0 | > ε) + P (|S21 − S20 | ≤ 2ε), which is the stated bound. The empirical AUROC is the average of hc over all pairs of a positive and a negative stay of the roster, that is the same expectation under the empirical class-conditional laws, and the argument applies unchanged. □ Numerical checks. The following are descriptive re-reads of the aggregates of the validation isolation run (job 13007), converted to latent correlations with Aπ of Lemma 1 (quadrature, checked against a trivariate normal orthant probability to 1.5 × 10−5 ). Table S12 converts the five-seed mean AUROC of each arm once; averaging per-seed latent correlations gives the same values, and averaging per-seed ratios gives ratios of 0.878 to 1.090 (median 0.987) instead of 0.878 to 1.084 (median 0.983). • The contrast on three scales. ∆LV is 0.252 to 0.381 on the latent scale and 0.115 to 0.188 on the AUROC scale in all 12 settings, so (A4) holds. The pre-specified difference-in-differences on the AUROC scale is positive in 5 of 12 settings (−0.017 to +0.012); after Holm correction within the 12 settings none is positive and two are negative (SaO2 and heart rate, both √ at (12, 6)). On the latent scale it is positive in 4 of 12 (−0.040 to +0.025), and on the binormal scale 2 Φ−1 (AUROC) in 9 of 12 (−0.050 to +0.057). The sign of the contrast therefore depends on its scale, as (a)(iv) allows. • Null value. Under linear CVR and exact matching, (a)(i) gives (ρS2 − ρSH )/∆LV = wV . The observed ratio is 0.878 to 1.084 (median 0.983). Its AUROC-scale interval excludes 1 in 3 of 12 settings, all below 1, which is the sign (a)(i) gives for a linear-CVR forecaster with wV < 1; the four settings with a ratio above 1 have intervals that cover 1. In simulations with the donor geometry calibrated to the shipped donor summaries, nearest-neighbor matching, the truncated roster and ties together moved the latent

37

difference-in-differences by about 0.02 (assumption (A2)), so a negative difference-in-differences of that size does not reject CVR. • Attainability. R1 and R2 hold strictly in 12 of 12 settings (Table S12; slacks 0.051 to 0.099 and 0.008 to 0.059), so by (b)(iv) every seed-averaged quadruple is attained both by a law that satisfies HCS and by one that violates it. Per seed, R1 and R2 hold in 59 of 60 seed-by-setting cells; SpO2 at (30, 24) with seed 31 lies just outside both (slacks −0.0071 and −0.0035). In a parametric bootstrap the standard deviation of the R2 slack is 0.002 to 0.014, and R2 is violated in 7% of replicates for SpO2 (30, 24), 0.5% for SaO2 (30, 24) and never elsewhere. A quadruple outside R1 or R2 would falsify the idealized class (ideal forecaster, exact matching, LT), not HCS , and a learned forecaster need not satisfy R1 or R2. • Size of a departure, if the twin were ideal. Computed from the quadruples as if the twin were the ideal forecaster, (b)(v) allows at most 0.078 to 0.447 (median 0.302) of the new-hour latent variance to lie outside the current value, which bounds the AUROC of personalization beyond the current state by 0.007 to 0.029 (median 0.022) and predicts corr(V, V ′ ) of 0.249 to 0.483 under HCS . These are properties of an ideal forecaster. The twin is not one, or LT or exact matching fails: the implied ideal hour-6 correlation (ρSH ρS2 )1/2 exceeds the latent correlation of the twin’s own hour-6 arm in 12 of 12 settings, by 0.003 to 0.071 (last column of Table S12). The values are therefore neither bounds on the twin nor evidence for HCS . The share 1 − ρSH /ρS2 , which equals the new-hour share of the latent variance only for an ideal forecaster under exact matching, is 0.464 to 0.771 and is reported as a model-based description only. • The proofs. The example of (a)(iv) was evaluated by adaptive quadrature at 30-digit working precision. On 9,929 random LT models with exact matching and nonsingular covariance, ι2 = 1 − corr(V, V ′ ) held to 10−16 , ς = ι20 /ι2 to 6 × 10−11 and the formula for ρCS∗ to 2 × 10−16 , with ι ≤ ιmax in every model; on 20,000 models with matching on a linear summary, corr(V, V ′ ) = Var E[V | z(w)]/ Var V held to 8 × 10−12 and the formula for s2SH to 4 × 10−15 ; the bound of (c)(ii) held in 200 random cases. • What (c) and (b)(v)(1) need. The shipped aggregates contain no per-stay predictions, and no own-value arm was run. The isolation run writes per-stay S2, SH, last-value and donor identifiers, from which the tied-pair check of (c)(i), corr(V, V ′ ) and a regression of logit S2 − logit SH on V − V ′ could be computed without a new model read; the own-value splice needs one. None of these was computed for this note: the per-stay values are not in the shipped aggregates, and by (b)(v)(5) matching on a summary of the first six hours biases 1 − corr(V, V ′ ) upwards, so corr(V, V ′ ) alone would not settle HCS . Exact CVR is false a priori for a Transformer that reads every channel of hours 7 to L, so a single-stay rejection would be uninformative; the informative quantities are δOV and AUROCS2 − AUROCOV , each against a tolerance fixed before reading. Tied pairs are informative only for the saturation targets (independent-pair tie probability 0.29 to 0.34 for SpO2 , 0.09 to 0.12 for SaO2 , at most 0.005 for heart and respiratory rate). Supplementary Table S12: Latent correlations of the four arms of the difference-in-differences (Proposition 2). Validation isolation run (job 13007); each arm’s five-seed mean AUROC is converted to a latent correlation ρ = A−1 π (AUROC) at the roster prevalence π. Own and donor are the patient’s and the matched donor’s last measured target value. Ratio: (ρS2 − ρSH )/∆LV , which equals the current-value share wV under linear CVR and exact matching (Proposition 2(a)(i)). R1 and R2: slacks of the attainability conditions of Proposition 2(b)(ii), positive when the quadruple is attained both under HCS and under its failure. Share: the largest share of the new-hour latent variance that can lie outside the current value for an ideal forecaster (Proposition 2(b)(v)). Gap: (ρSH ρS2 )1/2 minus the latent correlation of the twin’s hour-6 arm S1; a positive gap shows that the twin’s arms are not those of an ideal forecaster in this idealized class. Share and the conversion assume the latent-threshold Gaussian model; the table is descriptive. Target

(L, H)

π

ρS2

ρSH

SaO2 SaO2 SaO2 SpO2 SpO2 SpO2 Heart rate Heart rate Heart rate Respiratory rate Respiratory rate Respiratory rate

(12, 6) (18, 12) (30, 24) (12, 6) (18, 12) (30, 24) (12, 6) (18, 12) (30, 24) (12, 6) (18, 12) (30, 24)

0.268 0.394 0.565 0.340 0.490 0.664 0.125 0.220 0.343 0.192 0.308 0.442

0.498 0.501 0.424 0.491 0.510 0.414 0.584 0.583 0.480 0.471 0.516 0.422

0.248 0.211 0.097 0.220 0.273 0.134 0.257 0.210 0.141 0.210 0.194 0.113

ρown ρdonor

Ratio

R1

0.437 0.458 0.375 0.385 0.334 0.304 0.565 0.544 0.399 0.361 0.415 0.318

0.885 0.878 1.084 0.943 0.939 0.985 0.942 0.980 1.059 1.036 0.991 1.043

0.070 0.051 0.071 0.077 0.095 0.056 0.090 0.086 0.083 0.099 0.083 0.065

38

0.154 0.127 0.074 0.098 0.082 0.020 0.218 0.164 0.079 0.109 0.090 0.021

R2 Share 0.020 0.010 0.009 0.031 0.059 0.017 0.008 0.014 0.019 0.036 0.028 0.015

0.205 0.114 0.242 0.319 0.447 0.299 0.078 0.147 0.306 0.413 0.309 0.321

Gap 0.013 0.018 0.026 0.004 0.003 0.071 0.020 0.021 0.008 0.013 0.028 0.020

5.3 The realism audit as a two-sample test Proposition 3 (The realism audit is a first-moment classifier two-sample test). Standard and specific parts. Part (a) is elementary: an AUC is bounded by the Kolmogorov distance between the two laws of the score, and that by the total-variation distance. Parts (c)(i) and (c)(ii) are textbook properties of equalcovariance Gaussian discrimination, stated here for the audit’s quantities. Part (b), the first-moment blindness of the penalized population audit, and the calibration of its penalized direction in (c)(iii) and (c)(iv) are specific to this audit. Part (d) is a property of the generator’s construction under the stated data conditions, checked on the audited pairs, rather than a mathematical result. In this section the subscripts R and G mean real and generated; elsewhere R is the roster event. Setting. The window of a stay is its first min(T, 336) hourly rows. For each of 161 channels the audit computes four summaries on the window: the mean and the population standard deviation of the channel’s finite values (both undefined when there is none); the fraction of window rows whose value is not finite; and the lag-1 autocorrelation, the Pearson correlation of the pairs of consecutive finite values, which is defined only when the window has at least 3 rows, there are at least 3 such pairs, and both lagged series have population standard deviation at least 10−8 . An undefined summary is replaced by the pooled median, and every coordinate is then centered and scaled by pooled constants. This gives the feature vector f ∈ Rd , d = 644, one measurable function of the window applied identically to real and generated records. Let PR and PG be the laws of f for the real and for the generated record of an audited stay, let fR ∼ PR and fG ∼ PG be independent, and put δ := EfG − EfR when it is defined. For π ∈ (0, 1) let P̄ := (1 − π)PR + πPG . Call the pair (PR , PG ) standardized if P̄ has mean 0 and variance 0 or 1 in every coordinate. Under (A1), with π = 21 , the laws of the audit features form a standardized pair, and the audit’s standardization leaves a standardized pair unchanged; every construction below is a standardized pair. For a Borel score s : Rd → R let SR = s(fR ), SG = s(fG ) and AUC(s) := P(SG > SR ) + 12 P(SG = SR ). Let FRs and FGs be the distribution functions of SR and SG , let dKS (s) := supx |FRs (x) − FGs (x)|, and let TV(P, Q) := supA |P (A) − Q(A)| over Borel sets A. Put g(t) := t − t2 /2 and σ(t) := 1/(1 + e−t ). For λ ≥ 0 let

⊤ ⊤ Jλ (b, w) := (1 − π) E log 1 + e b+w fR + π E log 1 + e−(b+w fG ) + λ2 ∥w∥2 , b ∈ R, w ∈ Rd , the population risk of L2 -penalized logistic regression with an unpenalized intercept, π being the share of generated rows. A minimizer (b∗ , w∗ ) defines the population audit score s∗ (f ) = b∗ + w∗⊤ f and the population audit value AUC(s∗ ). The implemented audit has π = 12 and λ = 1/64 (A2). (a) Soundness, and no certificate of closeness. For every Borel score s, |AUC(s) − 12 | ≤ g(dKS (s)) ≤ dKS (s) ≤ TV(PR , PG )

and

|AUC(s) − 21 | ≤ g TV(PR , PG ) .

The bounds are sharp: for every t ∈ [0, 1] there is a standardized pair with dKS (s) = TV = t and AUC(s) = 1 1 1 2 + g(t) for the score s(f ) = f1 . A population AUC of 2 ± ε (0 ≤ ε ≤ 2 ) therefore certifies TV(PR , PG ) ≥ 1/2 dKS (s) ≥ 1 − (1 − 2ε) ≥ ε, and the same lower bound holds for the laws of the underlying records. No AUC value certifies that the laws are close: (i) for every affine score s(f ) = b + w⊤ f with w ̸= 0 and every a ∈ [0, 1], there is a standardized pair of mutually singular laws (TV = 1) with AUC(s) = a; (ii) for every a ∈ [ 21 , 1), π ∈ (0, 1), λ > 0 and d ≥ 2, there is a standardized pair of mutually singular laws with finite second moments for which Jλ has a unique minimizer and AUC(s∗ ) = a. (b) First-moment blindness. Assume E∥fR ∥ + E∥fG ∥ < ∞. (i) If λ > 0, then Jλ has a unique minimizer, and w∗ = 0 if and only if δ = 0. In that case b∗ = logit π, s∗ is constant and AUC(s∗ ) = 12 . (ii) If λ = 0 and δ = 0, then (logit π, 0) minimizes J0 . Every minimizer has a score equal to logit π almost surely under PR and under PG , so its AUC is 12 . The minimizer is unique when the covariance of f under P̄ is nonsingular. (iii) Alignment. If δ ̸= 0, every minimizer of Jλ , for any λ ≥ 0, satisfies δ ⊤ w∗ > 0. In particular w∗ ̸= 0 and AUC(s∗ ) > 0. (iv) Hence, for λ > 0 (and for λ = 0 whenever a minimizer exists), the population audit value departs from 1 1 2 only if δ ̸= 0. It equals 2 whenever the 644 feature means agree, whatever other differences coexist with equal means. These include differences in the variances and dependence of the summaries, and differences in properties of the records that f does not summarize (cross-channel dynamics, lags beyond 1, within-stay distributions beyond the standard deviation). The mean of an imputed coordinate combines the probability that the summary is undefined with its mean where it is defined. (v) Strong penalty. If δ ̸= 0, then
λw∗ → π(1 − π) δ as λ → ∞, so w∗ /∥w∗ ∥ → δ/∥δ∥, and AUC(s∗ ) → AUC(δ ⊤ f ) whenever P δ ⊤ (fG − fR ) = 0 = 0. (vi) A mean difference does not force AUC(s∗ ) > 21 : for every a ∈ (0, 12 ], π ∈ (0, 1), λ > 0 and d ≥ 1, there is a standardized pair with δ ̸= 0 and TV(PR , PG ) = 1 for which AUC(s∗ ) = a. (c) Gaussian calibration. Suppose PR = N (µR , Σ) and PG = N
(µG , Σ) with Σ ≻ 0, and let DM := (δ ⊤ Σ−1 δ)1/2 . (i) A linear score w⊤ f has AUC = Φ w⊤ δ/(2w⊤ Σw)1/2 , and 12 if w = 0. The largest AUC of 39

√ any score, linear or not, is Φ(DM / 2), and it is attained by w ∝ Σ−1 δ. (ii) TV(PR , PG ) = 2Φ(DM /2) − 1. ∗ (iii) For λ > 0 the minimizer satisfies w∗ = κ(αΣ + λI)−1 δ, where κ = (1 − π)E σ(s √ (fR )) > 0 and α = 1 1 ∗ ′ ∗ ′ ∗ (1 − π)E σ (s (fR )) + π E σ (s (fG )) ∈ (0, 4 ]. If δ ̸= 0, then 2 < AUC(s ) ≤ Φ(DM / 2), with equality if and only if δ is an eigenvector of Σ. For λ = 0 the minimizer is unique, w∗ = Σ−1 δ, and equality holds. √ −1 √
(iv) 1 Consequently a population audit value a ∈ [ 2 , 1) implies DM ≥ 2 Φ (a) and TV ≥ 2Φ Φ−1 (a)/ 2 − 1. Equality holds exactly when the audit direction is efficient, that is, when λ = 0 or δ is an eigenvector of Σ. For the audit values 0.5716 (full dataset), 0.6409 (3,000-patient probe) and the threshold 0.75, these lower bounds are DM ≥ 0.255, 0.510, 0.953 and TV ≥ 0.101, 0.201, 0.366 (rounded down). The distribution-free bound of (a) gives TV ≥ 0.074, 0.152, 0.292. A pass (a ≤ 0.75) bounds DM by 0.954 and TV by 0.367 (rounded up) only when the audit direction is efficient. For the penalized audit no such bound holds: for every a ∈ ( 12 , 1), π ∈ (0, 1), λ > 0 and d ≥ 3 there are standardized Gaussian pairs with a common covariance whose population audit value lies in ( 12 , a], while DM is arbitrarily large and TV is arbitrarily close to 1. (v) If δ = (0, δV ) in a partition −1 2 ⊤ −1 f = (fU , fV ), then DM = δV⊤ Σ−1 V |U δV ≥ δV ΣV V δV , where ΣV |U := ΣV V − ΣV U ΣU U ΣU V . (d) Scope by construction. Assume (A4) and (A4′ ). Let U be the 608 coordinates formed by the four summaries of the 149 channels the generator does not write (the 144 channels other than the 12 written outcome channels and the 5 treatment channels that drive the generator, together with those 5 treatment channels) together with the missing fractions of the 12 written outcome channels. Let V be the remaining 36 coordinates: the mean, within-stay standard deviation and lag-1 autocorrelation of the 12 written channels. Let f˜G be the feature vector of the generated record after its 5 treatment columns are replaced by the real ones, computed with the same imputation and scaling constants; let P̃G be its law and δ̃ := Ef˜G − EfR . (i) In every matched pair f˜G,U = fR,U . Hence PR and P̃G have the same marginal law on U , δ̃U = 0, every score that depends on f only through fU has AUC exactly 12 between them, and the two laws can differ only through the conditional law of fV given fU . (ii) For the pair (PR , P̃G ), ∇w Jλ (logit π, 0) = −π(1 − π)δ̃, which is supported on V . If δ̃V ̸= 0, then P̃G ̸= PR , and for λ > 0 (implemented 1/64) the minimizer exists, is unique and satisfies 2 δ̃V⊤ wV∗ > 0. In the Gaussian model of (c), AUC(s∗ ) > 21 and DM = δ̃V⊤ Σ−1 V |U δ̃V . Without such a model, δ̃V ̸= 0 1 ∗ does not by itself force AUC(s ) > 2 (by (b)(vi)). (iii) The generated treatment values are the single-precision roundings of the real doses, with a rounded dose not exceeding 10−9 set to 0. So PG can differ from P̃G , but only in the 20 treatment coordinates. The threshold 0.75 is a tolerance on how well this audit detects the difference. By (a), passing it certifies neither PG = PR nor any upper bound on TV(PR , PG ). All statements concern the audited generated dataset and its real source. The washout panels were audited separately under a registered protocol with the same features, discriminator and threshold (Methods), with PR and PG redefined as the laws of the 644 summaries of hours 24 to 163 of stays drawn as the panel rosters were. Parts (a) to (c) apply to that audit unchanged. Part (d)(i) holds for it because every channel other than the 12 written outcome channels is copied from the real record and the written channels’ missing cells are restored, although the copied treatment channels did not drive the generated values; the treatment-rounding exception of (d)(iii) does not arise. The panels passed (audit AUC 0.714 to 0.727), but the audit did not flag measurement noise of 0.15 reference standard deviations added to the real records. Assumptions. (A1) Population feature map. The median-imputation values and the centering and scaling constants are fixed at their population values under the pooled law with π = 21 , and they act coordinatewise and identically on both groups. So f = Ψ(X) for one measurable map Ψ of the stay record X restricted to its window, and (PR , PG ) is a standardized pair. PR and PG are the laws of f for the real and the generated record of a stay drawn uniformly from the matched stays of the audited dataset; the audit’s 400 stays are a uniform random subset of the 200,825 matched stays. In the implementation (icare_dt/audit_semisynth_realism.py, l. 80–100 and 155–182), imputation and scaling are refitted on each pooled training fold. The population statement idealizes this. (A2) Population target of the audit. The audit’s statistic is the pooled out-of-fold AUC of LogisticRegression(C=0.1) with the default lbfgs solver, after SimpleImputer(median) and StandardScaler. The solver penalizes only the slope vector, so the intercept is unpenalized, and its objective is the mean log loss plus ∥w∥2 /(2Cn). Five person-grouped folds on 400 matched pairs (each person contributes exactly one real and one generated row) give ntrain = 640, π = 12 exactly and λ = 1/(0.1 · 640) = 1/64 in every training fold. The imputer removes a column undefined for every training row. In the population idealization such a coordinate is the constant 0, which enters Jλ only through the penalty and receives weight 0 (Lemma 3), so the two conventions give the same score. On the probe, 76 of the 644 columns are undefined for all 800 rows (72 lag-1 autocorrelations, 2 means, 2 standard deviations), all in U . The population target is AUC(s∗ ), and parts (a) to (c) are population statements. The shipped number pools the out-of-fold scores of five fitted models over 4002 comparisons, including

40

the 400 within-pair ones, with d = 644 close to ntrain = 640. So it matches AUC(s∗ ) only up to estimation error. The null band and the reproduction are given in the numerical checks. (A3) Moments. E∥fR ∥ + E∥fG ∥ < ∞ wherever the risk Jλ or δ is used. This holds for standardized features. (A4) Generator configuration, for (d). The audited dataset is blended_data_semisynth_ eicu_complex_v2_seed13_faithful_v3_a434a5c. It was generated at commit a434a5c, and its configuration records generator SHA-256 f29b4eea..., which equals that of generate_semisynth.py at a434a5c. The configuration has preserve_observed_missingness, hold_carried_forward_outcomes and preserve_treatment_missingness on, factual_treatment_policy=observed_dose, treatment_off_raw= 0 for all five treatments, max_len= 336 with a head window, and measurement_base_rate= 1 with the other synthetic measurement parameters at 0. The boundedmemory route is off (trait_response_route=legacy, cap 0). The generator changes between a434a5c and the paper commit 82587202 affect only the bounded-memory route and optional counterfactual path outputs, so the factual path used here is unchanged. Line numbers in the proof refer to 82587202. The same configuration governs the 3,000-patient probe used for verification. (A4′ ) Data conditions, for (d). (1) Source identity. For every audited stay, the real frame the generator read (preprocessed_timeseries under its data root, G l. 2489) equals, on the 161 audited columns and the window, the real frame the audit read (preprocessed_timeseries under real_root=.../blended_icu/blended_data). The generator’s configuration does not record its data root, so this is an assumption. (2) Every real cell of the 161 audited columns in the window is finite or missing; none is ±∞. (3) Every finite real treatment value is nonnegative. Conditions (1) to (3) are not verified over all 200,825 stays. They are consistent with the audited pairs. On the probe, the 144 other channels agree exactly, the treatment summaries agree to 1.1 × 10−5 with identical missingness, and the written channels’ missing fractions agree exactly. On the full dataset’s audit summary, the missingfraction difference is 0 on all 161 channels and the 149 unwritten channels have |SMD| ≤ 1.3 × 10−9 . (A5) Gaussian idealization, for (c). The equal-covariance Gaussian model is a calibration device and does not describe the data. Several coordinates are fractions, are bounded, or carry atoms after imputation. (A6) Scope. PR and PG refer to the audited generated dataset and its real eICU source only. The washout panels come from a separate parameterization, with simulated dosing, mean-filled cells and an injected memory route, and their separate audit (Methods) is not covered by (A1)’s sampling clause, (A4) or (A4′ ). Proof. Conventions. Real rows carry the label y = 0 and generated rows the label y = 1. Ē denotes expectation under the mixture in which y ∼ Bernoulli(π) and, given y, f ∼ PR if y = 0 and f ∼ PG if y = 1. The law of f under this mixture is P̄ , and PR ≪ P̄ , PG ≪ P̄ because 0 < π < 1. With ℓ(y, t) := log(1 + et ) − yt we have ℓ(0, t) = log(1 + et ) and ℓ(1, t) = log(1 + e−t ), so Jλ (b, w) = Ē ℓ(y, b + w⊤ f ) + λ2 ∥w∥2 . Moreover ∂t ℓ(y, t) = σ(t) − y ∈ (−1, 1) and ∂t2 ℓ(y, t) = σ ′ (t) = σ(t)(1 − σ(t)) ∈ (0, 41 ], so ℓ(y, ·) is strictly convex. Also ℓ(0, t) ≥ t, ℓ(1, t) ≥ −t and 0 ≤ ℓ(y, t) ≤ log 2 + |t|. Write mR := E∥fR ∥, mG := E∥fG ∥ and θ0 := (logit π, 0). The function g(t) = t − t2 /2 is increasing on [0, 1] and satisfies g(t) ≤ t. Standardizing a construction. Applying one strictly increasing affine map to a coordinate in both groups leaves unchanged the TV, and the AUC and dKS (s) of any score that is a strictly increasing function of that coordinate. Adjoining coordinates that have the same law in both groups and, within each group, are independent of the others also leaves the TV unchanged: TV(µ ⊗ ν, µ′ ⊗ ν) = TV(µ, µ′ ). Indeed, sets A × Rk give R ≥, and for ≤, |(µ ⊗ ν − µ′ ⊗ ν)(A)| = | (µ − µ′ )(Az ) ν(dz)| ≤ TV(µ, µ′ ), with Az the section of A at z. Each construction below is either standardized by these two operations or written directly in standardized form. Lemma 1 (mid-distribution form of the AUC). For a real random variable S with distribution function F put M (x) := P(S < x) + 21 P(S = x) = 21 {F (x) + F (x−)}. Let MR and MG be these functions for SR and SG . Then AUC(s) = E MR (SG ) = 1 − E MG (SR ),

E MR (SR ) = E MG (SG ) = 21 .

Proof. By independence and Fubini, P(SG > SR | SG )+ 21 P(SG = SR | SG ) = MR (SG ), and taking expectations gives the first identity. The events {SG > SR }, {SG < SR } and {SG = SR } partition the sample space, so AUC(s) = 1−{P(SR > SG )+ 21 P(SR = SG )} = 1−EMG (SR ). If S ′ is an independent copy of S, exchangeability gives P(S ′ < S) = P(S ′ > S), so EM (S) = P(S ′ < S) + 12 P(S ′ = S) = 12 . □ 41

Lemma 2 (the population logistic risk). continuously differentiable on R1+d , with ∂b Jλ = Ē[σ(s) − y],

Assume mR + mG < ∞ and λ ≥ 0. Then Jλ is finite, convex and ∇w Jλ = Ē[(σ(s) − y)f ] + λw,

s = b + w⊤ f.

Hence (b, w) minimizes Jλ if and only if both derivatives vanish there. At θ0 , ∂b Jλ (θ0 ) = 0,

∇w Jλ (θ0 ) = −π(1 − π) δ,

Jλ (θ0 ) = H(π) := −π log π − (1 − π) log(1 − π).

If λ > 0, Jλ has exactly one minimizer. Proof. Finiteness follows from 0 ≤ ℓ ≤ log 2 + |b| + ∥w∥ ∥f ∥. The partial derivatives of ℓ(y, b + w⊤ f ) in (b, w) are bounded by 1 and by ∥f ∥, both integrable. Differentiation under Ē is therefore justified by dominated convergence, and the same bound with the continuity of σ makes the gradient continuous. Jλ is convex because ℓ(y, ·) is convex and s is affine in (b, w). For a convex differentiable function on R1+d , a point is a minimizer exactly when the gradient vanishes. At θ0 the score is the constant logit π and σ(s) = π, so ∂b Jλ (θ0 ) = π − Ēy = 0 and ∇w Jλ (θ0 ) = Ē[(π − y)f ] = (1 − π)π EfR − π(1 − π) EfG = −π(1 − π)δ; the value H(π) follows by substitution. Now let λ > 0. From ℓ(0, t) ≥ t and ℓ(1, t) ≥ −t,  Jλ (b, w) ≥ max λ2 ∥w∥2 , (1 − π)(b − ∥w∥mR ), π(−b − ∥w∥mG ) . This tends to ∞ as ∥(b, w)∥ → ∞: if ∥w∥ stays bounded, then |b| → ∞ and one of the last two terms diverges. A continuous coercive function attains its minimum. Let (b1 , w1 ) and (b2 , w2 ) be minimizers with value J ∗ . By convexity their midpoint is also a minimizer. If w1 ̸= w2 , the penalty is strictly convex along the segment and the loss term is convex, so the midpoint would have value < J ∗ . Hence w1 = w2 =: w. The function b 7→ Jλ (b, w) has second derivative Ē σ ′ (b + w⊤ f ) > 0 (the integrand is bounded by 41 ), so it is strictly convex and b1 = b2 . □ Lemma 3 (coordinates without mean information). Write f = (fA , fN ) and w = (wA , wN ), and suppose that within each group fN is independent of fA and has mean 0, with mR + mG < ∞. Then Jλ (b, wA , wN ) ≥ Jλ (b, wA , 0) + λ2 ∥wN ∥2 . ∗ ∗ ) minimizes Jλ (·, ·, 0), which is the risk = 0, and (b∗ , wA Consequently, if λ > 0, every minimizer has wN computed from fA alone. Proof. Given (y, fA ), independence within each group gives E[fN | y, fA ] = E[fN | y] = 0. Conditional ⊤ ⊤ ⊤ Jensen for the convex function ℓ(y, ·) gives Ē[ℓ(y, b + wA fA + wN fN ) | y, fA ] ≥ ℓ(y, b + wA fA ). Take expecλ λ 2 2 ∗ tations and add the penalties, which split as 2 ∥wA ∥ + 2 ∥wN ∥ . If λ > 0 and a minimizer had wN ̸= 0, then ∗ ∗ ∗ ∗ Jλ (b , wA , 0) < Jλ (b , w ), a contradiction. □

Proof of (a): the bounds. Write Fc := Fcs . Scores without atoms. Suppose first that SR and SG have no atoms. Then Mc = Fc , and Lemma 1 gives both AUC(s) − 12 = E[FR (SG ) − FG (SG )] and AUC(s) − 12 = E[FR (SR ) − FG (SR )]. R Average the two. With D := FR − FG and Q := 21 {L(SR ) + L(SG )}, this gives AUC(s) − 12 = D dQ. The distribution function of Q is FQ = 21 (FR + FG ), and pointwise |D| ≤ dKS (s), |D| ≤ max(FR , FG ) ≤ 2FQ , |D| ≤ max(1 − FR , 1 − FG ) ≤ 2(1 − FQ ). R So |AUC(s) − 12 | ≤ h(FQ ) dQ with h(u) := min{dKS (s), 2u, 2(1 − u)}. Since Q has no atoms, FQ (Z) is uniform on (0, 1) for Z ∼ Q. Hence Z Z 1 2 2 h(FQ ) dQ = h(u) du = dKS4(s) + dKS (s)(1 − dKS (s)) + dKS4(s) = g(dKS (s)). 0

General scores. Let UR and UG be uniform on (0, 1), independent of each other and of (fR , fG ). For ϵ > 0 R1 put Scϵ := Sc + ϵUc . These variables have no atoms, and their distribution functions Fcϵ (x) = 0 Fc (x − ϵu) du ϵ ϵ > SR ) − 21 | ≤ g(dKS (s)). As ϵ ↓ 0, the indicator satisfy |FRϵ − FGϵ | ≤ dKS (s). By the atomless case, |P(SG ϵ ϵ 1{SG > SR } converges to 1{SG > SR } on {SG ̸= SR }. On {SG = SR } it equals 1{UG > UR }, which is ϵ ϵ independent of that event and has probability 21 . By dominated convergence, P(SG > SR ) → AUC(s). Hence 1 |AUC(s) − 2 | ≤ g(dKS (s)) for every score. The chain. For every x, FR (x) − FG (x) = PR (Ax ) − PG (Ax ) with the Borel set Ax := s−1 ((−∞, x]), so dKS (s) ≤ TV(PR , PG ). As g is increasing on [0, 1] with g(t) ≤ t, we get g(dKS (s)) ≤ dKS (s) ≤ TV and g(dKS (s)) ≤ g(TV). 42

Attainment. Fix t ∈ [0, 1]. Let a raw variable r have law (1 − t) U(0, 1) + t δ−1 in the real group and (1 − t) U(0, 1) + t δ2 in the generated group. Let f1 be the pooled standardization of r, an increasing affine map (the pooled variance of r is positive), let f2 , . . . , fd be independent N (0, 1) in both groups, and take s(f ) = f1 . The pair is standardized. Two uniform draws compare with value 21 . Every comparison that involves the generated atom at 2 or the real atom at −1 is won by the generated draw. Hence AUC = 21 (1−t)2 +2t(1−t)+t2 = 12 +g(t). In the scale of r, FR −FG = t on [−1, 2) and 0 elsewhere, so dKS (s) = t, and the two laws of r differ by t(δ−1 −δ2 ), so their TV is t. By the conventions these values carry over to f . Certificate. If |AUC(s) − 12 | = ε, then g(dKS (s)) ≥ ε. With rε := (1 − 2ε)1/2 , g(1 − rε ) = (1 − rε )(1 + rε )/2 = ε. Since g is increasing on [0, 1], dKS (s) ≥ 1 − rε . Moreover 1 − rε ≥ ε because (1 − ε)2 ≥ 1 − 2ε. For the records, f = Ψ(X) with Ψ measurable (A1), and {f ∈ A} = {X ∈ Ψ−1 (A)}. So TV(PR , PG ) ≤ TV(L(XR ), L(XG )). □ Proof of (a)(i). Let s(f ) = b + w⊤ f with w ̸= 0 and let a ∈ [0, 1]. Put vj := 1 if wj ≥ 0 and vj := −1 otherwise, so vj2 = 1 and w⊤ v = ∥w∥1 > 0. Let f = Xv for a scalar X whose law in each group is given below. Then every coordinate fj = vj X has pooled mean vj ĒX and pooled variance VarP̄ (X), and s(f ) = b + ∥w∥1 X is strictly increasing in X. In every case below X takes finitely many values, with disjoint supports in the two groups, so the laws of f have disjoint finite supports and TV = 1. For a ∈ (0, 1), let X = 0 in the real group, and in the generated group let X take the values t(1−a) and −ta with probabilities a and 1−a, where t := (πa(1−a))−1/2 . Then ĒX = πt{a(1 − a) − (1 − a)a} = 0, VarP̄ (X) = πt2 {a(1 − a)2 + (1 − a)a2 } = πt2 a(1 − a) = 1, and AUC(s) = P(XG > 0) = a. For a = 1, let X = −(π/(1 − π))1/2 in the real group and X = ((1 − π)/π)1/2 in the generated group. The pooled mean is −(π(1 − π))1/2 + (π(1 − π))1/2 = 0, the pooled variance is π + (1 − π) = 1, and AUC(s) = 1. For a = 0 exchange the two signs. In every case the pair is standardized. □ Proof of (b). (i) Let λ > 0. By Lemma 2, ∇Jλ (θ0 ) = (0, −π(1 − π)δ). If δ = 0, then θ0 is a stationary point. It is therefore the minimizer, which is unique by Lemma 2, so w∗ = 0. Conversely, suppose w∗ = 0. Then b∗ minimizes b 7→ Jλ (b, 0) = (1 − π) log(1 + eb ) + π log(1 + e−b ), whose derivative σ(b) − π vanishes only at logit π. So (b∗ , w∗ ) = θ0 , and 0 = ∇w Jλ (θ0 ) = −π(1 − π)δ forces δ = 0. With w∗ = 0 the score is the constant logit π, so SG = SR surely and AUC(s∗ ) = 12 . (ii) Let λ = 0 and δ = 0. By Lemma 2, θ0 is stationary, and hence a minimizer of the convex function J0 . Let θ1 = (b1 , w1 ) be any minimizer, and put ∆ := (b1 − logit π) + w1⊤ f and θt := θ0 + t(θ1 − θ0 ). By convexity J0 (θt ) = J ∗ for t ∈ [0, 1]. Suppose P̄ (∆ ̸= 0) > 0. For t ∈ (0, 1), strict convexity of ℓ(y, ·) gives ℓ(y, logit π + t∆) < (1 − t)ℓ(y, logit π) + t ℓ(y, logit π + ∆) on {∆ ̸= 0}, with equality elsewhere. Taking expectations gives J0 (θt ) < J ∗ , a contradiction. Hence b1 + w1⊤ f = logit π holds P̄ -almost surely, and therefore almost surely under PR and under PG . The score is almost surely constant and its AUC is 21 . If CovP̄ (f ) is nonsingular, an almost surely constant w1⊤ f forces w1 = 0, and then b1 = logit π. (iii) Let δ ̸= 0, λ ≥ 0, and let θ∗ = (b∗ , w∗ ) be a minimizer. By Lemma 2, ∇Jλ (θ0 ) ̸= 0, so θ0 is not a minimizer and Jλ (θ∗ ) < Jλ (θ0 ). Convexity and differentiability give Jλ (θ∗ ) ≥ Jλ (θ0 ) + ∇Jλ (θ0 )⊤ (θ∗ − θ0 ) = Jλ (θ0 ) − π(1 − π) δ ⊤ w∗ . Hence π(1 − π) δ ⊤ w∗ ≥ Jλ (θ0 ) − Jλ (θ∗ ) > 0. Then E[w∗⊤ (fG − fR )] = w∗⊤ δ > 0, so P(w∗⊤ fG > w∗⊤ fR ) > 0, and AUC(s∗ ) ≥ P(SG > SR ) > 0. (iv) The population audit value depends on the laws of the records only through (PR , PG ). By (i) and (ii) it equals 21 on the set {δ = 0}, which restricts the laws only through their means. For the imputation remark, suppose the raw summary rj is undefined with probability νc,j in group c, is imputed by mj , and is standardized as (r − cj )/τj . Then Efc,j = {(1 − νc,j ) E(rj | rj defined, c) + νc,j mj − cj }/τj . (v) Let θλ = (bλ , wλ ) be the minimizer for λ > 0. From Jλ (θλ ) ≤ Jλ (θ0 ) = H(π) we get ∥wλ ∥2 ≤ 2H(π)/λ → 0. Once ∥wλ ∥ ≤ 1, the lower bounds in the proof of Lemma 2 give −H(π)/π − mG ≤ bλ ≤ H(π)/(1 − π) + mR . Take any subsequence along which bλ → b̄. Dominated convergence in Ē[σ(bλ + wλ⊤ f ) − y] = 0 gives σ(b̄) = π, so bλ → logit π. Dominated convergence in the second first-order condition, with the integrand bounded by ∥f ∥, then gives   λwλ = Ē (y − σ(bλ + wλ⊤ f ))f −→ Ē[(y − π)f ] = π(1 − π) δ. If δ ̸= 0, then wλ ̸= 0 for large λ, and uλ := wλ /∥wλ ∥ = λwλ /∥λwλ ∥ → u := δ/∥δ∥. The AUC of s∗ equals that ∗ of u⊤ λ f , because the intercept and a positive factor do not change the ordering. With ∆ := fG − fR , AUC(s ) = 1 ⊤ ⊤ ⊤ ⊤ ⊤ E[1{uλ ∆ > 0} + 2 1{uλ ∆ = 0}]. On the event {u ∆ ̸= 0}, uλ ∆ → u ∆, so the integrand converges to 1{u⊤ ∆ > 0}. If this event has probability one, dominated convergence gives AUC(s∗ ) → P(u⊤ ∆ > 0) = AUC(δ ⊤ f ). (vi) Let a ∈ (0, 21 ], π ∈ (0, 1), λ > 0, d ≥ 1 and xa := 2(1 − a)/a. Let a raw variable r equal 0 in the real group, and in the generated group equal 1 with probability 1 − a and −xa with probability a. Its pooled mean is r̄ := π{(1−a)−axa } = −π(1−a), and its pooled variance v is positive because r is not constant in the generated group. Put f1 := (r−r̄)/v 1/2 , and let f2 , . . . , fd be independent N (0, 1) in both groups. The pair is standardized, 43

δ1 = −(1 − a)/v 1/2 < 0 and δk = 0 for k ≥ 2, so δ ̸= 0. The point −r̄/v 1/2 carries all the mass of f1 in the real group and none in the generated group, so TV = 1. By Lemma 3, wk∗ = 0 for k ≥ 2. By (iii), δ1 w1∗ > 0, so w1∗ < 0 and s∗ is strictly decreasing in f1 . Hence AUC(s∗ ) = P(fG,1 < fR,1 ) + 12 P(fG,1 = fR,1 ) = P(rG = −xa ) = a, for every λ > 0. □ Proof of (c). We use Stein’s lemma: if X ∼ N (µ, Σ) and h : Rd → R is continuously differentiable with bounded gradient, then E[h(X)(X − µ)] = Σ E∇h(X). (i) For w with w⊤ Σw > 0, independence gives w⊤ fG − w⊤ fR ∼ N (w⊤ δ, 2w⊤ Σw). This law has no atoms,
⊤ ⊤ 1/2 so AUC = Φ w δ/(2w Σw) . For w = 0 the score is constant and AUC = 12 = Φ(0). By Cauchy–Schwarz, −1 w⊤ δ = (Σ1/2 w)⊤ (Σ−1/2 δ) ≤ (w⊤ Σw)1/2 DM , with equality √ if and only if w = c Σ δ for some c ≥ 0. As Φ is increasing, the largest AUC over linear scores is Φ(DM / 2). Now let s be any score. If δ = 0, then PR = PG and Lemma 1 gives AUC(s) = EMR (SR ) = 21 = Φ(0). Let δ ̸= 0. Let U be uniform on (0, 1) and independent of f , and put T := FR (s(f )−) + U {FR (s(f )) − FR (s(f )−)}. Under PR , T is uniform on (0, 1) (the randomized probability integral transform). Also E[T | f ] = MR (s(f )). By R1 Lemma 1, AUC(s) = E T (fG , U ) = 0 P(T (fG , U ) > u) du. The randomized test 1{T > u} has size 1 − u under PR . By the Neyman–Pearson lemma, its power under PG is at most the power β(1 − u) of the likelihood-ratio test of that size. The likelihood ratio dPG /dPR is a strictly increasing function of L(f ) := δ ⊤ Σ−1 f , whose law has no atoms under either group. For the score L, the test 1{T > u} is therefore the likelihood-ratio test of size √ R1 1 − u. The same identity gives AUC(L) = 0 β(1 − u) du ≥ AUC(s), and AUC(L) = Φ(DM / 2) by the linear formula. (ii) The affine bijection x 7→ Σ−1/2 (x − µR ) maps Borel sets onto Borel sets and so preserves TV. It sends the two laws to N (0, I) and N (Σ−1/2 δ, I), and a rotation, which also preserves TV, sends Σ−1/2 δ to DM e1 . The laws then have densities φ(x1 )ρ(x2:d ) and φ(x1 − DM )ρ(x2:d ), with ρ the standard normal density on Rd−1 . R 1 Using TV(P, Q) = 2 |p − q| for densities, and the fact that φ(x) > φ(x − DM ) if and only if x < DM /2, Z Z 1 TV = 2 |φ(x1 ) − φ(x1 − DM )| dx1 = {φ(x1 ) − φ(x1 − DM )} dx1 = 2Φ(DM /2) − 1. x1 <DM /2

(iii) Let λ > 0. By Lemma 2 (Gaussian laws have all moments) there is a unique minimizer (b∗ , w∗ ); write s∗ = b∗ + w∗⊤ f . Apply Stein’s lemma under PR to hR (x) = σ(b∗ + w∗⊤ x), and under PG to hG (x) = σ(b∗ + w∗⊤ x) − 1. Both have gradient σ ′ (b∗ + w∗⊤ x) w∗ , so E[hc (fc )fc ] = µc E hc (fc ) + Σw∗ E σ ′ (s∗ (fc )). The first-order condition in b reads (1 − π)Eσ(s∗ (fR )) = π E{1 − σ(s∗ (fG ))} =: κ > 0. Substituting into the condition for w, 0 = (1 − π)E[hR (fR )fR ] + π E[hG (fG )fG ] + λw∗ = κµR − κµG + (αΣ + λI)w∗ , so w∗ = κ(αΣ + λI)−1 δ, with α ∈ (0, 41 ] because 0 < σ ′ ≤ 41 . If δ ̸= 0, then w∗⊤ δ = κ δ ⊤ (αΣ + λI)−1 δ > 0, √ so by (i) 12 < AUC(s∗ ) ≤ Φ(DM / 2). By (i), equality holds if and only if w∗ is a positive multiple of Σ−1 δ. Put c := λ/α > 0. The condition (Σ + cI)−1 δ = t Σ−1 δ for some t > 0 is equivalent to (1 − t)δ = tc Σ−1 δ. Since δ ̸= 0 and c > 0, this forces Σ−1 δ, and hence Σδ, to be a multiple of δ. Conversely, if Σδ = νδ, then (Σ + cI)−1 δ = δ/(ν + c) and Σ−1 δ = δ/ν. Now let λ = 0. Under the mixture, P(y = 1 | f ) = σ(β0 + β ⊤ f ) with β = Σ−1 δ and β0 = logit π
− 1 ⊤ −1 −1 µG −µ⊤ µR ). For any (b, w), J0 (b, w)−J0 (β0 , β) = Ē KL Bern(σ(β0 +β ⊤ f )) ∥ Bern(σ(b+w⊤ f )) ≥ RΣ 2 (µG Σ 0. Equality holds if and only if b + w⊤ f = β0 + β ⊤ f P̄ -almost surely, that is, (b, w) = (β0 , β), because CovP̄ (f ) = Σ + π(1 − π)δδ ⊤ ≻ 0. So w∗ = Σ−1 δ and equality holds√in (i). (iv) The audit score is linear, so by (iii) a√= AUC(s∗ ) = Φ(Dλ / 2) with Dλ := w∗⊤ δ/(w∗⊤ Σw∗ )1/2 ≤ √ DM (and Dλ := 0 when δ = 0). Hence DM ≥ 2 Φ−1 (a), and by (ii), TV = 2Φ(DM /2) − 1 ≥ 2Φ(Φ−1 (a)/ 2) − 1. Both hold with √ equality exactly when Dλ =√DM , which by (iii) means λ =√0 or δ an eigenvector of Σ. The numbers are 2 Φ−1 (0.57163) = 0.2553, 2 Φ−1 (0.64095) = 0.5105 and 2 Φ−1 (0.75) = 0.9539, with corresponding 2Φ(D/2) − 1 = 0.1016, 0.2015, 0.3666. The distribution-free values 1 − (1 − 2ε)1/2 are 0.0744, 0.1526 and 0.2929. Lower bounds are rounded down in the statement and the upper limits 0.954 and 0.367 are rounded up. For the last claim, let a ∈ ( 21 , 1), π ∈ (0, 1), λ > 0, d ≥ 3, ρ ∈ [0, 1) and η > 0. Put ση2 := 1 + π(1 − π)η 2 , δ̃ := ση−1 η (e1 − e2 + e3 ) and
P ⊤ ⊤ ⊤ ⊤ ⊤ Σ̃ := ση−2 e1 e⊤ 1 + e2 e2 + ρ(e1 e2 + e2 e1 ) + e3 e3 + k≥4 ek ek , and let PR = N (−π δ̃, Σ̃) and PG = N ((1 − π)δ̃, Σ̃). Then Σ̃ ≻ 0 and the mean difference is δ̃. The pooled mean is 0. The pooled covariance Σ̃ + π(1 − π)δ̃ δ̃ ⊤ has diagonal ση−2 (1 + π(1 − π)η 2 ) = 1 in coordinates 1 to 3 and 44

√ 1 elsewhere, so the pair is standardized. The vectors e± := (e1 ± e2 )/ 2 and e3 are eigenvectors of Σ̃ with √ eigenvalues ν± = ση−2 (1 ± ρ) and ν3 = ση−2 , and δ̃ = ση−1 η( 2 e− + e3 ). Hence 2 DM =

 2η 2 ση−2 η 2 ση−2 2  . + = η2 1 + ν− ν3 1−ρ

By (iii), w∗ ∝ x := (Σ̃ + cI)−1 δ̃ with c := λ/α ≥ 4λ, so Dλ2 = (x⊤ δ̃)2 /(x⊤ Σ̃x). In the eigenbasis, x⊤ δ̃ =

2η 2 ση−2 η 2 ση−2 3η 2 ση−2 + ≤ , ν− + c ν3 + c c

x⊤ Σ̃x ≥

ν3 η 2 ση−2 . (ν3 + c)2

Therefore, using ν3 = ση−2 ≤ 1,   ν3 2 9η 2 (ν3 + c)2 1  2 2 2 1 + = 9η ≤ η B , B := 3 1 + , λ λ c2 c 4λ √ uniformly in ρ (B1/64 = 51). Choose η := 2 Φ−1 (a)/Bλ ; ση does not depend on ρ. Then for every ρ the √ √ population audit value is Φ(Dλ / 2) ≤ Φ(ηBλ / 2) = a, and it exceeds 12 by (iii) because δ̃ ̸= 0. As ρ ↑ 1, DM → ∞ and TV = 2Φ(DM /2) − 1 → 1. ⊤ −1 (v) By the block-inverse formula, (Σ−1 )V V = Σ−1 δ = δV⊤ Σ−1 V |U , so δ Σ V |U δV when δU = 0. Since −1 ΣV U ΣU U ΣU V ⪰ 0, we have 0 ≺ ΣV |U ⪯ ΣV V , and inversion reverses the Loewner order on positive definite matrices. □ Dλ2 ≤

Proof of (a)(ii). Let a ∈ [ 12 , 1), π ∈ (0, 1), λ > 0 and d ≥ 2, and put q := Φ−1 (a) ≥ 0, s20 := 1/{1 + 2π(1 − π)q 2 } √ and η := 2 q s0 . Within each group let the coordinates be independent, with: f1 ∼ N (−πη, s20 ) in the real group and N ((1 − π)η, s20 ) in the generated group; f2 uniform on {−hR , hR } in the real group and on {−hG , hG } in the generated group, where hR := 2−1/2 and hG := {(1 + π)/(2π)}1/2 ; fk ∼ N (0, 1) in both groups for k ≥ 3. All pooled means are 0. The pooled variances are s20 + π(1 − π)η 2 = s20 {1 + 2π(1 − π)q 2 } = 1, 1+π 2 2 (1 − π)h2R + πh2G = 1−π 2 + 2 = 1, and 1. So the pair is standardized. Since hG − hR = 1/(2π) > 0, the set R × {−hR , hR } × Rd−2 has probability 1 under PR and 0 under PG , so the laws are mutually singular. Both have finite second moments, and Lemma 2 gives a unique minimizer. Within each group (f2 , . . . , fd ) is independent of f1 and has mean 0. So by Lemma 3, wk∗ = 0 for k ≥ 2, and (b∗ , w1∗ ) minimizes the risk of f1 alone, a Gaussian ∗ 2 pair with variance s20 and √mean difference η. If q > 0, (c)(iii) with d = 1 gives w1 = κη/(αs0 + λ) > 0, and (c)(i) gives AUC(s∗ ) = Φ(η/( 2s0 )) = Φ(q) = a. If q = 0, then η = 0, w1∗ = 0 by (b)(i), and AUC(s∗ ) = 12 = a. □ Proof of (d). Line numbers refer to icare_dt/generate_semisynth.py (G) and icare_dt/audit_semisynth_realism.py (A) at 82587202. By (A4), the factual path they describe is the one that produced the audited dataset. Fix an audited stay, and write fl(x) for the single-precision rounding of x. 1. Window. The generator truncates the real frame to its first min(T, 336) rows (G l. 2956–2958; max_len= 336, head window) and generates exactly these rows. The audit reads the first 336 rows of both frames (A l. 88). For the generated frame these are all its rows, and for the real frame they are the same min(T, 336) rows. By (A4′ )(1), the real frame is the one the generator read. 2. Other columns. Missing cells of the real frame are filled with channel means (G l. 3000). simulate_patient copies this frame (G l. 5078) and overwrites only the 12 outcome columns and the 5 treatment columns (G l. 5330–5333). It then maps ±∞ to missing and fills missing cells: forward and backward in non-treatment columns, and with 0 in treatment columns (G l. 5335–5341). By (A4′ )(2) no real cell is infinite. So every cell that is finite in the real frame keeps its real value in the 144 columns that are neither outcomes nor treatments. 3. Treatments. Under observed_dose, the real dose is converted to single precision (G l. 2989), declared on when that value is finite and exceeds the off level plus 10−9 (G l. 2993; off level 0), and replayed as the rounded dose when on and as the off level otherwise (G l. 2996, 5210), in a single-precision array (G l. 5182). By (A4′ )(3), a finite real dose x ≥ 0 therefore becomes fl(x) when fl(x) > 10−9 and 0 otherwise. A missing real dose becomes 0 until step 5. 4. Written outcomes. The written values are finite, being clipped to fixed bounds (G l. 5255, 5283). With the hold on, a written cell takes a new value only in hours in which the standardized real value changed (G l. 5275). This fixes the values of the written channels and does not affect their missingness.

45

5. Missingness. With measurement_base_rate= 1 and the other measurement parameters 0, _apply_measurement_process returns the frame unchanged (G l. 4806–4829; called at l. 3095). restore_observed_missingness (G l. 4373–4406, called at l. 3097) then makes every column missing exactly where the real frame is missing, including the treatment columns under preserve_treatment_missingness, and leaves the other cells unchanged. Hence, on the window: the 144 other columns equal the real ones cell by cell; the treatment columns are missing exactly where the real ones are and elsewhere carry the values of step 3; and each written column is finite exactly where the real column is finite (steps 4 and 5 with (A4′ )(2)). After the treatment columns are replaced by the real ones, the generated frame equals the real frame on all 149 unwritten columns and has the real finiteness pattern on the 12 written columns. Each channel’s four summaries depend only on that channel’s column on the window (A l. 90–96), and its missing fraction depends only on the finiteness pattern. Imputation and standardization act coordinatewise with the same constants in both groups (A1), and f˜G uses these constants. So f˜G,j = fR,j for every j ∈ U in every pair. (i) For a uniformly drawn audited stay, the coupling (fR , f˜G ) has f˜G,U = fR,U surely. So the marginal laws on U coincide and δ̃U = 0. If s depends on f only through fU , then s(f˜G ) and s(fR ) have the same law, and Lemma 1 gives AUC(s) = EMR (SG ) = EMR (SR ) = 12 . Disintegrating both laws with respect to the common marginal law of fU shows that they can differ only through the conditional law of fV given fU . (ii) The gradient identity is Lemma 2 at θ0 , applied to (PR , P̃G ); it is supported on V because δ̃U = 0. If δ̃V ̸= 0, the means differ and so do the laws. For λ > 0, Lemma 2 gives existence and uniqueness, and (b)(iii) gives δ̃ ⊤ w∗ = δ̃V⊤ wV∗ > 0. In the Gaussian model of (c), (c)(iii) gives AUC(s∗ ) > 21 and (c)(v) gives DM . The number of finite consecutive pairs is the same in both members of a pair, so the definedness of a written channel’s lag-1 autocorrelation can differ only through the condition that both lagged series have standard deviation at least 10−8 . This affects only coordinates in V . The last sentence of (ii) is (b)(vi). (iii) By steps 3 and 5, the generated frame and the frame with real treatment columns differ only in finite cells of the 5 treatment columns, by the rounding of step 3. Since the constants are shared, fG and f˜G differ only in the 20 coordinates of the treatment channels. The closing sentences of (d) restate (a) and the scope (A6). □ Verification on shipped summaries. On the passing 3,000-patient probe (g2_hold_probe_ptm_a434a5c_20260918_v1, 400 pairs): • the 144 other channels agree exactly on all four summaries; • the treatment summaries agree to at most 1.1 × 10−5 absolute (5.1 × 10−8 relative), with FiO2 , norepinephrine and phenylephrine exactly equal and missingness identical; • the written channels’ missing fractions agree exactly; • the definedness of the written lag-1 autocorrelation differs in 21 of 4,800 cells (creatinine 15, bilirubin 5, lactate 1); • the linear audit restricted to the coordinates in U has out-of-fold AUC 0.5000. On the full dataset’s audit summary (g2_realism_full_root_a434a5c_20260918_v1), the missing-fraction difference is 0 on all 161 channels. Sixteen channels have a nonzero standardized mean difference: the 12 written channels (absolute values 0.025 to 0.217) and 4 treatment channels at the rounding scale (at most 1.3 × 10−9 ; FiO2 exactly 0). The other 144 channels have difference 0, except ventilator mode, whose difference is undefined because it has no finite value in any audited window. Numerical checks. The following are descriptive re-reads of the shipped audit outputs and of the probe’s per-patient summaries (the full dataset ships only aggregate summaries, so nothing below can be checked at the value 0.572 itself). Parts (a) to (c) were checked on the constructions of the proof: the attainment example of (a), the constructions of (a)(i), (a)(ii) and (b)(vi), and the three-coordinate construction of (c)(iv) reproduce their stated values exactly in population computations. For example, at λ = 1/64, π = 21 and a = 0.75, the construction of (c)(iv) with ρ = 0.9999 has population audit value 0.673 with DM = 2.65 and TV = 0.814, and with ρ = 1 − 10−8 it has 0.678 with DM = 264.5 and TV = 1.000. Table S13 gives the calibration of (a) and (c)(iv) for the audit values of Fig. 4b of the main text. • Null band and reproduction. Within-pair label swaps on the probe’s features give an out-of-fold AUC of 0.5011 ± 0.0088 (30 swaps, maximum 0.520) and 0.5005 ± 0.0087 (60 swaps). The band belongs to the probe’s features, so its use for the full-dataset value is a proxy. In stored row order the audit pipeline reproduces 0.630 against the shipped 0.641, and 30 random pair-to-fold assignments give 0.630 to 0.646. The probe and the full dataset should not be treated as one audit population: their values differ by 0.069, 46

√ which is 3.8 times 2 times the pair-bootstrap standard deviation (0.0129), and the root sum of squared standardized mean differences over the 12 written-channel means is 0.62 on the probe and 0.40 on the full dataset. • Efficiency of the ridge direction. On the probe, a smaller penalty (C = 0.01; out-of-fold AUC 0.647) and shrinkage linear discriminant analysis (0.645) exceed the audit’s linear classifier on the same folds (0.630), and a debiased full-covariance estimate of DM on the coordinates in V is 0.563, against the 0.510 implied by 0.641. This is consistent with (c)(iv), which reads the audit value as a lower bound. On the full dataset the root sum of squared standardized mean differences over the 12 written-channel means is 0.404; it equals DM only if those means were uncorrelated, so it is neither a bound nor a calibration. • A flexible classifier separates more. Histogram gradient boosting (scikit-learn defaults, 200 iterations, the same person-grouped folds; one configuration chosen after the audit) reaches out-of-fold AUC 0.735 on all 644 probe features, 0.730 after the dose rounding is removed by copying the real treatment values, and 0.696 on the 36 coordinates in V alone (linear audit on V : 0.640; label-swap null for boosting on V : 0.489 ± 0.014 over 12 swaps). By (a), a fitted score whose population AUC is near 0.73 would certify TV ≥ 0.26 for the probe’s laws, against 0.152 from 0.641. This shows that passing the audit does not certify closeness in distribution. It is not a new audit value. • Scope of (d). Boosting reaches 0.570 on the 20 treatment features alone, which (d)(iii) permits through the dose rounding, and exactly 0.5000 on the coordinates in U once the real treatment values are copied in, as (d)(i) predicts; the linear audit gives 0.5000 on U either way. • What the linear signal is. On the probe the summed squared standardized mean difference over V is 0.382 for the 12 means, 0.041 for the within-stay standard deviations and 0.011 for the lag-1 autocorrelations, and the linear audit on the 12 means alone gives 0.629 against 0.640 on all 36 coordinates. Paired t statistics (generated minus real) reach 18.5 for the respiratory-rate mean; among within-stay standard deviations the largest are SpO2 (−10.6), platelets (+4.8), temperature (+4.0) and invasive mean arterial pressure (+3.1); lag-1 differences are smaller (largest |t| = 2.94, temperature). Hotelling’s T 2 on the paired differences in V gives F (36, 364) = 10.7. On the full dataset the within-stay standard deviation of the generated records is 33% lower for lactate and 12% higher for creatinine; all other written channels change by at most 5%, and lag-1 autocorrelations by at most 0.015. Supplementary Table S13: Calibration of realism-audit values (Proposition 3). For an audit value a, the distributionfree lower bound on the total-variation distance between the real and generated laws of the 644 summaries, 1 −√ (1 − 2|a − 1 1/2 |) (Proposition 3(a)), and the lower bounds implied under an equal-covariance Gaussian idealization, D ≥ 2 Φ−1 (a) M 2 √ −1 and TV ≥ 2Φ(Φ (a)/ 2)−1 (Proposition 3(c)(iv)); the Gaussian bounds are attained only if the audit’s penalized direction is efficient. All lower bounds are rounded down. No audit value gives an upper bound on the distance (Proposition 3(a)(i), (a)(ii) and (c)(iv)). Audit values of the generated dataset are those of Fig. 4b of the main text; probe values use 3,000patient probes. The audit values are sample AUCs with sampling error (pair-bootstrap standard deviation about 0.013 on the probe), and the bounds are evaluated at them as if they were population values. Generator version No correction (probe) Measurement hold only (probe) Treatment missingness only (probe) Both corrections (probe) Both corrections (full dataset) Tolerance

Audit AUC

TV ≥ (any law)

DM ≥ (Gaussian)

TV ≥ (Gaussian)

0.963 0.956 0.767 0.641 0.572 0.750

0.727 0.704 0.317 0.152 0.074 0.292

2.525 2.417 1.029 0.510 0.255 0.953

0.793 0.773 0.393 0.201 0.101 0.366

5.4 Recoverability, amplitude and retention Proposition 4 (Recoverability: the target is trait recovery, amplitude enters through an exact law, and a held record keeps only what its draws see). Standard and specific parts. Parts (a)(i) to (a)(iii) are properties of the generator code, (a)(iv) restates the definition of R∗2 (σ(ζi )) for this target, and (e) follows at once from design exogeneity. Parts (b) to (d), the exact amplitude scaling of the record, the amplitude law of readers affine in the record and the retention of a held record, are specific to the washout panels. Setting. A patient i is an independent draw of the source tape, the generator’s person-level random numbers (propensity parameters, factual and counterfactual noise, dose-harvest and top-up draws) and the trait vector (ζij )5j=1 . E, Var, Cov, corr and SD are moments under this law. Rows t = 0, . . . , 89 precede the decision row t0 = 90, and h = 12. The scored treatment j ∗ has binary doses Ai (t) ∈ {0, 1}, with Ai (t) := 0 for t < 0. 47

The generator’s memory is ei (t) = λ0 ei (t − 1) + Ai (t − 2) with ei (0) = 0 and λ0 = 0.8, and its occupancy is qi (t) := φ1 (ei (t)), where φκ (e) := e/(κ + e). Write ζi := ζij ∗ (mean µζ , variance σζ2 > 0) and Λk := Lkj ∗ . Let k ∗ be the scored channel: heart rate in the registered phenylephrine cell; the benchmark also scores heart rate for norepinephrine and creatinine for NaCl. On channel k at row t the generator outputs, in standardized units, P5 yik (t) = ỹik (t) + c αik (t) tanh Γik (t), Γik (t) := j=1 Lkj ζij qij (t). Here ỹik (t) is the value after every clip that precedes the memory route, qij is the occupancy of treatment j, αik (t) ∈ [0, 1] is the attenuation near a physiological bound and c > 0 is the amplitude cap. 0 0 The draw set is Dik := {0} ∪ {1 ≤ t ≤ 89 : yik (t) ̸= yik (t − 1)}, where y 0 is the standardized clipped source tape. Its elements are ordered d0 = 0 < d1 < · · · < dn with n = nik , and dik (t) := max{d ∈ Dik : d ≤ t}. The held record is Zik (t) := yik (dik (t)) for t ≤ 89, and the hourly record is the case Dik = {0, . . . , 89}. The design is Si := (Ai (0), . . . , Ai (89), Di1 , . . . , Di,12 ). The record is Ri := (Zi , Si ) with Zi := (Zik (t))k≤12, t≤89 . In the counterfactual rollout from t0 under regime a ∈ {0, 1}, j ∗ is fixed at a at every step and the other treatments are drawn by the generator’s policy with common random numbers.PLet eais,j be the memory of a a . The target τi := φ1 (eais,j ) and Γais := j Lk∗ j ζij qis,j treatment j read by the outcome at row t0 + s + 1, qis,j ∗ is the frame-by-regime interaction of Definition 4 on the latent path of channel k . We assume throughout that Var(Ti ) > 0, with Ti as in (a)(i). For a predictor τ̂ , R2 (τ̂ ) := 1 − E(τ − τ̂ )2 /Var(τ ). The population R2 of a feature vector is the R2 of its population least-squares predictor with an intercept. R∗2 (G) := Var E[τ | G]/Var(τ ) is the information content of G. A reader with kinetics K = (ℓ, λ, κ) uses the exposure profile pK (t) := φκ (eK (t)), where eK (t) = λeK (t − 1) + A(t − ℓ) and eK (0) = 0. At the generator’s kinetics K0 = (2, 0.8, 1), pK0 = qi on rows 0, . . . , 89. Let A be the set of amplitudes c > 0 at which (A1) and (R) hold. The assumptions (G1) to (G4), (A1), (R), (K), (P), (H), (E), (L) and (RW) are listed separately, and each part names the ones it uses. (a) Target structure [(G1), (G2), (A1) at the amplitude c].  Ph−1  (i) τi = c Ti with Ti := h−1 s=0 tanh Γ1is − tanh Γ0is . Ti does not depend on c, and its s = 0 summand is 0. s own (ii) eais,j ∗ = λs+1 (ζi , ei (t0 )) + νi , where 0 ei (t0 ) + a (1 − λ0 )/(1 − λ0 ). Hence τi = τ

τ own (ζ, e) :=

h−1

i c Xh tanh Λk∗ ζ qs1 (e) − tanh Λk∗ ζ qs0 (e) , h s=0

 1 − λs0  qsa (e) := φ1 λs+1 . e + a 0 1 − λ0

The map τ own is shared by all patients and dosing schedules. The remainder νi := τi − τ own (ζi , ei (t0 )) is 0 whenever the counterfactual policy, under either regime, gives no treatment j ̸= j ∗ with Lk∗ j ̸= 0 at steps 0, . . . , h − 2. Ph−1 (iii) Let D(e) := h−1 s=0 [qs1 (e) − qs0 (e)] be the mean own-occupancy increment. Then D(0) = D∗ :=
Ph−1 h−1 s=1 φ1 (1 − λs0 )/(1 − λ0 ) = 0.6842 (to four decimals), and for every e ≥ 0 0 ≤ D∗ − D(e) ≤

λ0 (1 − λh0 ) e ≤ 0.3105 e. h(1 − λ0 )

(iv) Among predictors that are functions of ζi alone, τ̄ (ζi ) := E[τi | ζi ] has the smallest mean squared error. 2 Its R2 is Rtrait := R2 (τ̄ ) = 1 − E Var(τ | ζ)/Var(τ ), which is R∗2 (σ(ζi )) by definition, and its recovery is 2 1/2 1 − (1 − Rtrait ) . This bounds only σ(ζi )-measurable predictors. The record Ri is not a function of ζi , 2 so for predictors built on the record Rtrait is a reference level. (b) Exact scaling of the record [(G1), (G2), (G4), (A1)]. For every amplitude c at which (A1) holds, every k ≤ 12 and every t ≤ 89,
0 0 Zik (t; c) = Zik (t) + c Uik (t), Zik (t) := ỹik (dik (t)), Uik (t) := tanh Λk ζi qi (dik (t)) , where Zi0 is the held record of the cap-0 run. Zi0 , Ui , Ti and Si do not depend on c. (c) Amplitude law [(b), (R), (K), (H)]. Fix the reader’s kinetics K. Let its features be (Xi (c), XiS ). Here Xi (c) = aSi + ΦSi Zi (c) ∈ Rp , where the vector aS and the matrix ΦS depend only on K and on the design through its true draw sets; that is, the reader is affine in the record for fixed design. XiS = xS (Si ) collects the reader’s design-only features and a constant, with XiS = 1 if it has none.

48

For square-integrable V let Ṽ := V − Proj(V | X S ) be the residual of the population linear projection on span(X S ). Put X 0 := aS + ΦS Z 0 , UX := ΦS U , Σ0 := Var(X̃ 0 ), ΣU := Var(ŨX ) and θ := Cov(ŨX , T ). Let RS2 be the population R2 of T on X S , and define C0 :=

Var(T ) , θ ⊤ Σ+ Uθ

C1 :=

Var(T ) , θ ⊤ Σ−1 0 θ

L(c) :=

θ ⊤ (Σ0 /c2 + ΣU )−1 θ Var(T )

(c > 0),

0 where Σ+ U is the Moore–Penrose inverse. These objects are built from Z , U , T and the design alone, so none of them depends on c. The law L is defined at every c > 0, whether or not c ∈ A.

(i) For c ∈ A, X̃(c) = X̃ 0 + c ŨX , and the population R2 of τ on (X(c), X S ) is R2 (c) = RS2 + L(c). RS2 does not depend on c. Every R-measurable predictor, linear or not, has R2 ≤ R∗2 (σ(R)); in particular R2 (c) ≤ R∗2 (σ(R)). (ii) Properties of the law. 1/C0 ≤ 1 is the population R2 of T on ŨX , and C1 > 0. The function H(x) := 1/L(x−1/2 ) is concave and strictly increasing on (0, ∞), and H(x) ≥ C0 + C1 x

(x > 0),

lim H(x) = C0 , x↓0

lim H(x)/x = C1 .

x→∞

Equivalently, 1/L(c) ≥ C0 + C1 /c2 , L(c) → 1/C0 as c → ∞, and L(c)/c2 → 1/C1 as c → 0. Equality H(x) = C0 + C1 x holds at one x > 0, equivalently at every x > 0, iff some v ̸= 0 has both Σ0 v and ΣU v in span{θ}. Such a v is a nonzero multiple of Σ−1 0 θ, and the population least-squares coefficient on X̃(c) is then proportional to v at every amplitude; we say that one statistic carries the signal. In that case C0 = 1/corr2 (v ⊤ ŨX , T ),

C1 = Var(T ) Var(v ⊤ X̃ 0 )/Cov(v ⊤ ŨX , T )2 .

One statistic always carries the signal when p = 1. (iii) Consequences for the reader. • For every c ∈ A, R2 (c) − RS2 = L(c) < 1/C0 . The ceiling 1/C0 = supc>0 L(c) is not attained. The limit c → ∞ is a property of the law, which describes the reader only on A. √ • The bar. When RS2 = 0, the bar is a gain of 0.2 (recovery 1− 0.8). If R2 (c)−RS2 ≥ 0.2 at some c ∈ A,
1/2 then C0 < 5 and c ≥ c∗ := C1 /(5 − C0 ) . When one statistic carries the signal, L(c∗ ) = 0.2, so ∗ c is the reader’s exact crossing amplitude if c∗ ∈ A. If 1/C0 ≤ 0.2, the gain is below 0.2 at every c ∈ A. For the total R2 , replace 0.2 by 0.2 − RS2 . • Low signal. If (0, c̄] ⊆ A for some c̄ > 0, then (R2 (c) − RS2 )/c2 → 1/C1 as c ↓ 0. • Change of sampling. Take two samplings of the same target, read at the same kinetics and each satisfying (H), with laws Lold , Lnew and constants C1old , C1new . For ϱ ∈ (0, min(1/C0old , 1/C0new )), let cold (ϱ) and cnew (ϱ) be the amplitudes at which the two laws equal ϱ. Then cnew (ϱ)/cold (ϱ) → (C1new /C1old )1/2 as ϱ ↓ 0. If both sets A contain an interval (0, c̄], then for all small ϱ these are the amplitudes at which the readers’ gains equal ϱ. (iv) Ceiling in the linear regime [(L), (G3), (E)]. Let p = 1 and UX = Λk ζ Ξ for a design-only scalar Ξ with E Ξ ̸= 0. Write T = kT ζ + ε with kT := Cov(T, ζ)/σζ2 ̸= 0, and let Cov(ŨX , ε) = 0 (assumption (E)). Put CV(Ξ) := SD(Ξ)/|E Ξ|, Ξ̃ := Ξ − Proj(Ξ | X S ) and ρ2lin := corr2 (T, ζ) = kT2 σζ2 /Var(T ). Then C0 =

1 + CV2 (Ξ) + (µ2ζ /σζ2 ) Var(Ξ̃)/(E Ξ)2 . ρ2lin

Two single-channel readers of channel k are special cases; sums over j run over that channel’s draws. P • Cadence-aware reader at kinetics K: statistic Cik (K) := j ∆j Zik ∆j pK , design-only feature P P (0) (0) 2 Qik (K) := j (∆j pK ) , and Ξ = Ξik (K) := j ∆j pK ∆j qi . At K = K0 , Ξ = Qik is itself a (0)

feature, so Ξ̃ = 0 and C0 = (1 + CV2 (Qik ))/ρ2lin . At other K the general formula applies with Ξ̃ (0) the residual of Ξik (K) on (1, Qik (K)); then Var(Ξ̃) > 0 unless Ξik (K) is almost surely an affine (0) function of Qik (K). P89 S • Plain reader at kinetics P K: statistic Oik (K) := t=0 (ΠpK )(t)Zik (t), no design-only feature (X
2= 2 2 2 1), and Ξ = Mik (K) := t (ΠpK )(t) qi (dik (t)). At every K, C0 = 1+(1+µζ /σζ ) CV (Mik (K)) /ρlin , and Mik (K0 ) = Mik . 49

(v) Two-point constants [(ii)]. For 0 < c2 < c1 put ϱℓ := L(cℓ ). Let Ĉ0 + Ĉ1 /c2 be the line in 1/c2 through the −2 points (c−2 1 , 1/ϱ1 ) and (c2 , 1/ϱ2 ). Then: • Ĉ0 ≥ C0 and Ĉ1 ≥ C1 ; • the two-point prediction 1/(Ĉ0 + Ĉ1 /c2 ) is at least L(c) for c ∈ [c2 , c1 ] and at most L(c) for c ∈ / [c2 , c1 ]; • all of these relations hold with equality when one statistic carries the signal. So the two-point ceiling 1/Ĉ0 is at most the population ceiling 1/C0 , and the two-point low-signal constant is at least C1 . When c1 , c2 ∈ A, ϱ1 and ϱ2 are the reader’s population gains. (d) What a held record retains [(G2), (G4), (A1), (L), (RW)]. Fix a channel k with Λk ̸= 0, condition on Si and drop the indices i, k. By (A1), (L) and (RW), y(t) = m(t) + b(t) + cΛk ζq(t) for t ≤ 89. Here m is a known S-measurable mean path (m ≡ 0 is allowed) and b is the random walk of (RW). For 0 ≤ j < n write ∆j f := f (dj+1 ) − f (dj ) and gj := dj+1 − dj . Put Q◦ :=

P89

Q(0) :=

P

2 t=1 (q(t) − q(t − 1)) ,

2 j (∆j q) , Pdj+1 Ej := t=d (q(t) − q(t − 1))2 , j +1

C eff :=

J :=

P

2 j (∆j q) /gj ,

Q(1) :=

P

2 j gj (∆j q) ,

Etail :=

P89

2 t=dn +1 (q(t) − q(t − 1)) ,

P

j ∆j (Z − m) ∆j q/gj .

(i) The held record (Z(t))t≤89 and the draw values (y(dj ))j≤n generate the same σ-field. (ii) C eff is sufficient for ζ, and C eff ∼ N (cΛk ζJ, σb2 J). The Fisher information about ζ is (cΛk /σb )2 J; for the hourly record it is (cΛk /σb )2 Q◦ . (iii) Let Q◦ > 0. For Ej > 0 put wj := Ej /Q◦ and ψj := (∆j q)2 /(gj Ej ). Then ψj ∈ [0, 1]; ψj = 1 iff q is affine on {dj , . . . , dj+1 }; ψj = 0 iff q(dj+1 ) = q(dj ); and χ :=

X J = wj ψj , ◦ Q j: Ej >0

X j: Ej >0

wj = 1 −

Etail . Q◦

So χ ∈ [0, 1]. Interval j contributes ψj Ej = (∆j q)2 /gj , which depends only on the net change of exposure between its two draws, and exposure energy after the last draw is lost. For the generator’s profile (binary doses, λ0 = 4/5) more holds. For 0 ≤ t < t′ ≤ 89, q(t) = q(t′ ) iff q(t′ ) = 0 iff A(u) = 0 for every u ≤ t′ − 2. Hence ψj > 0 for every j with Ej > 0, so no interval over which the exposure moves is fully aliased. This exact statement carries little practical weight: alignment factors as small as 1.6 × 10−9 on heart rate and 1.1 × 10−13 on creatinine occur (numerical checks), so near-total aliasing does occur, and the retention χ is the quantity that measures it. Moreover J = 0 iff Q(0) = 0 iff q(dn ) = 0 iff no dose is given on rows 0, . . . , dn − 2. (iv) Let J > 0. The held record at amplitude c and the hourly record at amplitude c′ have equal Fisher information about ζ iff c′ = c χ1/2 , and this is also the condition under which the standardized laws of their sufficient statistics C eff coincide for every ζ. If J = 0, the held record carries no information about ζ at any amplitude. (v) Pooled over patients [(E)]. Let S be random and let (RW) hold given S, with the baseline independent of (ζ, ε) given S. Let T = kT ζ + ε with kT ̸= 0 and S ⊥ (ζ, ε), and let EJ > 0. On hourly records J = Q(0) = Q(1) = Q◦ . (1) Efficient statistic. The reader with features (C eff , J) satisfies (H) and has RS2 = 0, C0 = (1 + CV2 (J))/ρ2lin ,

C1 = σb2 /(Λ2k σζ2 ρ2lin EJ).

By (c)(iii), its low-signal amplitude factor from hourly (EQ◦ /EJ)1/2 . WithP to held records is raw eff out the mean correction, the reader uses C := + mq , where mq := j ∆j Z ∆j q/gj = C P raw ∆ m ∆ q/g . The reader with features (C , J) also satisfies (H), has the same C0 , and has j j j j

C1 = σb2 EJ + Var(m̃q ) / Λ2k σζ2 ρ2lin (EJ)2 , where m̃q is the residual of mq on (1, J).

50

(2) Information content. If moreover ζ ∼ N (µζ , σζ2 ), the channel’s record R(k) := ((Z(t))t≤89 , S) has trait   information content Var E[ζ | R(k) ]/σζ2 = E ςJ/(1 + ςJ) , where ς := c2 Λ2k σζ2 /σb2 . The amplitudes at which held and hourly records reach a common level ϱ satisfy cheld (ϱ)/chourly (ϱ) → (EQ◦ /EJ)1/2 as ϱ ↓ 0. P (3) Implemented cadence-aware statistic at K0 [(K)]. Let C m := j ∆j (Z − m) ∆j q, which is Cik when m ≡ 0. The reader with features (C m , Q(0) ) satisfies (H) and has RS2 = 0, C0 = (1 + CV2 (Q(0) ))/ρ2lin ,

C1 = σb2 EQ(1) /(Λ2k σζ2 ρ2lin (EQ(0) )2 ).

Its low-signal amplitude factor from hourly to held records is (EQ◦ EQ(1) )1/2 /EQ(0) ≥ (EQ◦ /EJ)1/2 . (e) Exposure is necessary [(G1) to (G4); (A1) is not needed]. Let Ni := {qi (d) = 0 for every d ∈ Dik and every k with Λk ̸= 0}. This event is determined by Si . Under binary dosing it is the event that Ai (u) = 0 for every u ≤ max Dik − 2 and every k with Λk ̸= 0. On Ni , Zi = Zi0 , and for every bounded measurable f , E[f (ζi ) | Ri ] = Ef (ζi ) almost surely; in particular E[τ̄ (ζi ) | Ri ] = E τ̄ (ζi ) there. Suppose P (Ni ) = 1, as in a panel with no pre-decision dose (Ai ≡ 0 on rows 0, . . . , 89, so that qi ≡ 0 there). Then Zi = Zi0 for every patient, and every R-measurable predictor τ̂ of that panel’s target has R2 (τ̂ ) ≤ R∗2 (σ(R)) ≤

E Var(τ | ζ) 2 = 1 − Rtrait , Var(τ )

with both moments computed for that panel. The target still depends on the trait: by (a), under (A1), its own-treatment part is τ own (ζi , 0). Assumptions. (G1) Shared paths and no feedback (code fact). The memory-route shift c α tanh Γ is added after every state-dependent clip. The state transition, the policy and the memory recursion do not read it, and it draws no random numbers (response_memory_outcome_shift, apply_response_memory_bounded_shift, simulate_patient and simulate_counterfactuals in icare_dt/generate_semisynth.py). The factual runs at cap 0 and cap c use the same tape, doses and seed. Both counterfactual frames start from the cap-0 factual trace with the same seed (simulate_person in icare_dt/revision_v3_g2_washout_occupancy_benchmark_v1.py, which checks that the two factual traces agree except for the recorded outcome). Doses are harvested with the generator’s policy, which reads only the latent state, and are topped up with a seed fixed by the person index. Common random numbers across regimes are implemented but the proofs do not need them. (G2) Dosing and memory (code fact). In the factual record only j ∗ is dosed, and doses are binary. NATMIN gives the policy’s natural doses on rows 6 to 53 and, if fewer than 8 rows are dosed, adds dosed rows to reach 8. FIX19 doses rows 6 to 24. Neither schedule doses at or after row 54, so Ai (89) = 0. Factual memory: e(t) = 0.8 e(t − 1) + A(t − 2) with e(0) = 0 and saturation constant 1. Counterfactual memory: eai,−1,j = eij (t0 ) and eais,j = 0.8 eai,s−1,j + Aij (t0 + s − 1). Here Aij (t0 − 1) is the factual dose and, for s ≥ 1, Aij (t0 + s − 1) is the action at step s − 1 (a for j ∗ , the policy draw otherwise). The outcome at row t0 + s + 1 reads eais,· . (G3) Trait independence (code fact plus a modeling convention). The traits come from a separate random stream and enter only the shift. We model distinct pseudo-random streams and the tape draw as independent random variables. Then the trait vector (ζij )j is independent of the tape, the factual noise and the dose-harvest and top-up draws, hence of (Zi0 , Si ) (Definition 5; all 800 NATMIN dose columns were reconstructed from the shipped u0 ). Tapes are drawn without replacement from a finite list, so treating patients as independent draws is an idealization. (G4) Hold (code fact). At a row outside Dik the record repeats the previous recorded value, and row 0 is always recorded. Dik is defined by an equality test on the standardized clipped source tape, so it depends on the tape only. (A1) Attenuation inactive (a property of an amplitude c; used by (a) to (d)). αik (t) = 1 wherever the shift is nonzero, both on the pre-decision record and on the h counterfactual rows of channel k ∗ in frame A under both regimes, and the final raw-unit clip does not bind. It holds when at least c standardized units of room remain in the direction of the shift.

51

• Heart rate: in the 240-person scaling check (A1) never failed, and the shipped per-patient targets are affine in c to 1.6 × 10−7 for phenylephrine and 2.4 × 10−7 for norepinephrine. • Phenylephrine’s other carriers: in the same check (A1) failed for 9 persons on white cells and 6 on platelets (maximum deviation 0.084). • NaCl on creatinine: the shipped targets are not affine in c for 56 to 62 of 400 persons (maximum residual 0.028, root mean square 0.0019 to 0.0024, against a target SD of 0.248). So (a) to (c) hold only approximately for that pair. Statements are in exact arithmetic; the generator computes in single precision. (R) Draw detection (a property of an amplitude c; used by (c)). The reader takes as draws the rows at which the recorded value changes, row 0 included, and this set equals Dik . Then aS and ΦS depend only on the design and on K. Consecutive draw values differ with probability one when no state-dependent clip binds at either draw, because ỹ contains independent Gaussian outcome noise at every row. Such a clip can bind at two consecutive draws on a channel whose values sit at a physiological bound; this is plausible for SpO2 , a carrier of NaCl’s response. There the cap-0 values coincide while the cap-c values can differ, so the detected set, and with it ΦS , can depend on c. This bears only on the NaCl numbers and was not quantified. (K) Fixed kinetics (used by (c) and (d)(v)(3)). The reader’s kinetics K are fixed in advance. The shipped readers choose (ℓ, λ) by cross-validation with κ = 1. • Cadence-aware reader on heart rate: it picked (2, 0.95) in all five folds on hourly records (job 10679) and (2, 0.8) in all five on held records (job 10678). • Plain reader, job 10679: it picked (2, 0.65) twice and (2, 0.8) three times. Fitted constants compared across cadences can therefore mix kinetics. (P) Population readers (used by (c)). (c) is a statement about population least squares with an intercept at fixed kinetics. The shipped readers use ridge regression with a nested cross-validated penalty and kinetics chosen from a grid, and their held-out R2 estimates the population value with a finite-sample loss. Take ordinary least squares with an intercept on p features (design-only features included) and n training rows, and suppose the features are Gaussian and the regression residual is independent of them with constant variance. Then the expected held-out loss relative to Var(y) is exactly (1 − R2 )(p + 1 − 2/n)/(n − p − 2). With ntr = 320, p = 24 and R2 = 0.2 this is 0.068. For the shipped ridge readers the expression is an approximation. (H) Orthogonality (used by (c)). Cov(X̃ 0 , T ) = 0, Cov(X̃ 0 , ŨX ) = 0, Σ0 ≻ 0 and θ ̸= 0. Exact sufficient conditions under (G3): • the second condition holds if ΦS E[U | S] ∈ span(X S ) or E[X 0 | S] ∈ span(X S ); • the first holds if E[T | Z 0 , S] ∈ span(X S ). In the generator’s linear regime, E[U | S] = Λ µζ q(d(·)) channelwise, and E[T | Z 0 , S] is a combination of the trait means of the treatments that load the scored channel. So (H) holds up to terms proportional to trait means and to the tanh nonlinearity. For phenylephrine the population trait mean is about 0.057 and the SD about 0.58, so µ2ζ /σζ2 ≈ 0.01 (panel estimates 0.012, 0.004 and 0.010 for phenylephrine, norepinephrine and NaCl). Reader-specific remarks: (0)

• Cadence-aware reader at K0 : the per-channel energies Qik are features and contain ΦS E[U | S] in their span, which removes the first kind of term exactly. At other kinetics the term is removed only (0) to the extent that Ξik (K) is affine in Qik (K). • Plain reader: it has no design-only feature. In the scaling check, response-free record features explained an adjusted R2 of −0.007 of T . (E) Design-noise orthogonality (idealization; used by (c)(iv) and (d)(v)). With T = kT ζ + ε as in (c)(iv), Cov(ŨX , ε) = 0. A sufficient condition is S ⊥ (ζ, ε), which (d)(v) assumes. In the generator, ζ ⊥ S holds by (G3), but S ⊥ ε holds only approximately. ε contains the part of τ own that is nonlinear in ζ, which depends on ei (t0 ), a function of the doses. It also contains νi , which depends on the counterfactual 52

policy; that policy shares the person’s propensity parameters and latent state with the harvest run that produced the doses, and it runs on the tape that produced the draw sets. Measured on a 400-person simulated panel (synthetic tapes, NATMIN, c = 0.75): • Cov(ŨX , ε)/θ = −0.007 (bootstrap SD 0.031) for Ξ = Q(0) ; • Cov(ŨX , ε)/θ = −0.006 (bootstrap SD 0.028) for Ξ = M ; • corr(ε, number of dosed rows) = −0.018. (L) Linear regime (idealization; used by (c)(iv) and (d)). tanh x is replaced by x in the record’s shift, so Uik (t) = Λk ζi qi (dik (t)). The argument reaches |Λk ζq| ≤ 2 · 5/6 ≈ 1.67 at the trait ceiling, where tanh(1.67) = 0.93. (RW) Random-walk baseline (idealization; used by (d) only). There is one carrier k. Given Pt S, its trait-free path is m(t) + b(t), where m is a known S-measurable mean path and b(t) = b(0) + u=1 ξu with ξu i.i.d. N (0, σb2 ), σb > 0 known, independent of b(0) and of the traits. The generator’s trait-free action route is linear time-invariant in the doses when its clips are inactive, so it is such an m. This idealizes four things: • It treats one carrier, whereas the generator writes phenylephrine’s response into heart rate, pH, white cells and platelets, and NaCl’s into mean arterial pressure, SpO2 , lactate, respiratory rate, urine output, creatinine, platelets and bilirubin. • The real baseline is persistent but not a random walk: the detrended heart-rate lag-1 autocorrelation is 0.69. The white-noise baseline of Proposition 5 is the other bracket. • Draw sets are not independent of the baseline, since both come from the real tape. • (d)(v)(2) also treats the trait as Gaussian, whereas the generator’s trait is a censored Gaussian with an atom of about 15% at −2/3. Proof. Conventions. Arguments are in exact arithmetic. Every generator variable used is bounded (|Ti | < 2, |yik (t)| ≤ 8 + c, 0 ≤ q < 5/6 under binary dosing, gaps at most 89), and (RW) adds Gaussian variables. So all moments below exist. Statements conditional on Si hold almost surely. Proof of (a). (i) Fix i and a. By (G1), the rollouts of frames A and 0 under regime a start from the same factual trace with the same seed. The only operation that reads the cap is the shift, which nothing else reads and which draws no random numbers. By induction over the steps s, the two frames have the same latent state, the same actions of all five treatments and the same noise, and hence by (G2) the same memories eais,· at every step. In particular, the pre-shift value ỹisa of channel k ∗ at row t0 + s + 1 is common to the two frames. Frame 0 has cap 0 and adds nothing. Frame A adds c α tanh Γais with α = 1 by (A1), and by (A1) the raw-unit 0,a A,a clip is inactive. On the latent path in standardized units, therefore, yis = ỹisa + c tanh Γais and yis = ỹisa . The summand of Definition 4 is
A,1 A,0
0,1 0,0
yis − yis − yis − yis = c tanh Γ1is − tanh Γ0is . Averaging over s = 0, . . . , h − 1 gives τi = cTi . Ti is a function of the traits, the loadings and the memories. The memories are functions of the factual doses and of the counterfactual actions, and the actions depend on the latent state and the policy draws, neither of which depends on c by (G1). Hence Ti does not depend on c. At s = 0, (G2) gives eai0,j = 0.8 eij (t0 ) + Aij (t0 − 1). Both terms are factual and therefore equal for a = 0 and a = 1. Hence Γ1i0 = Γ0i0 , and the s = 0 summand is 0. (ii) Unrolling the recursion of (G2) from eai,−1,j ∗ = ei (t0 ) gives s eais,j ∗ = λs+1 0 ei (t0 ) + λ0 Ai (t0 − 1) +

s X

λs−m a. 0

m=1

Ps

a a By (G2) Ai (89) = 0, and m=1 λs−m = (1 − λs0 )/(1 − λ0 ), so qis,j ∗ = qs (ei (t0 )). 0 P s For j ̸= j ∗ the factual memory and dose are 0 (G2), so eais,j = m=1 λs−m ãai,m−1,j , where ãai,m,j is the 0 a policy action at step m. Suppose ãi,m,j = 0 for all m ≤ h − 2, both regimes and every j ̸= j ∗ with Lk∗ j ̸= 0. a Then qis,j = 0 for these j and all s ≤ h − 1, so Γais = Λk∗ ζi qsa (ei (t0 )) and cTi = τ own (ζi , ei (t0 )), that is, νi = 0. The map τ own involves only c, h, λ0 , Λk∗ and φ1 , so it is the same for every patient and dosing schedule.
Ph−1 (iii) Since qs0 (0) = φ1 (0) = 0 and q01 (0) = φ1 (0) = 0, we have D(0) = h−1 s=1 φ1 (1 − λs0 )/(1 − λ0 ) , which evaluates to 0.684222.

53

1 0 Put υs := (1 − λs0 )/(1 − λ0 ) and xs := λs+1 0 e ≥ 0. Then qs (e) = φ1 (υs + xs ) and qs (e) = φ1 (xs ), so, using φ1 (0) = 0, h−1  1 X ∗ φ1 (υs + xs ) − φ1 (υs ) − φ1 (xs ) . D(e) − D = h s=0

φ1 is concave on [0, ∞) with φ1 (0) = 0, hence subadditive, so every bracket is at most 0. Since φ1 is nondecreasing and φ1 (x) ≤ x, every bracket is at least −φ1 (xs ) ≥ −xs . Summing, 0 ≤ D∗ − D(e) ≤ h−1

h−1 X

h λs+1 0 e = λ0 (1 − λ0 )e/ h(1 − λ0 ) ,

s=0

and 0.8(1 − 0.812 )/2.4 = 0.310427 ≤ 0.3105. (iv) For measurable f with Ef (ζi )2 < ∞, the tower property removes the cross term in
2
2 E τ − f (ζ) = E Var(τ | ζ) + E τ̄ (ζ) − f (ζ) , which is minimized exactly when f = τ̄ almost surely. Hence R2 (τ̄ ) = 1 − E Var(τ | ζ)/Var(τ ). Since 2 E[τ − τ̄ (ζ)] = 0, the root mean squared error of τ̄ is (E Var(τ | ζ))1/2 and its recovery is 1 − (1 − Rtrait )1/2 . The argument ranges over σ(ζi )-measurable f only. Proof of (b). Fix c at which (A1) holds. By (G2) only j ∗ is dosed in the factual record, so qij ≡ 0 for j ̸= j ∗ and Γik (t) = Λk ζi qi (t). By (G1) the factual runs at caps 0 and c share ỹik (t) at every row. By (A1), yik (t; c) = ỹik (t) + c tanh(Λk ζi qi (t)), while the cap-0 run has yik (t; 0) = ỹik (t). At a draw row the recorded raw value is µk + σk yik (t), unclipped by (A1), and by (G4) every other row repeats the previous recorded value. Hence the standardized held record is Zik (t; c) = yik (dik (t); c), and the identity follows by substitution. Finally, ỹ depends on the tape, the noise and the doses, and qi depends on the doses. The doses are harvested by the policy, which reads only the latent state (G1), and are topped up with a seed fixed by the person index. Dik depends on the tape only (G4). None of these depends on c, and Ti does not by (a). Proof of (c). Two lemmas are used. Lemma 1 (partial R2 ). Let X S contain a constant, let Y be square integrable with Var(Y ) > 0, and let X be square integrable with Var(X̃) ≻ 0. Then the population R2 of Y on (X, X S ) equals R2 (Y | X S ) + Cov(X̃, Y )⊤ Var(X̃)−1 Cov(X̃, Y )/Var(Y ). Proof. In L2 , span(X, X S ) = span(X S ) ⊕ span(X̃) with orthogonal summands, because every component of X̃ is orthogonal to span(X S ). The projection of Y on an orthogonal sum is the sum of the projections, so the explained variances add. The projection of Y on span(X̃) is β ⊤ X̃ with β = Var(X̃)−1 E[X̃Y ], and E[X̃Y ] = Cov(X̃, Y ) since EX̃ = 0. □ Lemma 2 (constrained minimum). Let N ⪰ 0 be symmetric and θ ∈ range(N) \ {0}. Then min{v ⊤ Nv : ⊤ v θ = 1} = 1/(θ ⊤ N+ θ) > 0, and v attains the minimum iff v ⊤ θ = 1 and Nv ∈ span{θ}. If N ≻ 0, the minimizer is unique: v = N−1 θ/(θ ⊤ N−1 θ). Proof. Write θ = Nw. Then w⊤ Nw = w⊤ NN+ Nw = θ ⊤ N+ θ, and this is positive, since w⊤ Nw = 0 would give Nw = 0. For v ⊤ θ = 1, the Cauchy–Schwarz inequality for the semi-inner product u⊤ Nv gives 1 = (v ⊤ Nw)2 ≤ (v ⊤ Nv)(w⊤ Nw). Equality holds iff N1/2 v and N1/2 w are proportional, that is, iff Nv ∈ span{Nw} = span{θ}. The vector v = N+ θ/(θ ⊤ N+ θ) satisfies v ⊤ θ = 1 and Nv = θ/(θ ⊤ N+ θ), so the bound is attained. If N ≻ 0, the conditions Nv = βθ and v ⊤ θ = 1 determine v. □ (i) Let c ∈ A. By (R) the reader’s detected draw sets are the Dik , so its features are aS + ΦS Z(c) and X S with the objects of the statement. By (A1) and (b), X(c) = X 0 + c UX . Since τ = cT with c > 0 fixed, the R2 of τ equals that of T on the same features. The residual map V 7→ Ṽ is linear and X S does not depend on c, so X̃(c) = X̃ 0 + c ŨX . By (H), Cov(X̃(c), T ) = c θ and Var(X̃(c)) = Σ0 + c2 ΣU ≻ 0. Lemma 1 gives R2 (c) = RS2 + c2 θ ⊤ (Σ0 + c2 ΣU )−1 θ/Var(T ) = RS2 + L(c). RS2 involves only T and X S , which do not depend on c. For any R-measurable τ̂ with finite variance, E(τ − τ̂ )2 ≥ E(τ − E[τ | R])2 = Var(τ ) − Var E[τ | R]. Hence 2 R (τ̂ ) ≤ R∗2 (σ(R)). The population least-squares predictor on (X(c), X S ) is R-measurable. (ii) First, θ ∈ range(ΣU ). If ΣU u = 0, then Var(u⊤ ŨX ) = 0, so u⊤ ŨX is almost surely constant and u⊤ θ = Cov(u⊤ ŨX , T ) = 0; thus θ ⊥ null(ΣU ). Directions with v ⊤ θ = 0 have zero correlation with T . Hence, by Lemma 2, the population R2 of T on ŨX is θ ⊤ Σ+ 1 1 Uθ = = , ⊤Σ v Var(T ) v Var(T ) C ⊤ U 0 v θ=1

sup corr2 (v ⊤ ŨX , T ) = sup v

54

and it is at most 1. C1 > 0 because Σ0 ≻ 0 and θ ̸=
0. For x > 0, H(x) = Var(T )/ θ ⊤ (xΣ0 + ΣU )−1 θ . Since xΣ0 + ΣU ≻ 0, Lemma 2 gives   H(x) = Var(T ) min x v ⊤ Σ0 v + v ⊤ ΣU v . v ⊤ θ=1

For fixed v the bracket is affine in x with slope v ⊤ Σ0 v > 0. A pointwise minimum of affine functions is concave, and here it is nondecreasing. It is strictly increasing: for 0 < x1 < x2 and v2 the minimizer at x2 , H(x1 ) ≤ Var(T )[x1 v2⊤ Σ0 v2 + v2⊤ ΣU v2 ] < H(x2 ). The minimum of a sum over a common set is at least the sum of the minima. Lemma 2, applied to Σ0 and to ΣU , therefore gives h 1 i x H(x) ≥ Var(T ) ⊤ −1 + ⊤ + = C0 + C1 x. θ Σ0 θ θ Σ U θ ⊤ + ⊤ −1 −1 For the limits, let vU := Σ+ U θ/(θ ΣU θ) and v0 := Σ0 θ/(θ Σ0 θ) be the minimizers of the two terms. Inserting them gives ⊤ H(x) ≤ C0 + x Var(T ) vU Σ0 v U , H(x) ≤ C1 x + Var(T ) v0⊤ ΣU v0 .

Together with the lower bound, H(x) → C0 as x ↓ 0 and H(x)/x → C1 as x → ∞. With x = c−2 , these are L(c) → 1/C0 as c → ∞ and L(c)/c2 = x/H(x) → 1/C1 as c → 0. Equality. Over a common constraint set, min(f + g ′ ) = min f + min g ′ iff some point minimizes f and g ′ simultaneously. (If v ∗ minimizes f +g ′ and equality holds, then f (v ∗ )−min f and g ′ (v ∗ )−min g ′ are nonnegative and sum to 0.) For x > 0, x v ⊤ Σ0 v is minimized on {v ⊤ θ = 1} only at v0 . By Lemma 2, v ⊤ ΣU v is minimized exactly by the v with v ⊤ θ = 1 and ΣU v ∈ span{θ}. So equality at some x > 0 holds iff ΣU v0 ∈ span{θ}, a condition free of x. This condition is equivalent to the stated one. If ΣU v0 ∈ span{θ}, then v = v0 is nonzero (because θ ̸= 0) and has Σ0 v0 = θ/(θ ⊤ Σ−1 0 θ). Conversely, let v ̸= 0 with Σ0 v = βθ and ΣU v ∈ span{θ}. Then β ̸= 0 because Σ0 ≻ 0, so v = β Σ−1 θ is a nonzero multiple of v0 , and ΣU v0 ∈ span{θ}. 0 The population least-squares coefficient on X̃(c) is (Σ0 + c2 ΣU )−1 cθ = c−1 (xΣ0 + ΣU )−1 θ with x = c−2 . By Lemma 2 applied to xΣ0 + ΣU ≻ 0, this is proportional to the unique minimizer at x. Under the condition, that minimizer is v0 at every x. Now take v = v0 , so that Cov(v ⊤ ŨX , T ) = v ⊤ θ = 1. Then C1 = Var(T ) v ⊤ Σ0 v = 2 ⊤ ⊤ 0 Var(T )Var(v X̃ )/Cov(v ŨX , T ) . Under the condition v0 also minimizes v ⊤ ΣU v, so C0 = Var(T ) v ⊤ ΣU v = 1/corr2 (v ⊤ ŨX , T ). Both ratios are invariant to rescaling v. For p = 1 every v ̸= 0 satisfies the condition. (iii) By (i), for c ∈ A the gain is L(c) = 1/H(c−2 ) ≤ 1/(C0 + C1 /c2 ) < 1/C0 . Since L(c) → 1/C0 as c → ∞ by (ii), supc>0 L(c) = 1/C0 , and the strict inequality shows that the supremum is not attained. If the gain at c ∈ A is at least 0.2, then 5 ≥ H(c−2 ) ≥ C0 + C1 /c2 . Since C1 /c2 > 0, this forces C0 < 5 and 2 c ≥ C1 /(5 − C0 ). In the one-statistic case H(x) = C0 + C1 x, so L(c∗ ) = 1/(C0 + (5 − C0 )) = 0.2. Because L is strictly increasing in c, L(c) < 0.2 for c < c∗ ; hence, if c∗ ∈ A, it is the reader’s crossing amplitude. If 1/C0 ≤ 0.2, then L(c) < 1/C0 ≤ 0.2 at every c. The total R2 is RS2 + L(c) with RS2 free of c, which gives the stated replacement of 0.2. If (0, c̄] ⊆ A, the gain equals L on that interval, and L(c)/c2 → 1/C1 by (ii). Amplitude ratio. H is concave, hence continuous on (0, ∞), and strictly increasing. So c 7→ L(c) = 1/H(c−2 ) is continuous and strictly increasing, with limits 0 as c → 0 (because L(c) ≤ c2 /C1 ) and 1/C0 as c → ∞. For ϱ in the stated range each law therefore takes the value ϱ at a unique c(ϱ), and c(ϱ) → 0 as ϱ ↓ 0. By the low-signal limit, ϱ = c(ϱ)2 (1 + o(1))/C1 , so c(ϱ)2 = C1 ϱ(1 + o(1)). Applying this to both laws and taking the ratio gives cnew (ϱ)/cold (ϱ) → (C1new /C1old )1/2 . If both sets A contain (0, c̄], then c(ϱ) ≤ c̄ for all small ϱ, and there each reader’s gain equals its law. (iv) Here p = 1. For a design-only w, ζ ⊥ S gives E[ζΞ] = µζ E Ξ and Cov(ζΞ, w) = µζ Cov(Ξ, w). The linear projection depends only on means and covariances, so Proj(ζΞ | X S ) = µζ Proj(Ξ | X S ) and   ŨX = Λk (ζ − µζ )Ξ + µζ Ξ̃ . Ξ̃ has mean 0 because X S contains  a constant. Using ζ ⊥ S and E(ζ − µζ ) = 0, the cross term vanishes, and Var(ŨX ) = Λ2k σζ2 E Ξ2 + µ2ζ Var(Ξ̃) . For the covariance: Cov((ζ − µζ )Ξ, ζ) = E Ξ · E[(ζ − µζ )ζ] = σζ2 E Ξ, and Cov(Ξ̃, ζ) = E Ξ̃ E ζ = 0. By (E), Cov(ŨX , ε) = 0. So θ = Cov(ŨX , T ) = Λk kT σζ2 E Ξ ̸= 0. With Var(T ) = kT2 σζ2 /ρ2lin and E Ξ2 /(E Ξ)2 = 1 + CV2 (Ξ), 2

C0 =

σζ2 E Ξ2 + µ2ζ Var(Ξ̃) 1 + CV (Ξ) + (µ2ζ /σζ2 )Var(Ξ̃)/(E Ξ)2 Var(T ) Var(ŨX ) = = . ρ2lin σζ2 (E Ξ)2 ρ2lin θ2

0 0 The two readers. In the linear regime, Zik (t) = Zik (t) + cΛk ζi qi (dik (t)), so ∆j Zik = ∆j Zik + cΛk ζi ∆j qi .

55

0 j ∆j Zik ∆j pK + cΛk ζi Ξik (K). So UX = Λk ζi Ξik (K), and the reader’s (0) (0) design-only features are (1, Qik (K)). At K = K0 , pK0 = qi on rows 0, . . . , 89, so Ξik (K0 ) = Qik (K0 ) = (0) Qik lies in span(X S ) and Ξ̃ = 0. At other K, Var(Ξ̃) = 0 iff Ξik (K) is almost surely an affine function of (0) Qik (K).

• Cadence-aware: Cik (K) =

• Plain: Oik (K) = CV2 (Ξ).

P

P

0 S 2 t (ΠpK )(t)Zik (t) + cΛk ζi Mik (K) and X = 1. So Ξ̃ = Ξ − E Ξ, and Var(Ξ̃)/(E Ξ) =

Finally, if S ⊥ (ζ, ε), then E[(ζ − µζ )Ξε] = E Ξ · Cov(ζ, ε) = 0 (by the definition of kT ) and Cov(Ξ̃, ε) = 0. Hence Cov(ŨX , ε) = 0. (v) By (ii), H is concave and strictly increasing on (0, ∞), H(x) → C0 as x ↓ 0 and H(x)/x → C1 as x → ∞. Put xℓ := c−2 ℓ , so x1 < x2 , and let L(x) := Ĉ0 + Ĉ1 x be the line through (x1 , H(x1 )) and (x2 , H(x2 )); thus Ĉ1 = (H(x2 ) − H(x1 ))/(x2 − x1 ). For a concave function and u < v < w, the chord slopes satisfy H(w) − H(u) H(w) − H(v) H(v) − H(u) ≥ ≥ . v−u w−u w−v We apply this three times. • For x ∈ (0, x1 ), take (u, v, w) = (x, x1 , x2 ): (H(x1 ) − H(x))/(x1 − x) ≥ Ĉ1 , so H(x) ≤ L(x). Letting x ↓ 0 gives C0 ≤ Ĉ0 . • For y > x2 , take (u, v, w) = (x1 , x2 , y): (H(y) − H(x2 ))/(y − x2 ) ≤ Ĉ1 , so H(y) ≤ L(y). As y → ∞ the left side tends to C1 , because H(y)/y → C1 and y/(y − x2 ) → 1; hence C1 ≤ Ĉ1 . • For x ∈ [x1 , x2 ], take (u, v, w) = (x1 , x, x2 ): (H(x) − H(x1 ))/(x − x1 ) ≥ Ĉ1 , so H(x) ≥ L(x). L is positive on (0, ∞): on [x1 , x2 ] it lies between H(x1 ) > 0 and H(x2 ) > 0, and outside that interval L ≥ H > 0. Taking reciprocals, 1/L(x) ≥ L(x−1/2 ) for x ∈ [x1 , x2 ], that is, for c ∈ [c2 , c1 ], and 1/L(x) ≤ L(x−1/2 ) outside. In the one-statistic case H is affine, so L = H and every relation is an equality. When c1 , c2 ∈ A, (i) identifies ϱℓ with the reader’s gains. Proof of (d). Condition on S throughout. By (A1), (L) and (RW), y(t) = m(t) + b(t) + cΛζq(t) for t ≤ 89, where m is known and S-measurable. (i) With j(t) := max{j : dj ≤ t}, Z(t) = y(dj(t) ) is a function of the draw values and S; conversely y(dj ) = Z(dj ). This part uses neither (L) nor (RW). (ii) Because e(0) = 0, we have q(0) = φ1 (0) = 0, so Y0 − m(0) = b(0), where Yj := y(dj ). The map from the draw values (Y0 , . . . , Yn ) to (Y0 , ∆0 Y, . . . , ∆n−1 Y ) is a bijection, and dj+1

∆j (Y − m) = cΛζ ∆j q +

X

ξu .

u=dj +1

The noise sums run over disjoint blocks, so they are independent N (0, σb2 gj ) variables, independent of b(0) and ζ. With ϕv the N (0, v) density, the joint density given ζ therefore factorizes as Y
ϕσb2 gj ∆j (Y − m) − cΛζ∆j q . pb(0) (Y0 − m(0)) j

Its logarithm equals (cΛζ/σb2 ) C eff − (cΛζ)2 J/(2σb2 ) plusP terms free of ζ, where ∆j Y = ∆j Z.P By the factorization theorem C eff is sufficient. It is Gaussian with mean cΛζ j (∆j q)2 /gj = cΛζJ and variance j (∆j q)2 σb2 gj /gj2 = σb2 J. Minus the second derivative of the log-likelihood in ζ is (cΛ)2 J/σb2 . On the hourly record dj = j, gj = 1 and J = Q◦ . For a general profile with q(d0 ) ̸= 0 and a diffuse level, write b(0) = β0 + ξ0 , with β0 an unknown real parameter and ξ0 ∼ N (0, s20 ). The information matrix for (ζ, β0 ) has entries Iββ = 1/s20 , Iζβ = cΛq(d0 )/s20 and 2 Iζζ = (cΛq(d0 ))2 /s20 + (cΛ)2 J/σb2 . So the efficient information Iζζ − Iζβ /Iββ is again (cΛ/σb )2 J. Pdj+1 (iii) ∆j q = t=dj +1 (q(t) − q(t − 1)) is a sum of gj terms. By the Cauchy–Schwarz inequality (∆j q)2 ≤ gj Ej , with equality iff the gj hourly increments are equal, that is, iff q is affine on the integers of [dj , dj+1 ]. Hence ψj ∈ [0, 1] with the stated equality cases, and ψj = 0 iff ∆j q = 0. If Ej = 0, then ∆j q = 0 and the interval contributes to neither J norP Q◦ . The intervals (dj , dj+1 ], 0 ≤ j < n, and (dn , 89] P partition {1, . . . , 89}. So P ◦ Q = j Ej + Etail and J = j:Ej >0 ψj Ej . Dividing by Q◦ gives the identity, and χ ≤ j wj ≤ 1. Generator’s profile. Let A be binary and λ0 = 4/5.

56

Pt−2 • Zero memory. For t ≥ 0, e(t) = u=0 λt−2−u A(u) (an empty sum is 0). This is a sum of nonnegative 0 terms with positive weights, so e(t) = 0 iff A(u) = 0 for all u ≤ t − 2. In particular e(t′ ) = 0 implies e(t) = 0 for every t ≤ t′ . • Equal memories are zero. Let 0 ≤ t < t′ with e(t) = e(t′ ). If t′ ≤ 1, both are 0. Otherwise let M := t′ − 2 ≥ 0 and δ := t′ − t ≥ 1. Then 0 = e(t′ ) − e(t) =

M X

cm λm 0 ,

cm := A(t′ − 2 − m) − A(t − 2 − m) ∈ {−1, 0, 1},

m=0

with A := 0 atPnegative rows. Suppose some cm ̸= 0, and let m∗ be the largest such index. Multiplying ∗ ∗ by 5M gives m≤m∗ cm 4m 5M −m = 0. Every term with m < m∗ is divisible by 5M −m +1 , so 5M −m +1 ∗ ∗ ∗ divides cm∗ 4m 5M −m ; that is, 5 divides cm∗ 4m . This is impossible, because |cm∗ | = 1 and 5 does not ∗ divide 4m . Hence every cm = 0, that is, A(v) = A(v − δ) for 0 ≤ v ≤ t′ − 2. For v < δ the right side is a dose at a negative row, hence 0; by induction on v, A(v) = 0 for all 0 ≤ v ≤ t′ − 2. So e(t) = e(t′ ) = 0. • Equivalences. φ1 is strictly increasing, so q(t) = q(t′ ) iff e(t) = e(t′ ). Together with the two previous items this proves: q(t) = q(t′ ) iff q(t′ ) = 0 iff A(u) = 0 for every u ≤ t′ − 2. • ψj > 0 when Ej > 0. If Ej > 0, then q is not constant on {dj , . . . , dj+1 }, so e(s) > 0 for some s ≤ dj+1 , and hence e(dj+1 ) > 0 by the zero-memory item. By the equivalences, q(dj ) ̸= q(dj+1 ) and ψj > 0. • When J = 0. J = 0 iff ∆j q = 0 for every j iff Q(0) = 0. By the equivalences each ∆j q = 0 forces q(dj+1 ) = 0, so J = 0 implies q(dn ) = 0; this also holds when n = 0, since q(d0 ) = 0. Conversely, q(dn ) = 0 gives e(dn ) = 0, hence q(dj ) = 0 for all j ≤ n, and J = 0. Finally, q(dn ) = 0 iff A(u) = 0 for every u ≤ dn − 2. (iv) Let J > 0. By (ii), C eff /(σb J 1/2 ) ∼ N (cΛζJ 1/2 /σb , 1) for the held record. For the hourly record at eff amplitude c′ , Chourly /(σb (Q◦ )1/2 ) ∼ N (c′ Λζ(Q◦ )1/2 /σb , 1). Since Λ ̸= 0, the two laws coincide for every ζ iff 1/2 ′ ◦ 1/2 cJ = c (Q ) , that is, iff c′ = cχ1/2 ; this is also the condition for equal Fisher information. If J = 0, then ∆j q = 0 for every j, the density in (ii) does not depend on ζ, and the Fisher information is 0 at every amplitude. (v) In the idealized model of (d), (A1) and (R) hold at every c, so A = (0, ∞) for every sampling. (1) The statistic C raw = C eff + mq is linear in the Pheld record for fixed design, and J and mq are design-only. Its zero-effect part is X 0 = mq + N , where N := j ∆j b ∆j q/gj ; its signal part is UX = ΛζJ; and X S = (1, J). Given (S, ζ, ε), N has mean 0 and variance σb2 J. Hence EN = 0 and Cov(N, J) = 0, so Proj(N | 1, J) = 0 and X̃ 0 = m̃q + N . The cross term E[m̃q N ] = E[m̃q E[N | S]] is 0, so Σ0 = Var(m̃q ) + σb2 EJ > 0. The orthogonality conditions of (H) hold:   • E[N T ] = E T E[N | S, ζ, ε] = 0. • m̃q is S-measurable with mean 0 and S ⊥ (ζ, ε), so E[m̃q T ] = E m̃q ET = 0. With the previous item, Cov(X̃ 0 , T ) = 0. • By (c)(iv) with Ξ = J ∈ span(1, J), ŨX = Λ(ζ − µζ )J, a function of (S, ζ). Then E[N ŨX ] = 0 as in the first item, and E[m̃q Λ(ζ − µζ )J] = Λ E[m̃q J] E(ζ − µζ ) = 0. So Cov(X̃ 0 , ŨX ) = 0. By (c)(iv), θ = ΛkT σζ2 EJ ̸= 0 and C0 = (1 + CV2 (J))/ρ2lin . By (c)(ii) with p = 1, and using Var(T ) = kT2 σζ2 /ρ2lin , C1 =

σb2 EJ + Var(m̃q ) Var(T ) Σ0 = . Λ2 σζ2 ρ2lin (EJ)2 θ2

RS2 = 0 because Cov(J, T ) = kT Cov(J, ζ) + Cov(J, ε) = 0. For C eff the term mq is absent, Var(m̃q ) is replaced by 0, and C1 = σb2 /(Λ2 σζ2 ρ2lin EJ). The hourly record is the case J = Q◦ , so C1held /C1hourly = EQ◦ /EJ for the mean-corrected reader, and (c)(iii) gives the factor. (2) Given S: ζ ∼ N (µζ , σζ2 ) is independent of the baseline; Y0 −m(0) = b(0) is independent of (ζ, ∆(Y −m)); and C eff , which is computable from R(k) because m is S-measurable, is sufficient by (ii). The posterior of ζ given R(k) is therefore its posterior given C eff . That posterior is normal with variance σζ2 /(1 + ςJ), a function of S only. Hence Var E[ζ | R(k) ]/σζ2 = 1 − E Var(ζ | R(k) )/σζ2 = E[ςJ/(1 + ςJ)]. For X ∈ {J, Q◦ } let FX (ς) := E[ςX/(1 + ςX)]. FX is continuous and FX (0) = 0. It is strictly increasing when P (X > 0) > 0, which follows from EJ > 0 and Q◦ ≥ J. By monotone convergence (X is bounded),

57

−1 FX (ς)/ς = E[X/(1 + ςX)] ↑ EX as ς ↓ 0. Hence ςX (ϱ) := FX (ϱ) satisfies ςX (ϱ)/ϱ → 1/EX. Since ς ∝ c2 with the same constant for both records,

cheld (ϱ)2 /chourly (ϱ)2 = ςJ (ϱ)/ςQ◦ (ϱ) → EQ◦ /EJ. P P Pdj+1 (3) C m = N ′ + cΛζQ(0) with N ′ := j ∆j b ∆j q = j ∆j q u=d ξu . Given (S, ζ, ε), N ′ has mean 0 and j +1 P variance σb2 j gj (∆j q)2 = σb2 Q(1) . Q(0) > 0 iff some ∆j q ̸= 0 iff J > 0, so EJ > 0 gives EQ(0) > 0. The argument of (1), with m ≡ 0 and (N ′ , Q(0) ) in place of (N, J), gives: • X̃ 0 = N ′ and Σ0 = σb2 EQ(1) ; • (H) holds, with ŨX = Λ(ζ − µζ )Q(0) and θ = ΛkT σζ2 EQ(0) ̸= 0; • RS2 = 0; • C0 = (1 + CV2 (Q(0) ))/ρ2lin by (c)(iv), with Ξ = Q(0) ∈ span(1, Q(0) ); • C1 = Var(T )Σ0 /θ 2 = σb2 EQ(1) /(Λ2 σζ2 ρ2lin (EQ(0) )2 ). On hourly records gj = 1 and Q(0) = Q(1) = Q◦ , so C1held /C1hourly = EQ(1) EQ◦ /(EQ(0) )2 , and (c)(iii) gives the factor. For the inequality, the Cauchy–Schwarz inequality per patient gives X 1/2
−1/2
Q(0) = gj |∆j q| gj |∆j q| ≤ (Q(1) J)1/2 , j

and the Cauchy–Schwarz inequality in L2 then gives EQ(0) ≤ E[(Q(1) J)1/2 ] ≤ (EQ(1) EJ)1/2 . EQ(1) EQ◦ /(EQ(0) )2 ≥ EQ◦ /EJ.

Hence

Proof of (e). By (G2) only j ∗ has nonzero factual memory, so the factual shift on channel k at row d is c α tanh(Λk ζi qi (d)). On Ni this is 0 at every draw d ∈ Dik of every channel, because Λk = 0 or qi (d) = 0; this 0 uses no property of α. Hence yik (d) = ỹik (d) at every draw, and by (G1) and (G4) Zik (t) = ỹik (dik (t)) = Zik (t) 0 for all k and t ≤ 89. So Ri 1Ni = (Zi , Si )1Ni . For bounded measurable f and ϕ, since Ni is determined by Si and by (G3),       E f (ζi ) ϕ(Ri )1Ni = E f (ζi ) ϕ(Zi0 , Si )1Ni = Ef (ζi ) E ϕ(Ri )1Ni . Since 1Ni is σ(Ri )-measurable, this says 1Ni E[f (ζi ) | Ri ] = 1Ni Ef (ζi ) almost surely. With f = τ̄ (bounded) this is the second claim. Characterization. Under binary dosing, qi (d) = 0 iff ei (d) = 0 iff Ai (u) = 0 for every u ≤ d − 2, by the zero-memory item in the proof of (d)(iii). Moreover ei (d′ ) = 0 implies ei (d) = 0 for every d ≤ d′ . So, for each k, qi vanishes at every draw of channel k iff it vanishes at max Dik , which gives the stated description of Ni . Now let P (Ni ) = 1. Write τ = τ̄ (ζ) + ν ′ with ν ′ := τ − τ̄ (ζ), so that Eν ′2 = E Var(τ | ζ). Then E[τ | R] = E τ̄ (ζ) + E[ν ′ | R], and by Jensen’s inequality Var E[τ | R] = Var E[ν ′ | R] ≤ Eν ′2 . Combined with R2 (τ̂ ) ≤ R∗2 (σ(R)) from the proof of (c)(i), this gives R2 (τ̂ ) ≤ E Var(τ | ζ)/Var(τ ), with every moment computed for the panel in question. With no dose on rows 0, . . . , 89, ei (t) = 0 for t ≤ 90, so qi ≡ 0 on rows 0, . . . , 89, ei (t0 ) = 0 and Ai (89) = 0. Then (a)(ii), under (A1), gives the own-treatment part τ own (ζi , 0). ■ Numerical checks. All checks below are descriptive re-reads of shipped panel outputs, reconstructions of the panels’ doses from shipped inputs, or simulations; none reads a new label. • Target structure (a). A replica of the generator reproduces the shipped targets of the synthetic-record run (job 10692), which are the same targets as in the real-record discovery panels, to 8.5×10−8 . D∗ = 0.6842, and the s = 0 increment is exactly 0. In the registered phenylephrine cell SD(τ ) = 0.2322, corr(τ, ζ) = −0.964 and the R2 of τ on τ own is 0.941; τ − τ own correlates +0.559 with ζ, which is why E[τ | ζ] predicts better than τ own alone. The cross-fitted R2 of an estimate of E[τ | ζ] is 0.959 to 0.960 (recovery 0.797 to 0.800), and the shipped oracle rung scores 0.802 to 0.805; this is the reference level of 0.80 in the main text. The target standard deviations of the discovery, replication and fixed-schedule panels are 0.2322, 0.2317 and 0.2323, which is consistent with (a)(ii)–(iii); those parts bound only the own-treatment part and do not predict the standard deviations. The largest retained memory at the decision is 2.03 × 10−3 under NATMIN, so (a)(iii) bounds the change of the own-occupancy increment by 6.3 × 10−4 ; under FIX19 the largest retained memory is 3.09 × 10−6 . • Exact scaling (b). In a 240-person scaling check the held heart-rate record is affine in c to 2.6 × 10−6 and T to 1.0 × 10−6 ; the exceptions to (A1) are listed under (A1). 58

• Amplitude law (c). Two-point fits through the shipped recovery map at c = 0.5 and c = 1 predict the value at c = 0.75 with mean absolute error 0.007 (maximum 0.014; 17 series) when the fitted values are first corrected for the expected finite-sample loss of (P), and 0.006 (13 series) without the correction; the mean signed errors are +0.004 and −0.004. Extrapolated to c = 0.25, the mean signed errors are −0.019 (corrected; 21 series) and +0.014 (uncorrected; 16 series). By (c)(v) the population law gives two-point predictions that are too high inside [0.5, 1] and too low outside; the corrected fits show these signs and the uncorrected fits do not. (The correction uses the expression (1 − R2 )p/(n − p − 1), which is 4% to 8% smaller than the exact value in (P), so it slightly under-corrects.) In the registered cell the corrected fit predicts recovery 0.185 at c = 0.75 (0.177 uncorrected) against the observed 0.181 (95% interval 0.137 to 0.221). The chord crossing amplitude c∗ of (c)(iii) is 0.53 from the uncorrected fit and 0.44 from the corrected fit; the observed recovery at c = 0.5 is 0.095, below the bar of 0.106. In 2 corrected and 7 uncorrected effect-scale fits the estimated Ĉ0 is below 1, which is impossible for the population constant and reflects estimation error. A series is one reader (window-mean, assimilating, plain or cadence-aware) on one treatment-response pair at one cadence. It enters a count when the map has recoveries at c = 0.5, at c = 1 and at the predicted amplitude, and when the two fitted R2 gains, after the correction where it is used, both exceed 0.005, so that the line in 1/c2 is defined. The correction raises every gain, so more series qualify with it (17 against 13 for interpolation, 21 against 16 for extrapolation), and not every reader was run at every amplitude (the cadence-aware reader on hourly records only at c = 1, and at c = 0.75 for phenylephrine). Interpolation inside [0.5, 1] is a weak test, because any smooth increasing curve passes close to its chord there; the size and sign of the extrapolation errors are the more informative check, and the uncorrected fits have the sign opposite to the one the law predicts. • Ceilings. By (c)(v), two-point ceilings bound the population ceiling from below before sampling error. For the plain reader the corrected chord ceilings fall from 0.79 (hourly) to 0.53 (real cadence) for phenylephrine on heart rate, from 0.79 to 0.36 for norepinephrine on heart rate, and from 0.86 to 0.13 for NaCl on creatinine. The single-channel linear-regime ceiling of (c)(iv) at K0 changes much less on heart rate: from 0.855 to 0.788 (0.774 at the panel’s change rate) for phenylephrine and from 0.792 to 0.731 (0.718) for norepinephrine for the plain reader, and from 0.822 to 0.825 for the cadence-aware reader. The real-cadence plain chord ceilings on heart rate lie below this single-channel value but within sampling error (bootstrap probability that the chord ceiling is at least the theory value: 0.31 for phenylephrine, 0.17 for norepinephrine), so on heart rate the sampling acts mainly through C1 for both readers. The theory predicts a ceiling collapse only for the plain reader on creatinine, where 1 + (1 + µ2ζ /σζ2 )CV2 (M ) is heavy-tailed across seeds (28 to 42 at the proxy change rates and 71 to 161 at the panel’s rate), giving a single-channel ceiling of about 0.006 to 0.03, while the cadence-aware single-channel ceiling on creatinine is 0.54 to 0.60. The multi-channel creatinine chord ceiling (0.13 corrected, 0.32 uncorrected) does not settle whether the population ceiling lies below the bar: the bootstrap probability that it is below 0.2 is 0.75 (corrected) and 0.56 (uncorrected). • Amplitude factors. The low-signal amplitude factors from hourly to real cadence estimated from the map are 2.68, 2.44 and 2.20 for the plain reader on the three pairs, and 1.64 (corrected; 1.90 uncorrected) for the cadence-aware reader on phenylephrine and heart rate, with 90% Monte Carlo ranges [1.18, 2.54] and [1.34, 3.10]. The cadence-aware estimate mixes kinetics, because the hourly fits selected (2, 0.95) and the held fits (2, 0.8) (assumption (K)), so it does not test (c)(iii) cleanly. Table S14 gives the single-carrier theory values; the map’s 1.64 lies between the implemented statistic’s white-noise and random-walk values. • Formula (c)(iv). A Monte Carlo of the plain reader gives C0 = 2.254 against 2.252 from the formula with the µζ term and 1.962 without it. • Retention (d). Table S14 gives retention under the idealization (RW) with Bernoulli draw patterns at each real stay’s own change rate (400 stays of the realism diagnostic), reported at these proxy rates and P rescaled to the panels’ own mean rates. The identity χ = j wj ψj holds to 8 × 10−16 . The alignment factors of intervals with Ej > 0 are at least 1.6 × 10−9 on heart rate and 1.1 × 10−13 on creatinine, positive as (d)(iii) requires; the zero-memory lemma was also verified exhaustively over 8,190 short binary dose sequences and on 300 random 90-row sequences in exact rational arithmetic. • Exposure (e). In a replica panel with no pre-decision dosing, Z A = Z 0 exactly and a label-free transport predictor is constant (centered PEHE divided by SD(τ ) equals 1.000), while the target still correlates 0.9986 with the NATMIN target. The bound of (e) must be computed with that panel’s own moments; the 2 NATMIN value of Rtrait is not a bound for it.

59

Supplementary Table S14: Retention of trait information and amplitude factors on a single carrier (Proposition 4(d)). Idealized single-carrier random-walk baseline (assumption (RW)), generator kinetics, the panels’ reconstructed natural doses (phenylephrine for heart rate, NaCl for creatinine) and Bernoulli draw patterns at each real stay’s own change rate (proxy; mean rate 0.677 for heart rate and 0.046 for creatinine), then rescaled to the panels’ own mean rates (0.572 and 0.037). Retention χ = J/Q◦ is the share of the hourly record’s Fisher information about the trait that the held record keeps. The low-signal amplitude factor is the ratio of the amplitudes that give held and hourly records the same small R2 : (EQ◦ /EJ)1/2 for the efficient statistic (Proposition 4(d)(v)(1)), and (EQ◦ EQ(1) )1/2 /EQ(0) for the implemented cadenceaware statistic under a random walk (Proposition 4(d)(v)(3)); the white-noise value for the implemented statistic is from Proposition 5. These are properties of the idealized model, not measurements of the panels.

Median retention χ Mean retention χ Patients with χ < 0.05 (%) Energy in intervals with ψj < 0.1 (%) Energy after the last draw (%) Patients with J = 0 (%) Amplitude factor, efficient statistic Amplitude factor, implemented statistic, random walk Amplitude factor, implemented statistic, white noise

Heart rate proxy panel rate

Creatinine proxy panel rate

0.806 0.702 2 10.8 0.5 0.4 1.20 1.84 0.97

0.048 0.070 51 72.1 12.8 8.6 4.09 5.25 1.15

0.607 0.610 2 13.7 0.6 0.4 1.29 1.97 0.96

0.036 0.058 59 70.6 17.3 12.4 4.50 5.71 1.22

5.5 Which reader is efficient Proposition 5 (Levels or increments: the baseline decides which reader is efficient). Standard and specific parts. Parts (a)(i) and (a)(ii) are the Gauss–Markov theorem for generalized least squares with a nuisance mean (Aitken), and (b) is its white-noise case, in which partialling out the trend (Frisch and Waugh) gives the plain contrast. Parts (a)(iii) and (c) to (e) are specific to the pooled readers and the held records of this evaluation, and (d) is partly numerical. The weight vectors v of this section are unrelated to the slope variance v of Sections 5.6 and 5.7. Setting. Symbols not defined here are those of the notation and definitions at the start of this note. Assumptions N1 to N4 hold throughout; N5, N6 and N7 are used where stated. Readers use the generator kinetics K0 = (2, 0.8, 1), so that pK0 = qi . For a set K of outcome channels, yiK stacks the latent paths of those K channels on rows t = 0, . . . , 89, and q̄iK stacks the vectors Λk qi , k ∈ K; by N1, yiK = bK i + c ζi q̄i . For one channel k with loading Λ := Λk ̸= 0 the superscript is dropped, so yi = bi + cΛζi qi and q̄i = Λqi . The held record of channel k is Zi = Hi yi , where the 90 × 90 hold matrix has Hi (t, dik (t)) = 1 and is zero elsewhere; on hourly records Hi = I. A linear statistic is vi⊤ yiK with a weight vector vi that is a measurable function of the design Si ; a statistic of channel k’s held record has vi = Hi⊤ wi , which is supported on the draw set Dik . K Let µi := E[bK i | Si ] and let Σi := Cov(bi | Si ), including the covariances between channels, on the rows the statistic reads, positive definite there. Given (Si , ζi ), vi⊤ yiK has mean vi⊤ µi + c ζi vi⊤ q̄iK (its signal is c ζi vi⊤ q̄iK ) and variance vi⊤ Σi vi . Notation. For a matrix B of full column rank, possibly with no columns, MB := Σ−1 − Σ−1 B(B ⊤ Σ−1 B)−1 B ⊤ Σ−1 , and MB := Σ−1 when B has no columns. For the draws d0 = 0 < d1 < · · · < dn P89 Pn−1 of channel k: ∆j f := f (dj+1 ) − f (dj ), gj := dj+1 − dj , Q◦ := t=1 (q(t) − q(t − 1))2 , Q(m) := j=0 gjm (∆j q)2 for m ∈ {−1, 0, 1}, and J := Q(−1) . ∇ is the 89 × 90 first-difference matrix, (∇x)(t) = x(t) − x(t − 1); + ⊤ t := (0, 1, . . . , 89)⊤ and t2 is its elementwise square. Plain statistic O := P (Πq) Z with Π := I − B2 B2 and 2 B2 := [1, t, t ] (no companion feature); cadence-aware statistic C := j ∆j Z ∆j q with companion feature P (0) Q ; 1/gap-weighted statistic G := j ∆j Z ∆j q/gj with companion feature J. A pooled linear reader of the channels K is the population least-squares regression of T := τ /c on (1, FiS , Xi ), where FiS are design-only features and Xi = Vi⊤ yiK ∈ Rp with Vi design-measurable and E[V ⊤ ΣV ] nonsingular; R2 (c) is its population R2 at amplitude c and 1/C1 := limc→0 R2 (c)/c2 . Put Υ := kT2 σζ4 / Var(T ), with kT from N4 and µζ , σζ2 the mean and variance of the trait, and, for p = 1 and X = v ⊤ y K , I(v) :=

(E[v ⊤ q̄])2 , E[v ⊤ Σv] + Vm (v)

where Vm (v) is the residual variance of the design-measurable offset v ⊤ µ after linear regression on (1, F S ). The amplitude factor of a reader from hourly to held records of the same baseline is f := (C1held /C1hourly )1/2 . Unconditional expectations E are over patients. (a) Gauss–Markov bounds. (i) Per patient, baseline mean known: for v ̸= 0, (v ⊤ q̄)2 /(v ⊤ Σv) ≤ q̄ ⊤ Σ−1 q̄, with equality iff v and Σ−1 q̄ are linearly dependent. When q̄ ̸= 0 on the rows read this means v = αΣ−1 q̄ 60

for some α ̸= 0; when q̄ = 0 there, both sides are 0 for every v. (ii) Per patient, µ ∈ span(B) with free coefficients: the linear statistics whose law does not depend on the free coefficients are exactly those with B ⊤ v = 0. For them (v ⊤ q̄)2 /(v ⊤ Σv) ≤ q̄ ⊤ MB q̄, with equality iff v and MB q̄ are linearly dependent, that is, when MB q̄ ̸= 0, iff v = αMB q̄ with α ̸= 0. For Gaussian b, c2 q̄ ⊤ MB q̄ is the Fisher information for ζ with the free coefficients unknown, and generalized least squares attains it. The per-patient efficiency of v is eff(v) := (v ⊤ q̄)2 /(v ⊤ Σv · q̄ ⊤ MB q̄) ∈ [0, 1] when q̄ ⊤ MB q̄ > 0, and eff(v) := 0 otherwise. (iii) Pooled: if Bi⊤ Vi = 0 almost surely, then R2 (c) ≤ c2 Υ E[q̄ ⊤ MB q̄] at every amplitude c, where Σ is the joint covariance of the channels read. For p = 1, 1/C1 = Υ I(v) and I(v) ≤ E[q̄ ⊤ MB q̄]; if E[q̄ ⊤ MB q̄] = 0 both sides are 0, and otherwise equality holds iff Vm (v) = 0 and v = αMB q̄ almost surely for one constant α ̸= 0. When Vm (v) = 0 and Pr(a ̸= 0) > 0, with a := v ⊤ q̄ and s2v := v ⊤ Σv, E[eff(v) q̄ ⊤ MB q̄] (Ea)2 I(v) = · , ⊤ ⊤ 2 E[q̄ MB q̄] E[q̄ MB q̄] E[a /s2v ] E[s2v ] with a2 /s2v := 0 where sv = 0; the second factor is at most 1, with equality iff a/s2v is almost surely constant on {sv > 0}. If the nuisance basis is block-diagonal across channels, the per-patient bound q̄ K⊤ MB q̄ K , and hence the pooled bound, is nondecreasing in the set K of channels read. These are bounds for pooled linear readers; they are not bounds on estimators. (b) White noise plus a free quadratic trend (N5), hourly records of one channel. Here MB2 = σ −2 Π, so the plain statistic (Πq)⊤ y is efficient among trend-invariant linear statistics of the channel, and (Πq)⊤ y/∥Πq∥2 is the generalized least-squares estimate of cΛζ. At kinetics K the plain statistic (ΠpK )⊤ y has efficiency P89 2 ⊤ ⊤ ⊤ ⊤ (p⊤ K Πq) /(pK ΠpK · q Πq). The hourly increment statistic t=1 (∇y)(t) (∇q)(t) = (∇ ∇q) y is not efficient. For every dosing of the scored treatment with doses in [0, 1] (the implemented doses are binary) on P88 rows 6 to 53 only and at least one nonzero dose, (t2 )⊤ ∇⊤ ∇q = 177 q(89) − 2 s=1 q(s) < 0, so the law of the increment statistic depends on the free quadratic coefficient; and with the mean known its efficiency (Q◦ )2 /(∥∇⊤ ∇q∥2 ∥q∥2 ) is strictly below 1. (c) Random-walk baseline (N6), one channel. (i) The level-invariant linear statistics of the held record are exactly the linear functions of the draw-to-draw increments. Among them G is efficient, and on the draw ⊤ rows q̄D M1 q̄D = Λ2 J/σb2 for every positive finite level variance. (ii) C reads the held record only through ∆j Z, which equals the latent increment ∆j y; the implemented reader finds the draws as the rows where the recorded value changes, which recovers Dik except for exact ties. The hold therefore acts on C only through the draw set. If Q(0) > 0, the per-patient efficiency of C is (Q(0) )2 /(Q(1) J) ≤ 1, with equality iff gj is constant on {j : ∆j q ̸= 0}; on hourly records C = G. (iii) The pooled readers of the channel on (C, Q(0) ) and on (G, J) satisfy 1/R2 (c) = C0 + C1 /c2 exactly, with 1/C1 = ΥΛ2 (EQ(0) )2 /(σb2 EQ(1) ) and 1/C1 = ΥΛ2 EJ/σb2 respectively. The latter is the largest 1/C1 of any pooled linear reader of this channel’s held record whose statistics are level-invariant. From hourly records of the same baseline to held records,  EQ◦ 1/2  EQ(1) EQ◦ 1/2 ≥ f = . fC = G EJ (EQ(0) )2 (d) Pooled ordering and the AR(1) switch on hourly records. For pooled linear readers with single statistics v1 , v2 and I(v2 ) > 0, R12 (c)/R22 (c) → I(v1 )/I(v2 ) as c → 0; if I(v1 ) > I(v2 ), reader 1 has the larger R2 at every sufficiently small amplitude. Under N7 on hourly records of one channel let R(ϑ) := I(Πq)/I(∇⊤ ∇q), plain over cadence-aware. Then R is continuous on [0, 1); R(ϑ) → RRW as ϑ ↑ 1, where RRW is its value under N6 with every row a draw; and RRW < 1 whenever Pr{(Πq)(0) ̸= 0} > 0. For the reconstructed discovery doses R(0) = 9.22 and RRW = 0.110, so R − 1 has a zero in (0, 1) and the more informative reader on hourly records changes from the plain to the cadence-aware reader as ϑ rises. Numerically, on grids of step 0.01 and 0.001, R is strictly decreasing and R − 1 changes sign once, between 0.642 and 0.643; bisection gives ϑ∗ = 0.642 (population detrended lag-1 autocorrelation 0.583), and at the calibrated persistence 0.754 (N7) R = 0.55. For the shared FIX19 schedule the same computation gives R(0) = 23.57, RRW = 0.254, one sign change between 0.806 and 0.807, and R = 1.4 at persistence 0.754. These statements concern hourly records; they do not determine the ordering on held records. (e) Shared schedule. Let Zi ∈ R90×12 be the record of all twelve channels on rows 0 to 89 (hourly or held). If all patients receive the same doses, then at every fixed kinetics the features of the window-mean, assimilating and plain readers are one affine map of Zi , the same for all patients, on hourly and held records alike. Their population least-squares R2 is therefore at most that of the history-blind reader on all 1,080 values of Zi , at every amplitude and without N1; with kinetics chosen from a finite grid the same holds for the best kinetics; and with a next-draw target, where the same draw-step indicator is appended 61

to every rung including the history-blind one, the same holds. On hourly records without exact ties the cadence-aware features are also such a map. Under a shared schedule a population advantage of a record-based reader of the ladder (window-mean, assimilating, plain, cadence-aware) over the historyblind reader can therefore arise only from patient-specific draw sets, which exist only on held records. On hourly records under a shared schedule, weights that depend on the design only through the doses are the same for all patients; under N1 and N2, with Σ := Cov(bK ) and a common nuisance basis B, the efficient such weight MB q̄ K is one fixed vector for every set of channels K, and the efficient statistic is a fixed linear functional of vec(Zi ). Assumptions. N1 (linear regime, fixed kinetics). On rows 0, . . . , 89 the factual latent path of every outcome channel k is exactly yik (t) = bik (t) + c αik (t) tanh(Λk ζi qi (t)), because only j ∗ is dosed in the factual record, so that K Γik = Λk ζi qi . The linear regime sets α ≡ 1 and replaces tanh x by x; then yiK = bK i + cζi q̄i for every set of channels. Readers use the generator kinetics K0 , at which the implemented occupancy et /(1 + et ) with et = 0.8et−1 + At−2 equals qi . Idealizations: on heart rate |Λζq| reaches values near 5/3 (ζ ≤ 2, q < 5/6), where tanh is far from linear; the ladder chooses kinetics by nested cross-validation and fits a ridge, whereas population least squares at fixed kinetics is analyzed (Definition 6). Parts (b) to (d) concern statistics of one channel, whereas the implemented plain and cadence-aware readers read all twelve channels (12 and 24 features). Two features of the generator then change the numbers for the implemented readers, although not the statements (a)(iii) and (e), which cover several channels: the phenylephrine response is also written into pH, platelets and white cells (loadings about 0.75, −0.80, −0.22), and the channels’ baselines are correlated, because every channel’s generator contribution contains the shared latent state and the real tapes are measured jointly (the generator part of each other channel correlates with that of heart rate at 0.24 to 0.64 in absolute value). N2 (exogeneity). ζi is independent of (Si , bK i ) for every K. The design is trait-free (Definition 5, verified by reconstructing all dose columns from shipped u0 ); bi contains no trait because the bounded shift never feeds back into state, policy or memory, and the trait is drawn from a separate pseudo-random stream. N3 (second moments and nuisance). Patients are independent and identically distributed, and Σi is positive definite on the rows read. For per-patient statements the design is held fixed and µi is known up to span(Bi ) with free coefficients. Pooled statements keep the offset variance Vm . In the generator µi contains the scored treatment’s ordinary, trait-free action route, a time-varying function of the doses, so Vm > 0 unless a reader’s design-only features absorb it; the numerical AR(1) and random-walk comparisons omit it except where stated. N4 (target). T := τ /c, free of c by P4(a), satisfies T = const+kT ζ +ε with kT := Cov(T, ζ)/σζ2 ̸= 0, Cov(ζ, ε) = 0, Var(T ) > 0, and (ζ, ε) independent of (S, bK ). Idealization: ε contains the nonlinear part of the target and the policy-dosed cross-treatment term ν, which depends on the latent state at the decision and hence weakly on the pre-decision baseline; the retained memory e(t0 ) ≤ 2.03 × 10−3 is a design quantity inside T. N5 (used in (b)). Hourly rows 0, . . . , 89 of one channel; b = B2 β + b◦ with β ∈ R3 free and Cov(b◦ | S) = σ 2 I. This is the model under which the plain reader’s projector is efficient; no shipped panel follows it exactly. N6 (used in (c)). One channel. Given S, the hourly increments b(t) − b(t − 1) are uncorrelated with mean 0 and variance σb2 , and the level b(0) is a free nuisance with any positive finite variance, uncorrelated with the increments (level-invariant statistics do not depend on it). The draw-to-draw increments are then uncorrelated with variance σb2 gj (Proposition 4(d)). The “hourly record of the same baseline” observes this b at every row. Idealizations: real heart rate is not a random walk (N7 calibrates its persistence at 0.754); the ordinary treatment route gives the increments a nonzero mean; and the draw set is treated as independent of b, whereas in the shipped panels the draw set is the tape’s change set, so the tape component of b is constant between draws and the variance of its jumps need not grow with the gap. Statistics of the held record depend only on the draw values. The shipped real-hourly panel (hold off) observes a baseline whose tape component is piecewise constant between the tape’s change times; only the generator’s contribution moves every hour. ′

2 N7 (used in (d)). Hourly rows of one channel; Σ(t, t′ ) = σAR ϑ|t−t | /(1 − ϑ2 ) with ϑ ∈ [0, 1) and innovation 2 variance σAR , the same for all patients, and a constant mean, so Vm = 0 for level-invariant statistics. The family interpolates between white noise (ϑ = 0) and a random walk (ϑ ↑ 1). Calibration: the shipped pooled lag-1 autocorrelation of the quadratically detrended zero-effect heart-rate record on hourly real records is 0.687 (real-cadence probe with NaCl dosing, in which heart rate carries no

62

response; g2_washout_real_cadence_probe_20260918_v1/gls.json, entry real_hold0, projected, heart_rate, lag1). The population version of the same pooled estimator (ratio of expected numerator and denominator) equals 0.687 under N7 at persistence ϑ = 0.754. The corresponding creatinine value 0.602 gives 0.662. No calibration is taken from held records, which are a zero-order hold of the latent path and not an hourly AR(1). Data for the numerical parts. Doses: the 400 NATMIN dose columns of the discovery panel, reconstructed from the shipped u0 (400 of 400 dosed-row counts equal the shipped table); FIX19 doses on rows 6 to 24. Draw sets for the held-record numbers: row 0 plus independent Bernoulli draws at each stay’s own heart-rate change rate from the realism diagnostic’s real eICU stays (382 stays with a rate; mean 0.677, median 0.713), a proxy for the panels’ own draw masks, which are not shipped. Heart rate changes in 57% of hours on the panel stays (main text), so every number is also given with the rates rescaled by a common factor to a mean of 0.58. Proof. Conventions. For random vectors X and F with finite second moments, L(X | 1, F ) is the population linear projection of X on (1, F ) and X̃ := X − L(X | 1, F ). Unconditional E averages over patients. All second moments below are finite: 0 ≤ q < 5/6, every weight vector used is a bounded function of the design, and Σi is bounded. When a statistic reads only a design-dependent set of rows (all rows, or the draw rows of each channel read), Pv is supported on that set and q̄, Σ, B, MB are restricted to it. The memory of the scored treatment is e(t) = s≤t−2 0.8 t−2−s A(s) with A(s) = 0 for s < 0, and q = e/(1+e); in particular e(0) = e(1) = 0, hence q(0) = q(1) = 0, for every dosing. Lemma 1 (Gauss–Markov with a free nuisance). Let Σ ∈ RN ×N be positive definite, q̄ ∈ RN , B ∈ RN ×nB of rank nB ≥ 0, and MB as in the proposition. (1) B ⊤ MB = 0, MB ΣMB = MB and MB ⪰ 0. (2) If B ⊤ v = 0 then v ⊤ q̄ = v ⊤ ΣMB q̄. (3) If v ̸= 0 and B ⊤ v = 0 then (v ⊤ q̄)2 ≤ (v ⊤ Σv)(q̄ ⊤ MB q̄), with equality iff v and MB q̄ are linearly dependent. If MB q̄ = 0 both sides are 0; if MB q̄ ̸= 0, equality means v = αMB q̄ with α ̸= 0, and B ⊤ MB q̄ = 0. Hence q̄ ⊤ MB q̄ = sup{(v ⊤ q̄)2 /(v ⊤ Σv) : v ̸= 0, B ⊤ v = 0}, with the supremum of the empty set read as 0. (4) In the model y = Bβ + θq̄ + e with Ee = 0 and Cov e = Σ, the law of v ⊤ y does not depend on β ∈ RnB iff B ⊤ v = 0. (5) If e is Gaussian, the Fisher information for θ with β unknown is q̄ ⊤ MB q̄, and, when it is positive, θ̂ := q̄ ⊤ MB y/q̄ ⊤ MB q̄ is unbiased with variance 1/q̄ ⊤ MB q̄. Proof. Let P := Σ−1/2 B(B ⊤ Σ−1 B)−1 B ⊤ Σ−1/2 (P = 0 if nB = 0), the orthogonal projector onto the column space of Σ−1/2 B. Then MB = Σ−1/2 (I −P )Σ−1/2 , so MB ⪰ 0 and MB ΣMB = Σ−1/2 (I −P )2 Σ−1/2 = MB ; also B ⊤ MB = B ⊤ Σ−1 − B ⊤ Σ−1 B(B ⊤ Σ−1 B)−1 B ⊤ Σ−1 = 0. This is (1). For (2), put β̂ := (B ⊤ Σ−1 B)−1 B ⊤ Σ−1 q̄; then ΣMB q̄ = q̄ − B β̂, so v ⊤ ΣMB q̄ = v ⊤ q̄ − (B ⊤ v)⊤ β̂ = v ⊤ q̄. For (3), let x := Σ1/2 v and z := Σ1/2 MB q̄. By (2) and the Cauchy–Schwarz inequality, (v ⊤ q̄)2 = (x⊤ z)2 ≤ ∥x∥2 ∥z∥2 = (v ⊤ Σv)(q̄ ⊤ MB ΣMB q̄) = (v ⊤ Σv)(q̄ ⊤ MB q̄), with equality iff x and z are linearly dependent, which, Σ1/2 being invertible, holds iff v and MB q̄ are. If MB q̄ = 0 then v ⊤ q̄ = 0 by (2) and both sides vanish. If MB q̄ ̸= 0 and v ̸= 0, linear dependence means v = αMB q̄ with α ̸= 0, and MB q̄ is admissible by (1), so the supremum is attained. For (4), E[v ⊤ y] = (B ⊤ v)⊤ β + θv ⊤ q̄ is the same for all β iff B ⊤ v = 0, and β enters the law of v ⊤ y only through this mean. For (5), the Fisher information matrix of (θ, β) is [q̄ B]⊤ Σ−1 [q̄ B], and the information for θ with β unknown is its Schur complement q̄ ⊤ Σ−1 q̄ − q̄ ⊤ Σ−1 B(B ⊤ Σ−1 B)−1 B ⊤ Σ−1 q̄ = q̄ ⊤ MB q̄. By (1), Eθ̂ = (q̄ ⊤ MB Bβ + θ q̄ ⊤ MB q̄)/q̄ ⊤ MB q̄ = θ and Var θ̂ = q̄ ⊤ MB ΣMB q̄/(q̄ ⊤ MB q̄)2 = 1/q̄ ⊤ MB q̄ (Aitken 1936). □ Lemma 2 (Level-invariant statistics are functions of increments). Let d0 < · · · < dP n and x = P n−1 ⊤ ⊤ v (j = 0, . . . , n − 1), v x = x(d ) 1 v + (x(d0 ), . . . , x(dn )). For v ∈ Rn+1 and uj := i 0 i>j j=0 uj ∆j x. ⊤ ⊤ n Hence 1 v = 0 iff v x is a linear function of (∆0 x, . . . , ∆n−1 x), and every u ∈ R arises from exactly one v with 1⊤ v =P 0, namely vi = uP i−1 − uP i with u−1 = uP n := 0. n n−1 n i−1 ⊤ ⊤ ∆ x = Proof. u ∆ x = v i=1 i j=0 j i=1 vi (x(di ) − x(d0 )), and adding x(d0 )1 v gives v x. If j=0 j j 1⊤ v = 0 the statistic depends on the increments only; if 1⊤ vP̸= 0 it changes with x(d0 ) at fixed increments. For given u, vi := ui−1 − ui satisfies 1⊤ v = u−1 − un = 0 and i>j vi = uj − un = uj ; conversely ui−1 − ui = vi P for i ≥ 1 and v0 = − i≥1 vi = −u0 when 1⊤ v = 0, so v is unique. □ Lemma 3 (Pooled linear readers). Assume N1 to N4. Fix a set of channels K and write y := y K , q̄ := q̄ K , b := bK . Let V = V (S) with p columns and F = F S (S), X = V ⊤ y, a := V ⊤ q̄, m := V ⊤ µ, ξ := V ⊤ (b − µ), and suppose E[V ⊤ ΣV ] is nonsingular. (i) X̃ = c(ζ − µζ )a + cµζ ã + m̃ + ξ and Var(X̃) = c2 σζ2 E[aa⊤ ] + Var(cµζ ã + m̃) + E[V ⊤ ΣV ]. (ii) Cov(X̃, T ) = ckT σζ2 E[a], Cov(F, T ) = 0, and R2 (c) = Cov(T, X̃) Var(X̃)−1 Cov(X̃, T )/ Var(T ). (iii) R2 (c) ≤ c2 Υ E[a]⊤ E[V ⊤ ΣV ]−1 E[a] for every c, and R2 (c)/c2 → Υ E[a]⊤ Ω−1 E[a] as c → 0, with Ω := E[V ⊤ ΣV ] + Var(m̃). For p = 1 the limit is ΥI(v). If moreover Ea ̸= 0 and either ã = 0 (the companion a is a linear function of F ) or m̃ = 0, then exactly 1/R2 (c) = C0 + C1 /c2 63

with C1 = 1/(ΥI(v)) and C0 = Var(T ) [σζ2 Ea2 + µ2ζ Var ã]/(kT2 σζ4 (Ea)2 ). (iv) If B ⊤ V = 0 almost surely for a design-measurable B, then E[a]⊤ E[V ⊤ ΣV ]−1 E[a] ≤ E[q̄ ⊤ MB q̄]. If E[q̄ ⊤ MB q̄] = 0 both sides are 0. If E[q̄ ⊤ MB q̄] > 0, equality holds iff V w = αMB q̄ almost surely for some w ∈ Rp \ {0} and a constant α; then α ̸= 0. (v) Let K ⊆ K′ , let the nuisance basis of K′ be block-diagonal across channels with the blocks of K forming the basis of K, and let the covariance of K be the corresponding submatrix of that of K′ . Then ′ ′ q̄ K⊤ MB K q̄ K ≤ q̄ K ⊤ MB K′ q̄ K for every patient. Proof. Step 1. Write X = c(ζ − µζ )a + cµζ a + m + ξ. By N2, ζ is independent of S, so E[(ζ − µζ )a] = 0 and Cov((ζ − µζ )a, F ) = E[ζ − µζ ] E[aF ⊤ ] = 0; hence L((ζ − µζ )a | 1, F ) = 0. Since E[ξ | S] = V ⊤ (E[b | S] − µ) = 0, Eξ = 0 and Cov(ξ, F ) = E[E[ξ | S]F ⊤ ] = 0, so ξ˜ = ξ. Linearity of L gives the expression for X̃. Step 2. Put X1 := (ζ−µζ )a. Because ã, m̃ are functions of S and ζ is independent of (S, b): Cov(X1 , ã) = E[ζ−µζ ] E[aã⊤ ] = 0, likewise Cov(X1 , m̃) = 0, and Cov(X1 , ξ) = E[ζ − µζ ] E[aξ ⊤ ] = 0. Also Cov(ξ, ã) = E[E[ξ | S]ã⊤ ] = 0 and likewise for m̃. Finally Var(X1 ) = σζ2 E[aa⊤ ] and Var(ξ) = E[V ⊤ E[(b − µ)(b − µ)⊤ | S]V ] = E[V ⊤ ΣV ]. This proves (i). Step 3. By N4, T − ET = kT (ζ − µζ ) + ε − Eε with (ζ, ε) independent of (S, b). Then Cov(X1 , T ) = E[a] E[(ζ − µζ )(T − ET )] = E[a] (kT σζ2 + Cov(ζ, ε)) = kT σζ2 E[a], and every square-integrable function of (S, b) is uncorrelated with T , so Cov(ã, T ) = Cov(m̃, T ) = Cov(ξ, T ) = Cov(F, T ) = 0. The closed linear span of (1, F, X) equals that of (1, F, X̃), F and X̃ are uncorrelated, and Cov(F, T ) = 0; the projection of T − ET on this span is therefore its projection on X̃, whose squared norm divided by Var(T ) is the stated R2 (Var(X̃) ⪰ E[V ⊤ ΣV ] ≻ 0 by (i)). This proves (ii). Step 4. By (i), Var(X̃) ⪰ E[V ⊤ ΣV ], so Var(X̃)−1 ⪯ E[V ⊤ ΣV ]−1 and, by (ii), R2 (c) = c2 kT2 σζ4 E[a]⊤ Var(X̃)−1 E[a]/ Var(T ) ≤ c2 Υ E[a]⊤ E[V ⊤ ΣV ]−1 E[a]. Expanding (i), Var(X̃) = Ω+cX2 +c2 X3 with X2 := µζ (E[ãm̃⊤ ]+E[m̃ã⊤ ]) and X3 := σζ2 E[aa⊤ ]+µ2ζ E[ãã⊤ ]. Matrix inversion is continuous at the positive definite Ω, so R2 (c)/c2 → Υ E[a]⊤ Ω−1 E[a]; for p = 1, Ω = E[v ⊤ Σv] + Vm (v) and the limit is ΥI(v). If ã = 0 or m̃ = 0 then X2 = 0 and 1/R2 (c) = Var(T )(Ω + c2 X3 )/(c2 kT2 σζ4 (Ea)2 ), which is C0 + C1 /c2 with the stated constants. This proves (iii). Step 5. Fix w ∈ Rp \ {0} and let v := V w, so B ⊤ v = 0 almost surely and E[v ⊤ Σv] = w⊤ E[V ⊤ ΣV ]w > 0. Lemma 1(2), applied patient by patient, gives w⊤ E[a] = E[v ⊤ q̄] = E[x⊤ z] with x := Σ1/2 v and z := Σ1/2 MB q̄. The Cauchy–Schwarz inequality in L2 (Pr; RN ) and MB ΣMB = MB give (w⊤ E[a])2 ≤ E∥x∥2 E∥z∥2 = (w⊤ E[V ⊤ ΣV ]w) E[q̄ ⊤ MB q̄]. Since E[a]⊤ E[V ⊤ ΣV ]−1 E[a] = supw̸=0 (w⊤ E[a])2 /(w⊤ E[V ⊤ ΣV ]w), the inequality in (iv) follows, and if E[q̄ ⊤ MB q̄] = 0 the nonnegative left side is 0. Let E[q̄ ⊤ MB q̄] > 0. As E∥x∥2 > 0 and E∥z∥2 > 0, equality in the Cauchy–Schwarz step holds iff x = αz almost surely for a constant α, necessarily α ̸= 0, that is iff V w = αMB q̄ almost surely. If equality holds in (iv), then E[a] ̸= 0 (otherwise the left side is 0 < E[q̄ ⊤ MB q̄]), the maximizer w∗ := E[V ⊤ ΣV ]−1 E[a] ̸= 0 attains the supremum, and the Cauchy–Schwarz step is an equality at w∗ . Conversely, if V w = αMB q̄ almost surely for some w ̸= 0, the Cauchy–Schwarz step is an equality at w, so the supremum is at least E[q̄ ⊤ MB q̄]. Finally α = 0 is impossible, since then V w = 0 almost surely and w⊤ E[V ⊤ ΣV ]w = 0. This proves (iv). Step 6. By Lemma 1(3), q̄ K⊤ MB K q̄ K is the supremum of (v ⊤ q̄ K )2 /(v ⊤ ΣK v) over v ̸= 0 with B K⊤ v = 0. Extend such a v by zeros to v ′ on the channels of K′ . By ′ ′ ′ block-diagonality B K ⊤ v ′ consists of B K⊤ v = 0 and zeros, and v ′⊤ q̄ K = v ⊤ q̄ K , v ′⊤ ΣK v ′ = v ⊤ ΣK v. Every ratio ′ for K is therefore a ratio for K , and the supremum cannot decrease. This proves (v). □ Proof of (a). Given (S, ζ), N2 gives E[v ⊤ y | S, ζ] = v ⊤ µ + cζv ⊤ q̄ and Var(v ⊤ y | S, ζ) = v ⊤ Σv. Part (i) is Lemma 1(3) with nB = 0 (MB = Σ−1 , and Σ−1 q̄ = 0 iff q̄ = 0). Part (ii) is Lemma 1(3) to (5) with θ = cζ, β the free coefficients and e = b − µ recentered on the known part; eff(v) ≤ 1 is (3). For (iii), the bound on R2 (c) is Lemma 3(iii) combined with (iv). For p = 1, 1/C1 = ΥI(v) by Lemma 3(iii), and I(v) ≤ (Ea)2 /E[s2v ] ≤ E[q̄ ⊤ MB q̄] by Lemma 3(iv). If E[q̄ ⊤ MB q̄] = 0, then Ea = 0 and I(v) = 0. Otherwise the second inequality is an equality iff v = αMB q̄ almost surely with α ̸= 0, and then Ea = α E[q̄ ⊤ MB q̄] ̸= 0, so the first is an equality iff Vm (v) = 0. When µ ∈ span(B) and B ⊤ v = 0, v ⊤ µ = 0 and Vm = 0. For the factorization, a2 /s2v = eff(v) q̄ ⊤ MB q̄ holds everywhere: on {sv > 0, q̄ ⊤ MB q̄ > 0} by definition; on {sv > 0, q̄ ⊤ MB q̄ = 0} because then a = 0 by (ii); and on {sv = 0} because Σ ≻ 0 forces v = 0 and a = 0. Hence E[eff(v) q̄ ⊤ MB q̄] = E[a2 /s2v ], which is positive when Pr(a ̸= 0) > 0, and E[q̄ ⊤ MB q̄] ≥ E[a2 /s2v ] > 0. With Vm = 0, (Ea)2 E[a2 /s2v ] (Ea)2 I(v) = = · . ⊤ 2 ⊤ ⊤ 2 E[q̄ MB q̄] E[sv ] E[q̄ MB q̄] E[q̄ MB q̄] E[a /s2v ] E[s2v ] By Cauchy–Schwarz, (Ea)2 = (E[(a/sv ) sv 1{sv > 0}])2 ≤ E[a2 /s2v ] E[s2v ], and since both factors are positive, equality holds iff a/sv = λsv almost surely on {sv > 0} for a constant λ, that is iff a/s2v is almost surely constant there. Monotonicity in K is Lemma 3(v), and taking expectations carries it to the pooled bound. Lemma 3 concerns population least squares on features that are linear in y with design-measurable weights; the Bayes predictor E[T | record] is in general nonlinear in the record (the trait has a bounded, non-Gaussian law and the information varies across patients), so an estimator can exceed these bounds. 64

Proof of (b). Under N5, Σ = σ 2 I and B2 has rank 3 on 90 distinct rows, so MB2 = σ −2 (I−B2 (B2⊤ B2 )−1 B2⊤ ) = σ −2 Π, where Π = I − B2 B2+ is exactly the implemented projector (B2+ = (B2⊤ B2 )−1 B2⊤ ). With q̄ = Λq, Πq = σ 2 Λ−1 MB2 q̄ attains Lemma 1(3), and the generalized least-squares estimate q̄ ⊤ MB2 y/q̄ ⊤ MB2 q̄ of cζ, multiplied by Λ, is (Πq)⊤ y/∥Πq∥2 (the partialling of Frisch and Waugh 1933). At kinetics K, v = ΠpK satisfies 2 ⊤ 2 ⊤ ⊤ 2 −2 ⊤ B2⊤ v = 0, (v ⊤ q̄)2 = Λ2 (p⊤ q Πq, which gives the stated K Πq) , v Σv = σ pK ΠpK and q̄ MB2 q̄ = Λ σ ⊤ ⊤ ⊤ efficiency. For the increment statistic, vinc := ∇ ∇q and B2 vinc = (∇B2 ) ∇q. Since ∇1 = 0, (∇t)(t) = 1 and (∇t2 )(t) = 2t − 1, 89  ⊤ X B2⊤ vinc = 0, q(89) − q(0), (2t − 1)(∇q)(t) ,

89 88 X X (2t − 1)(∇q)(t) = 177 q(89) − 2 q(s) − q(0),

t=1

t=1

s=1

P by summation and q(0) = 0. Let the doses A(s) ∈ [0, 1] vanish outside rows 6 to 53 and s A(s) > 0. Pby parts, Then e(t) < k≥0 0.8k = 5, so q(t) = e(t)/(1 + e(t)) ≥ e(t)/6, and q(t) ≤ e(t). First, 177 q(89) ≤ 177 e(89) = P53 P P 177 s=6 0.887−s A(s) ≤ 177 · 0.834 s A(s) < 0.0898 s A(s). Second, 2

88 X t=1

q(t) ≥

88 88 X 1X 5X 1X e(t) = A(s) 0.8 t−2−s = A(s) (1 − 0.887−s ) 3 t=1 3 s 3 s t=s+2 X X 5 ≥ (1 − 0.834 ) A(s) > 1.665 A(s). 3 s s

P ⊤ Hence (t2 )⊤ vinc < (0.0898 − 1.665) s A(s) < 0, B2⊤ vinc ̸= 0, and by Lemma 1(4) the law of vinc y depends on the free quadratic coefficient; its mean is unbounded over β2 . With the mean known (nB = 0), its efficiency is ⊤ ⊤ (vinc q̄)2 /(σ 2 ∥vinc ∥2 · σ −2 ∥q̄∥2 ) = (Q◦ )2 /(∥vinc ∥2 ∥q∥2 ), because vinc q = ∥∇q∥2 = Q◦ . By (a)(i), equality would ⊤ require ∇ ∇q = αq for some α. Let t1 be the first row with A(t1 ) > 0. Then q(t1 ) = q(t1 + 1) = 0 and e(t1 + 2) = A(t1 ) > 0, so q(t1 + 2) > 0; row t1 + 1 ∈ [7, 54] is interior, so (∇⊤ ∇q)(t1 + 1) = 2q(t1 + 1) − q(t1 ) − q(t1 + 2) = −q(t1 + 2) ̸= 0 = αq(t1 + 1). The efficiency is therefore strictly below 1. Proof of (c). (i) Under N6 the level is a free nuisance with B = 1 on the draw rows; by Lemma 1(4) the admissible statistics satisfy 1⊤ v = 0, and by Lemma 2 they are the linear functions u⊤ ∆yD of ∆yD := (∆0 y, . . . , ∆n−1 y). Given (S, ζ), ∆yD = cζΛ ∆q + ∆b with E[∆b | S] = 0 and Cov(∆b | S) = G := σb2 diag(g0 , . . . , gn−1 ), because ∆j b sums the hourly increments over the disjoint blocks (dj , dj+1 ]. Lemma 1(3) with nB = 0 applied to (Λ∆q, G) gives (u⊤ Λ∆q)2 /(u⊤ Gu) ≤ Λ2 ∆q ⊤ G−1 ∆q = Λ2 J/σb2 , with equality iff u is a nonzero multiple of G−1 ∆q, that is iff the statistic is a nonzero multiple of G (uj = ∆j q/gj ). For any positive finite level variance uncorrelated with the increments, the covariance of the draw values is positive definite, ⊤ M1 q̄D and for 1⊤ v = 0 the variance v ⊤ Σv = u⊤ Gu does not involve the level variance. By Lemma 1(3), q̄D 2 ⊤ ⊤ is the supremum over 1 v = 0, which by Lemma 2 equals the supremum over u, so q̄D M1 q̄D = Λ J/σb2 . (ii) For every draw, dik (dj ) = dj , so Z(dj ) = y(dj ) and ∆j Z = ∆j y. The implemented reader takes as draws row 0 and the rows where the recorded value changes; this is Dik except where the latent value at a draw P ⊤ equals that at the previous draw. Hence C = ∆ y ∆ j j q = u ∆yD with u = ∆q, a function of the draw j values alone, and the hold enters only through Dik . Its signal is u⊤ Λ∆q = ΛQ(0) per unit cζ and its noise variance is u⊤ Gu = σb2 Q(1) , so by (i) its efficiency is Λ2 (Q(0) )2 /(σb2 Q(1) · Λ2 J/σb2 ) = (Q(0) )2 /(Q(1) J) ≤ 1. Equality holds iff ∆q is a multiple of G−1 ∆q, that is iff gj is the same on {j : ∆j q ̸= 0}. On hourly records every gj = 1 and C = G. (iii) Apply Lemma 3 with K = {k}, p = 1, v the weight vector of C on the latent rows and F S = Q(0) , so a = ΛQ(0) and ã = 0. Under N6, µ = E[b | S] is constant in t and 1⊤ v = 0, so m = 0 and Vm = 0. Hence 1/R2 = C0 + C1 /c2 exactly with 1/C1 = ΥI(v) = ΥΛ2 (EQ(0) )2 /(σb2 EQ(1) ). For G with ⊤ companion J: a = ΛJ and s2v = σb2 J, so 1/C1 = ΥΛ2 EJ/σb2 = Υ E[q̄D M1 q̄D ]. For any pooled linear reader of this channel’s held record whose statistics are level-invariant (1⊤ V = 0 on the draw rows), Lemma 3(iii) and ⊤ (iv) give 1/C1 ≤ Υ E[a]⊤ E[V ⊤ ΣV ]−1 E[a] ≤ Υ E[q̄D M1 q̄D ], with any number of statistics. (Under a free level any statistic that is not level-invariant has unbounded variance.) On the hourly record of the same baseline every row is a draw, g ≡ 1, Q(0) = Q(1) = J = Q◦ , and both readers have 1/C1 = ΥΛ2 EQ◦ /σb2 . Therefore fC2 = EQ(1) EQ◦ /(EQ(0) )2 and fG2 = EQ◦ /EJ. Finally (EQ(0) )2 ≤ EQ(1) EJ by the Cauchy–Schwarz inequal

1/2 P
1/2 P P √ √
E j (∆j q)2 /gj ; this is gj |∆j q| |∆j q|/ gj ≤ E j gj (∆j q)2 ity over patients and intervals, E j fC ≥ fG . Proof of (d). By Lemma 3(iii), Rℓ2 (c)/c2 → ΥI(vℓ ) for ℓ = 1, 2, with Υ > 0 by N4. As I(v2 ) > 0, R22 (c) > 0 for small c, the ratio converges to I(v1 )/I(v2 ), and if I(v1 ) > I(v2 ) there is c0 > 0 with R12 (c) > R22 (c) for ⊤ 0 < c < c0 . On hourly records every row is a draw, so the cadence-aware statistic is vC y with vC := ∇⊤ ∇q (its increment weights are (∇q)(t), t = 1, . . . , 89, by Lemma 2), and the plain statistic is vP⊤ y with vP := Πq. 65

Both are level-invariant: 1⊤ Πq = (Π1)⊤ q = 0 and 1⊤ ∇⊤ ∇q = (∇1)⊤ ∇q = 0. Under N7 the mean is constant, ⊤ so Vm = 0 and I(v) = (E[v ⊤ q̄])2 /E[v ⊤ Σϑ v], with E[vP⊤ q̄] = Λ E[q ⊤ Πq] and E[vC q̄] = Λ EQ◦ ; the factors Λ2 2 and σAR cancel in R. The entries of Σϑ are continuous on [0, 1) and the integrands are bounded, so R is continuous therePby dominated convergence. Random-walk limit. For 1⊤ v = 0 write the stationary AR(1) as ′ t 2 2 /(1 − ϑ2 ) and independent innovations ηt′ of variance σAR . x(t) = ϑt x(0) + t′ =1 ϑt−t ηt′ with Var x(0) = σAR Then
P 89  X 2 2 t 2 X σAR t vt ϑ 2 t−t′ ⊤ + σ v ϑ . v Σϑ v = t AR 1 − ϑ2 ′ ′ t =1

P

t

P

t

t≥t

2 Because (1 − t |, so the first term is at most σAR (1 − t vt = 0, | t vt ϑ | = | t vt (ϑ − 1)| ≤P P Pϑ) t t|v 2 2 2 ⊤ ϑ)( t t|vt |) /(1 + ϑ) → 0; the second tends to σAR t′ ( t≥t′ vt ) , which is the variance of v y under N6 with every row a draw and σb = σAR (Lemma 2: the weight on the increment from row t′ − 1 to row t′ is P t≥t′ vt ). Both terms are bounded by bounded functions of the design uniformly in ϑ, the numerators of I do not depend on ϑ, and the limiting denominators are positive, so R(ϑ) → RRW . Strict inequality at the limit. 2 Under N6 with every row a draw, (c)(iii) and Lemma 3(iv) show that I(vC ) = Λ2 EQ◦ /σAR = E[q̄ ⊤ M1 q̄] is the largest value of I over level-invariant weights, and that another level-invariant v attains it iff v = αM1 q̄ almost surely with α ̸= 0, that is iff its increment weights equal α′ P (∇q)(t) almost surely for one constant α′ ̸= 0. The weight of vP on the increment from row 0 to row 1 is t≥1 (Πq)(t) = −(Πq)(0), because 1⊤ Πq = 0, whereas (∇q)(1) = q(1) − q(0) = 0 for every dosing. Equality would therefore force (Πq)(0) = 0 almost surely, so Pr{(Πq)(0) ̸= 0} > 0 implies I(vP ) < I(vC ) under N6, that is RRW < 1. Numerical part. For the 400 reconstructed discovery dosings (equal weights), |(Πq)(0)| ≥ 2.29 × 10−4 for every dosing, R(0) = 9.221 and RRW = 0.110. By continuity, the limit and the intermediate value theorem, R−1 has a zero in (0, 1). On the grids {0, 0.01, . . . , 0.99} ∪ {0.9995} and {0, 0.001, . . . , 0.999} ∪ {0.9995}, R is strictly decreasing and R − 1 changes sign once, between 0.64 and 0.65 on the first grid and between 0.642 and 0.643 on the second; bisection gives ϑ∗ = 0.6421, whose population detrended lag-1 autocorrelation (the population version of the shipped pooled estimator) is 0.583. At ϑ = 0.7537, the persistence at which the population version of the shipped pooled estimator equals 0.687, R = 0.546. For the common FIX19 occupancy, R(0) = 23.57, RRW = 0.254, one sign change between 0.806 and 0.807, ϑ∗ = 0.8068, and R(0.7537) = 1.426. Monotonicity and uniqueness of the crossing are established on these grids only.

P

P

Proof of (e). Fix
kinetics K. With common doses pK is common to all patients. The plain feaThe assimilating features, tures (ΠpK )⊤ Z·k k are a linear map of Zi with common coefficients. ⊤ 2 −9 2 (ΠpK ) Z·k /(∥ΠpK ∥ + 10 ), ∥ΠpK ∥ , pK (89), the mean of pK , and the channel means of Zi (as implemented), are affine in Zi with common coefficients. The window-mean features average Z·k over four windows fixed by the first and last dose, which are common. Each feature vector is therefore Φ(Zi ) = AΦ vec(Zi ) + aΦ with (AΦ , aΦ ) the same for every patient, whether Zi is the hourly or the held record, so span{1, Φ(Zi )} ⊆ span{1, vec(Zi )} in L2 . Standardizing features is affine and does not change these spans. Population least squares is orthogonal projection, and projecting τ on a larger closed subspace cannot lower R2 ; this uses only finite second moments of (Zi , τ ) and not N1. Appending the same further regressors to both sides, such as the draw-step indicator of a next-draw target, preserves the inclusion. With kinetics from a finite grid Kgrid the best kinetics has R2 = maxK∈Kgrid R2 (K), and each term is at most P the history-blind value. On hourly records without exact ties every row is a draw for every channel, so Ck = t (∇Z·k )(t) (∇pK )(t) is linear in Zi with P (0) common coefficients and Qk = t ((∇pK )(t))2 is a constant. On held records the cadence-aware weights and companions depend on Dik , which differs across patients. The oracle and draw-step-only rungs are not functions of the record and are excluded. Hence any population advantage of a record-based reader over the history-blind reader under a shared schedule requires patient-specific draw sets. On hourly records under a shared schedule, weights that depend on the design only through the doses are common; with Σ := Cov(bK ) and common B, Lemma 1 applies to every patient with the same (Σ, B, q̄ K ), so MB q̄ K is a common vector and (MB q̄ K )⊤ y K , with y K read from Zi , lies in span{vec(Zi )}. These are population statements; the history-blind ridge estimates 1,080 coefficients from 320 training persons, so out-of-fold orderings can reverse. Numerical checks. The checks use reconstructed doses, the shipped per-channel summaries and simulation; none reads a new label. • (b). For 40,000 random dosings with doses in [0, 1] on rows 6 to 53, (t2 )⊤ ∇⊤ ∇q < 0 in every case, with e < 5 and q ≥ e/6 throughout; the constants are 177 · 0.834 = 0.0897 and 35 (1 − 0.834 ) = 1.6658, and the 400 reconstructed discovery dosings give at most −25.30. The closed form of (t2 )⊤ ∇⊤ ∇q matches direct computation to 7 × 10−13 . With the mean known, the efficiency of the hourly increment statistic under white noise has median 0.020 (maximum 0.063) over the discovery dosings.

66

• (c). J equals the level-invariant bound of (c)(i) for every patient. Under a random walk at heart-rate cadence (Bernoulli draw proxy at each real stay’s change rate, mean 0.677; 5 replicates), the per-patient efficiency of C is at most 1.0000, with median 0.915 and information-weighted mean 0.903; the second factor of (a)(iii) is 0.468, so the pooled efficiency of C is 0.422, and fC = 1.849, fG = 1.201. Rescaled to the panels’ mean change rate of 0.58, the median is 0.873, the information-weighted mean 0.849, the second factor 0.499, the pooled efficiency 0.424, fC = 1.979 and fG = 1.288. A Monte Carlo with 300,000 patients at heart-rate cadence under a random walk matches the exact law 1/R2 = C0 + C1 /c2 of (c)(iii) to the third decimal at c = 0.1, 0.3, 0.75 and 2 for both readers. • (d). The grids and crossings are as stated. The detrended lag-1 autocorrelation 0.687 of N7 corresponds to persistence 0.754, and ϑ∗ = 0.642 to a detrended lag-1 autocorrelation of 0.583. Adding the ordinary treatment route of the generator to the mean raises the plain reader’s noise by 1.2% and the cadenceaware reader’s by 5.2%, which moves R(0.754) from 0.546 to 0.568. An independent re-implementation reproduced these values. Comparison with the recorded reader orderings. The shipped recoverability runs order the plain and cadence-aware readers as follows; Table S15 collects the numbers. The comparison uses the ratio of R2 /(1 − R2 ) of the plain reader to that of the cadence-aware reader, which estimates the ratio of their low-signal slopes 1/C1 when C0 is close to 1. • Synthetic tapes. The synthetic records of the panels are smooth curves (an offset, a slope and a sinusoid per channel) plus small noise, far from a persistent random baseline, and the plain reader was better (observed ratio 2.14 to 2.26 at c = 0.75). The recipe covariance of the synthetic records gives 1.53 for heart rate alone, 2.02 for the four channels that carry phenylephrine’s response and 4.69 for all twelve channels, so the direction agrees and the single-channel value does not match quantitatively. • Real records observed hourly. At the persistence calibrated on real heart rate the single-carrier model gives R = 0.55, so the cadence-aware reader should be better on hourly phenylephrine records. It was: recovery 0.369 against 0.310 at c = 0.75 and 0.453 against 0.385 at c = 1, with two-point-fit ratios of 0.67 to 0.88. The direction agrees; the model overstates the cadence-aware advantage by a factor of about 1.2 to 1.6. For NaCl on creatinine the readers tie at c = 1 (0.345 and 0.342), which the model allows: creatinine’s detrended lag-1 autocorrelation of 0.602 corresponds to persistence 0.662, near ϑ∗ , where R = 0.90. For norepinephrine on heart rate the readers also tie at c = 1 (0.393 and 0.392), against a predicted R of about 0.54; that run has seven loaded channels and lies outside the low-signal regime, and no low-signal hourly run with the cadence-aware reader was shipped, so this mismatch is unexplained. The hourly ordering is thus confirmed in one of the three pairs, a near tie in a second, and unexplained in the third. • Held records at real cadence. Part (d) is an hourly statement and does not predict held orderings. The cadence-aware reader was better in all 10 informative held real-record runs across the three pairs. A twelve-channel model with a tape-plus-generator baseline, calibrated to the shipped per-channel background standard deviations and per-stay change rates, gives held ratios of 0.27, 0.34 and 0.28 against the observed 0.35, 0.30 and 0.38 at c = 1. On the fixed schedule the AR(1) model at persistence 0.754 predicts the plain reader to be better on held records (1.86), whereas the shipped held fixed-schedule cells show the cadence-aware reader at least as good, and the tape-plus-generator model agrees with the cells (0.87 on heart rate alone, 0.55 on twelve channels). No hourly fixed-schedule run was shipped, so ϑ∗ = 0.807 for FIX19 is untested. • The plain reader’s loss at real cadence. The ratio of the plain reader’s C1 on held records to that on hourly records is 5.9 (phenylephrine), 7.2 (norepinephrine) and 19.8 (NaCl) at c = 1. Sampling of the scored channel alone explains about 1.4 on heart rate in a tape-plus-generator model (1.38 and 1.40). With the other carriers and the cross-channel structure the twelve-channel model gives 7.36, 5.60 and 8.29, so it accounts for most of the loss in the two heart-rate cells and under-predicts NaCl on creatinine by a factor of about 2.4. A per-channel refit of the registered readers on the existing panels is the direct test; the attribution is not confirmed. • A 1/gap-weighted reader. The prediction that G needs less amplitude than C at real cadence holds only for a single carrier whose draw-to-draw increment variance grows with the gap (N6). If each jump has constant variance, C is the efficient level-invariant statistic by Lemma 1(3) applied to the increments, and in a tape model calibrated to the lag-1 autocorrelation of 0.687 the held information of G relative to C is 0.83 (against 1.42 with gap-proportional variance). In the twelve-channel model the heart-rate amplitude factor falls only from 2.09 to 1.81. The claim is also a low-signal one: in the random-walk Monte Carlo G has the larger C0 (1.43 against 1.20). The shipped aggregates cannot check whether jump variance grows with the gap; a regression of the squared draw-to-draw change on the gap in the real tapes would.

67

Supplementary Table S15: Plain against cadence-aware readers (Proposition 5). Entries below the first block are ratios of the plain reader’s information to the cadence-aware reader’s (values below 1 favor the cadence-aware reader): for the models, the ratio R of low-signal slopes; for the observed runs, the ratio of R2 /(1 − R2 ) at c = 1. Single carrier: the scored channel alone with an AR(1) baseline at the persistence calibrated from its detrended lag-1 autocorrelation (heart rate 0.754, creatinine 0.662). Twelve channels: a tape-plus-generator baseline calibrated to the shipped per-channel background standard deviations and per-stay change rates, with every channel that carries the response; scored channel only: the same model restricted to the scored channel. C1 ratio: held over hourly, for the plain reader, whose square root is its low-signal amplitude factor. All model values are idealized and descriptive. Phenylephrine on heart rate

Norepinephrine on heart rate

NaCl on creatinine

0.642

0.640

0.642

0.55 0.46 0.70

0.54 0.53 1.01

0.90 0.46 1.02

0.27 0.35

0.34 0.30

0.28 0.38

1.38 7.36 5.9

1.40 5.60 7.2

> 103 8.29 19.8

Crossing persistence ϑ∗ (single carrier) Hourly records Single carrier, calibrated persistence Twelve channels Observed (c = 1) Held records at real cadence Twelve channels Observed (c = 1) Plain reader, C1 held over hourly Scored channel only Twelve channels Observed (c = 1)

5.6 Calibration geometry and transport Proposition 6 (Calibration geometry and transport: how far a predictor can beat zero, and why structure-informed learners need no treated query). Standard and specific parts. Part (a1) is the decomposition of the mean squared error into a correlation term and a conditional-bias term that is standard in forecast verification (Murphy, 1988); (a2) to (a6) are elementary corollaries, and (b) collects standard properties of conditional expectations. Parts (c) and (d), transport to a query regime of propensity zero and the spread of an empirical-Bayes random slope under baseline overdispersion, are specific to this benchmark. The spread ratio u of this section is unrelated to the generator’s latent severity ut of Supplementary Note 4. Notation. τ is the washout target of a patient (Definition 4) and τ̂ a predictor of it. Both are square integrable and Var(τ ) > 0. Put: • στ := SD(τ ) and u := SD(τ̂ )/στ (the spread ratio); • r := corr(τ̂ , τ ) if SD(τ̂ ) > 0, r := 0 otherwise, and r+ := max(r, 0); • cPEHE := SD(τ̂ − τ ), R2 := 1 − E(τ̂ − τ )2 / Var(τ ) and rec := 1 − {E(τ̂ − τ )2 }1/2 /στ ; • for a σ-field G, R∗2 (G) := Var E[τ | G]/ Var(τ ). The constant prediction 0 has centered PEHE στ , and τ̂ beats zero if cPEHE < στ . Moments are population moments. Part (a) holds verbatim for the empirical distribution of n pairs (τ̂i , τi ) with divisor n. Part (a) is an identity. It states whether and by how much a predictor beats zero; it does not by itself explain why a given learner fails to. (a) Calibration geometry (any predictor). (a1) cPEHE2 / Var(τ ) = (1 − r2 ) + (u − r)2 , an information term plus a dispersion term. (a2) τ̂ beats zero if and only if 0 < u < 2r. (a3) minθ0 ,θ1 ∈R SD(θ0 + θ1 τ̂ − τ ) = στ (1 − r2 )1/2 . If u > 0 the minimum is attained at θ1 = r/u, and the recalibrated predictor has spread ratio and correlation both equal to |r|. Over θ1 ≥ 0 (recalibrations that keep the ranking) the minimum is 2 1/2 στ (1 − r+ ) . The dispersion term (u − r)2 of (a1) vanishes exactly when u = r, which requires r ≥ 0. (a4) |στ − cPEHE| ≤ SD(τ̂ ). Hence the gain over zero satisfies στ − cPEHE ≤ SD(τ̂ ) = u στ , a bound computable without labels. Also cPEHE ≥ |1 − u| στ . (a5) R2 = r2 − (u − r)2 − (Eτ̂ − Eτ )2 / Var(τ ) ≤ r2 , and rec = 1 − (1 − R2 )1/2 . (a6) If u > 0, the least-squares slope of τ on τ̂ equals r/u. Hence u = r if and only if Cov(τ − τ̂ , τ̂ ) = 0. (b) Conditional expectations. Let G be a σ-field. (b1) If τ̂ = E[τ | G], then Eτ̂ = Eτ , u = r, r2 = R2 = R∗2 (G) and rec = 1 − (1 − r2 )1/2 . (b2) Every square-integrable G-measurable f satisfies corr(f, τ )2 ≤ R∗2 (G),

R2 (f ) ≤ R∗2 (G),

cPEHE(f ) ≥ στ {1 − R∗2 (G)}1/2 .

If G ⊆ H then R∗2 (G) ≤ R∗2 (H). Let Xi be the data on patient i that a predictor reads. Suppose f is a function of Xi and of data T independent of (Xi , τi ), for example a rule fitted on other, independent patients. Then the 68

same bounds hold with G = σ(Xi ). For the supervised readers and the oracle-structure learner, Xi = Oi as defined in (c). For a learner that also reads the admission row, Xi includes that row. (b3) τ̂ = E[τ | G] for some σ-field G if and only if E[τ | τ̂ ] = τ̂ . The condition u = r is necessary for this but not sufficient. Consequently, u ̸= r shows that τ̂ is not the conditional expectation of τ given any information. (c) Transport. For patient i let: • Si be the design (doses of j ∗ and draw sets); • Zi be the held pre-decision record (rows 0 to 89, 12 channels), and Oi := (Si , Zi ); • Zi0 be the zero-effect record (same panel row, cap 0); • ζi be the trait, with law πζ ; • Oi+ be all factual data on patient i: the admission covariates, rows 0 to 89 of every channel and treatment, the design, and any factual rows recorded after the decision. Let Ri be the realized treatment regime on rows 90 to 101, and let 1̄ be the query regime, in which j ∗ is given on each of these rows. Under NATMIN and FIX19, Ai (t) = 0 for every t ≥ 54, so P (Ri = 1̄) = 0: the query regime has propensity zero. (c1) Non-identification. Let Yi be the post-decision potential outcomes of patient i under every regime in both frames. Let P be the set of joint laws of (Oi+ , ζi , Zi0 , Yi ) that • satisfy consistency (factual post-decision outcomes equal the frame-A potential outcomes under Ri ), and • give (Oi+ , ζi , Zi0 ) its actual law. Then: • For every measurable g with E|g(Oi+ )| < ∞, some law in P has E[τi | Oi+ ] = g(Oi+ ) almost surely. • For every fixed measurable f with 0 < Var f (Oi+ ) < ∞, every δ ≥ 0 and every ϱ ∈ [−1, 1], some law in P has cPEHE(f ) = δ, and some has corr(f, τi ) = ϱ. • Apply the construction to every patient with independent auxiliary draws. This keeps patients independent and identically distributed and leaves every patient’s factual data unchanged. Since training uses factual data only, the accuracy statements hold for a learner’s rule given any realized training set. Hence the factual law, even together with T2 and T3(i)–(ii), determines neither E[τi | Oi+ ] nor the accuracy of any predictor of τi . Every estimator relies on an assumption that links the query regime to the factual law, that is, it transports. In (c2) the assumptions that play this role are T1 andRT3(iii). (c2) Structural transport. Assume T2 and T3, and put m(z | s) := k(z | s, ζ ′ ) πζ (dζ ′ ). Define Πζ (dζ | Oi ) :=

k(Zi | Si , ζ) πζ (dζ) m(Zi | Si )

on {m(Zi | Si ) > 0},

and Πζ (· | Oi ) := πζ on the complement, which has probability zero. Then Πζ is a version of the conditional law of ζi given Oi , so Z E[τ̄ (ζi , Si ) | Oi ] =

τ̄ (ζ, Si ) Πζ (dζ | Oi ).

This is a known function of Oi that uses no outcome observed under the query regime. TheR query regime enters only through the map τ̄ , which T3(iii) assumes known. If T1 also holds, then E[τi | Oi ] = τ̄ (ζ, Si ) Πζ (dζ | Oi ). Without T1, take τ̄ (ζ, s) := E[τi | ζi = ζ, Si = s]. Then Z E[τi | Oi ] − τ̄ (ζ, Si ) Πζ (dζ | Oi ) 2 ≤ {E Var(τi | ζi , Si )}1/2 ≤ {E Var(τi | ζi )}1/2 . L

(c3) Exposure is necessary. Suppose the design s has qs (d) = 0 at every draw d of every channel k with Λ k R ̸= 0. Then Πζ (· | Oi ) = πζ almost surely on {Si = s}, and the transported prediction is the constant τ̄ (ζ, s) πζ (dζ). This is the Bayesian form of Proposition 4(e). (c4) Identification of the trait law. If πζ is not given, the law of Oi identifies it under two conditions: • for some design s with P (Si = s) > 0, some channel k has a draw d with Λk qs (d) ̸= 0 at which, almost surely on {Si = s}, the attenuation is inactive and no clip binds; 0 • the characteristic function of Zik (d) given Si = s vanishes only on a Lebesgue-null set.

69

The second condition holds for every bounded zero-effect value, such as that of the implemented, clipped record. Its characteristic function extends to an entire function equal to 1 at 0, so its zeros are isolated. (d) Empirical-Bayes random slopes under baseline overdispersion. Fix one channel (or a fixed loading direction), a kernel and the working baseline. For patient i let p̃i and r̃i be the exposure profile and the ⊤ residual, both whitened by the working covariance. Put A := p̃⊤ i p̃i (the same for every patient), Bi := p̃i r̃i and Si := Bi /A1/2 . • Truth. The Si are independent and identically distributed with Si = A1/2 b∗i + εi , where Var(b∗i ) = σ∗2 > 0, Var(εi ) = ω0 and Cov(b∗i , εi ) = 0. Put SNR := Aσ∗2 and V := SNR + ω0 = Var(Si ). • Working model. r̃i = p̃i βi + ei , with βi ∼ N (µ, v) independent of ei ∼ N (0, I). The pair (µ, v) ∈ R × [0, ∞) is fitted by maximum likelihood on n patients, with all other parameters fixed. • Prediction and target. τ̂i := D Ew [βi | r̃i ] at the fitted values, where Ew is the working-model expectation, and τi = Db∗i with a constant D ̸= 0. (d1) As a function of (µ, v), the working likelihood depends on the data only through S1 , . . . , Sn . The estimates are A1/2 µ̂ = S̄ and 1 + Av̂ = max(V̂n , 1), where V̂n is the sample variance with divisor n. As n → ∞, 1 + Av̂ → max(V, 1) almost surely. Moreover Ew [βi | r̃i ] = (µ + vBi )/(1 + Av). (d2) Take a patient independent of the fitting sample, with the fitted values held fixed, and put ŵ := A1/2 v̂/(1 + Av̂). If v̂ > 0, then exactly r = (SNR/V )1/2 and u = ŵV 1/2 /σ∗ . If v̂ = 0, the prediction is constant, u = 0, and it does not beat zero. As n → ∞ there are three cases. • V > 1. Almost surely u → (V − 1)/(SNR V )1/2 , so u/r → 1 + (ω0 − 1)/SNR. This limit equals the ratio of the limiting fitted shrinkage weight to the weight of the best linear predictor of b∗i from Si . In the limit: – τ̂ is calibrated (u = r) if and only if ω0 = 1; – cPEHE2 / Var(τ ) → {ω0 SNR + (ω0 − 1)2 }/(SNR V ); – the limiting predictor beats zero if and only if ω0 − 1 < SNR. Almost surely, the fitted predictor beats zero for all large n when ω0 − 1 < SNR, and fails to for all large n when ω0 − 1 > SNR. The mean error tends to D Eεi /A1/2 and affects neither r nor u. • V < 1. Almost surely v̂ = 0 for all large n, so the prediction is constant and does not beat zero. • V = 1. Almost surely u → 0. On the event {v̂ > 0} the predictor beats zero for all large n, by a margin of at most uστ → 0 (a4). If ESi4 < ∞ and |Si − ESi | is not almost surely constant, this event has probability tending to 1/2. (d3) Extension. Suppose instead that τi = τ0 + Dτ b∗i + νi , where τ0 and Dτ are constants with DDτ > 0, and νi is square integrable and uncorrelated with b∗i and εi . Let ρ2lin := Dτ2 σ∗2 / Var(τi ). For a fitted v̂ > 0 and a patient independent of the fitting sample, r = ρlin (SNR/V )1/2 exactly. If V > 1, then almost surely: • u → ρlin (D/Dτ )(V − 1)/(SNR V )1/2 ; • u/r → (D/Dτ ){1 + (ω0 − 1)/SNR}; • the limiting predictor beats zero if and only if (D/Dτ ){1 + (ω0 − 1)/SNR} < 2. Assumptions. (A0) Moments. τ, τ̂ ∈ L2 and Var(τ ) > 0. Parts (a) and (b) are exact statements about population moments. – Part (a) uses only the bilinearity of covariance. It therefore holds exactly for the empirical distribution of the n scored patients with divisor n, which is the convention of the shipped metrics tables. – The spread ratio u is shipped as prediction_to_target_sd_ratio in the metrics files and is read from there. – u cannot be recovered reliably from rounded (r, cPEHE) entries such as those of Supplementary Table S4. By (a1), u = r ± {r2 − 1 + cPEHE2 / Var(τ )}1/2 has two roots, and the inversion is illconditioned when u is close to r. On S4 the nearer root is off by up to 0.05, and the choice of root is ambiguous. (A1) Learned predictors. Given its training folds, a cross-fitted predictor is a function of the test patient’s data. (b2) applies when the training data are independent of the test patient’s (Xi , τi ), as they are with independent patients. Out-of-fold metrics pool several fitted predictors, so they estimate these population quantities with sampling error. 70

(C0) Potential outcomes and dosing. The target is the frame-by-regime interaction of Definition 4, built from post-decision potential outcomes. Factual post-decision outcomes, if recorded, obey consistency in frame A. – natmin_actions writes binary doses only on rows 6 to 53. It copies the natural column on rows [6, 54); the top-up block starts at 6 + randint(0, 48 − need) (exclusive upper end) and so ends by row 52; the fallback sets idle rows in [6, 54). Only the column of j ∗ is nonzero. – FIX19 doses rows 6 to 24. – Hence Ai (t) = 0 for all t ≥ 54 and every patient of both designs, and P (Ri = 1̄) = 0. – The supervised readers and factual_response read Oi only. The generic unit learners also read the archived admission row, and the sequence learners read static features. No learner reads a row at or after row 90. T1 Invariance. E[τi | ζi , Oi ] = τ̄ (ζi , Si ) almost surely: the record carries no information about the query effect beyond the trait that also sets the pre-decision shift. In the generator this holds only approximately. There τi = τ own (ζi , ei (t0 )) + νi , where νi depends on the other four traits and on the post-decision state (Proposition 4(a)). (c2) bounds the resulting error. T2 Exogeneity. (Si , Zi0 ) is independent of ζi . In code: – the trait enters only the bounded shift, which feeds back into no state, policy or memory; – doses are harvested from a policy run on the same real window, which depends on the stay’s tape and latent state but not on the trait (verified by reconstructing all 800 NATMIN dose columns from the shipped u0 ); the design therefore depends on the baseline, and T2 asserts only its independence from the trait; – draw sets are fixed by the source tape. The pseudo-random streams of one person are treated as independent draws, the usual idealization. T3 Known structure. (i) Zi = M (Zi0 , Si , ζi ) for a known measurable map M , and the conditional law P 0 (· | s) of Zi0 given Si = s is known. In the generator, M adds the shift c α tanh(Λk ζi qi (d)) to the held zero-effect record at each draw d of each channel; attenuation and raw clips are known functions of the pre-shift value; and in the factual record only j ∗ is dosed. With T2, the conditional law of Zi given (Si , ζi ) = (s, ζ) is then a known kernel. It is assumed to have a jointly measurable density k(z | s, ζ) with respect to a σ-finite measure µs that does not depend on ζ. * The existence of such a density is an idealization. It holds when the zero-effect record has no atom at an effective bound at an exposed draw. It fails if, for example, a heart-rate value sits at the source clip (−4.33 z, 20 bpm) with positive probability at an exposed draw, because attenuation then moves that atom to a trait-dependent location. * Without the density, the conditional law of ζi given Oi still exists; only the ratio formula for Πζ needs the density. * The design takes finitely many values (binary doses by row, and subsets of rows 0 to 89 as draw sets), so no measurability question arises in s. (ii) The trait law πζ is known, or identified as in (c4). (iii) The query map τ̄ (ζ, s) is known and E|τ̄ (ζi , Si )| < ∞. This map is the only place where the query regime enters. T3 is what the oracle-structure learner (factual_response) is given: the generator’s mechanism, kernel, bounded map and loadings, and a particle prior over traits and policy. It is not given P 0 . It replaces the zero-effect likelihood by a Huber pseudo-likelihood (δ = 1.345) on factual increments, with nuisance scales fitted on training persons, so it approximates Πζ . (C4) For (c4). At the chosen draw, almost surely on {Si = s}, the attenuation is inactive (α = 1) and no clip binds. Also c > 0, P 0 (· | s) is known, and its characteristic function at that coordinate is nonzero Lebesgue-almost everywhere; the last condition is automatic for a bounded record. The first condition holds only approximately in the generator, where the attenuation band near the physiologic bounds is reached with small positive probability. (D) For (d).

71

(D1) Equal exposure information A across patients. In the panels it varies with dosing and draw pattern. (D2) A response linear in the slope along the working exposure direction, and a constant query contrast D. On the reconstructed real NATMIN doses the generator-kernel contrast is 0.6838 to 0.6842. A fitted kernel can have a different, patient-varying contrast, which (d3) and Proposition 7(d) treat. (D3) A Gaussian working model with unit noise along p̃i after whitening. All other parameters are held at their fitted values: kernel, baseline covariance, fixed effects, and, for rs_dlm and rs_dlm_mech, the noise variance σ 2 , which both profile in closed form in the shared _profile. At an interior joint optimum the conditional maximizer in λ at σ̂ 2 is the joint one, so this reading is consistent. (D4) Patients independent and identically distributed with finite second moments, in the population limit n → ∞. Only second moments of (b∗i , εi ) enter; the truth need not be Gaussian. Code correspondence: – ICare-RH stage 2 at a fixed kernel (_stage2 and posterior in revision_v3_washout_response_hybrid_core_v1.py). The prior is bi ∼ N (m, 1) with loading γ, so v = γ 2 and µ = γm. The posterior mean is (m + hi )/(1 + ai ), and the prior mean is profiled. – rs_dlm_mech (in revision_v3_washout_standard_cate_dlm_v1.py). Laird–Ware slope β + bi with bi ∼ N (0, σb2 ). Each person’s intercept and linear trend are projected out, and λ = σb2 /σ 2 ∈ [0, 100] is profiled. The BLUP is λe⊤ (r − Xβ)/(1 + λe⊤ e) and the prediction is (β + b̂i ) dqi . – rs_dlm (same file). A random slope on the 12-lag dose sum; the prediction is a constant plus 6.5 b̂i , so its query scale differs from the target’s. 2 2 – For both random-slope learners, p̃i = ei /σ̂, Ai = e⊤ i ei /σ̂ and v = σb , so (D1) is an idealization.

– For both, (d3) is the relevant idealized form. It also needs equal A and νi uncorrelated with (b∗i , εi ), and neither holds exactly, since νi depends on the post-decision state. – The upper bound λ ≤ 100 is assumed not to bind. Proof. Conventions. For X ∈ L2 write X c := X − EX and ∥X∥ := (EX 2 )1/2 , so SD(X) = ∥X c ∥ and Cov(X, Y ) = E[X c Y c ]. Let s := SD(τ̂ ) = uστ . With the convention r := 0 when s = 0, in every case Cov(τ̂ , τ ) = r s στ = r u στ2 .

(3)

Part (a) uses only (3) and bilinearity, so it holds for any probability measure, in particular for the empirical distribution of n pairs with divisor n. Proof of (a).

(a1) By (3), cPEHE2 = Var(τ̂ − τ ) = s2 − 2 Cov(τ̂ , τ ) + στ2 = στ2 (u2 − 2ru + 1),

and u2 − 2ru + 1 = (1 − r2 ) + (u − r)2 . (a2) The zero prediction has cPEHE = SD(−τ ) = στ . By (a1), cPEHE < στ if and only if u2 − 2ru < 0, that is u(u − 2r) < 0. Since u ≥ 0, this holds if and only if u > 0 and u < 2r. (a3) For θ0 , θ1 ∈ R, Q(θ1 ) := Var(θ0 + θ1 τ̂ − τ ) = θ12 s2 − 2θ1 rsστ + στ2 , which does not depend on θ0 . • If s = 0, then r = 0 and Q ≡ στ2 = στ2 (1 − r2 ). • If s > 0, Q is a convex quadratic with minimizer θ1∗ = rστ /s = r/u and Q(θ1∗ ) = στ2 (1 − r2 ). The recalibrated predictor τ̃ := θ0 + θ1∗ τ̂ has SD(τ̃ ) = |θ1∗ |s = |r|στ . When r ̸= 0, corr(τ̃ , τ ) = sign(θ1∗ ) r = |r|; when r = 0 both equal 0. On θ1 ≥ 0: • if r ≥ 0, then θ1∗ ≥ 0 and the minimum is unchanged; • if r < 0, then Q′ (θ1 ) = 2θ1 s2 − 2rsστ > 0 on [0, ∞), and the minimum is Q(0) = στ2 . 72

2 In both cases the minimum equals στ2 (1 − r+ ). Finally, the term (u − r)2 in (a1) is zero exactly when u = r. Since u ≥ 0, this cannot happen when r < 0. (a4) We have cPEHE = ∥τ̂ c − τ c ∥, στ = ∥τ c ∥ and SD(τ̂ ) = ∥τ̂ c ∥. The triangle inequality in L2 gives c | ∥τ ∥ − ∥τ̂ c − τ c ∥ | ≤ ∥τ̂ c ∥, and also ∥τ̂ c − τ c ∥ ≥ | ∥τ̂ c ∥ − ∥τ c ∥ | = |u − 1|στ . (a5) E(τ̂ − τ )2 = Var(τ̂ − τ ) + (Eτ̂ − Eτ )2 . Dividing by Var(τ ) and using (a1) gives

R2 = r2 − (u − r)2 − (Eτ̂ − Eτ )2 / Var(τ ). Since {E(τ̂ − τ )2 }1/2 /στ = (1 − R2 )1/2 , we get rec = 1 − (1 − R2 )1/2 . (a6) If s > 0, the least-squares slope is Cov(τ, τ̂ )/ Var(τ̂ ) = rsστ /s2 = r/u. Also Cov(τ − τ̂ , τ̂ ) = rsστ −s2 = sστ (r − u), which is zero if and only if u = r. □ Proof of (b). Let τ̂ := E[τ | G]; τ̂ ∈ L2 by the conditional Jensen inequality. For every G-measurable f ∈ L2 , E[f τ ] = E[E[f τ | G]] = E[f τ̂ ], and f ≡ 1 gives Eτ̂ = Eτ . Hence Cov(f, τ ) = Cov(f, τ̂ ),

E[f (τ − τ̂ )] = 0

for every G-measurable f ∈ L2 .

(4)

(b1) Take f = τ̂ in (4): Cov(τ̂ , τ ) = Var(τ̂ ), that is ruστ2 = u2 στ2 by (3), so u(u − r) = 0. If u > 0 then u = r; if u = 0 then r = 0 = u by convention. Hence r2 = u2 = Var(τ̂ )/ Var(τ ) = R∗2 (G). The mean error is zero, so (a5) gives R2 = r2 and rec = 1 − (1 − r2 )1/2 . (b2) Let f ∈ L2 be G-measurable. • Correlation. If Var f = 0, then corr(f, τ ) = 0 by convention. Otherwise, by (4) and the Cauchy–Schwarz inequality, | Cov(f, τ )| = | Cov(f, τ̂ )| ≤ SD(f ) SD(τ̂ ), so corr(f, τ )2 ≤ R∗2 (G). • R2 . τ̂ − f is G-measurable, so by (4) E(τ − f )2 = E(τ − τ̂ )2 + E(τ̂ − f )2 ≥ E(τ − τ̂ )2 . Because E[τ̂ (τ − τ̂ )] = 0 and Eτ = Eτ̂ , we have E(τ − τ̂ )2 = Eτ 2 − Eτ̂ 2 = Var(τ ) − Var(τ̂ ). Dividing by Var(τ ) gives R2 (f ) ≤ R∗2 (G). • cPEHE. τ − τ̂ has mean zero and f − τ̂ is G-measurable, so (4) gives Cov(f − τ̂ , τ − τ̂ ) = 0. Hence Var(f − τ ) = Var(f − τ̂ ) + Var(τ − τ̂ ) ≥ Var(τ ){1 − R∗2 (G)}, which is the bound on cPEHE(f ). • Monotonicity. If G ⊆ H, then E[τ | G] = E[E[τ | H] | G]. By the conditional Jensen inequality, E(E[τ | G])2 ≤ E(E[τ | H])2 . Both have mean Eτ , so Var E[τ | G] ≤ Var E[τ | H]. • Independent training data. Let T be independent of (Xi , τi ). For bounded measurable φ, ψ, E[τi φ(Xi )ψ(T )] = E[τi φ(Xi )] Eψ(T ) = E[E[τi | Xi ]φ(Xi )] Eψ(T ) = E[E[τi | Xi ]φ(Xi )ψ(T )]. The sets {Xi ∈ A} ∩ {T ∈ B} form a π-system generating σ(Xi , T ). By the π–λ theorem, E[τi | Xi , T ] = E[τi | Xi ], so R∗2 (σ(Xi , T )) = R∗2 (σ(Xi )). A predictor f (Xi , T ) is σ(Xi , T )-measurable, so the bounds above apply with G = σ(Xi ). Which data Xi a given predictor reads is recorded in (C0). (b3) If τ̂ = E[τ | G], then σ(τ̂ ) ⊆ G and, by the tower property, E[τ | τ̂ ] = E[E[τ | G] | τ̂ ] = τ̂ . Conversely, if E[τ | τ̂ ] = τ̂ , take G = σ(τ̂ ). By (b1) every conditional expectation has u = r, so u ̸= r excludes every G. The condition u = r is not sufficient. Take X ∼ N (0, 1), τ̂ := X and τ := X + X 2 − 1. Then Cov(τ, τ̂ ) = 1 = Var(τ̂ ), so the slope in (a6) is 1 and u = r (= 3−1/2 ), while E[τ | τ̂ ] = X + X 2 − 1 ̸= τ̂ . □ Proof of (c1). Setup. Let P be the actual joint law. Let Yi contain the potential outcomes on rows 91 to 102 for every post-decision regime and both frames. τi is the fixed linear combination of Definition 4; its frame-A, regime-1̄ coordinates YisA (1̄) enter with coefficient 1/h each. Every coordinate of Oi+ is of one of two kinds: • pre-decision (admission covariates, rows 0 to 89, the design), which involves no post-decision potential outcome; or • a factual post-decision outcome, which by consistency is a frame-A coordinate of Yi under Ri .

73

Since P (Ri = 1̄) = 0 (C0), on a set of probability one no coordinate of Oi+ involves any YisA (1̄). Construction. Let ∆ be any integrable variable measurable with respect to (Oi+ , Yi , ξ). Here ξ is a standard normal variable independent of everything else, obtained by a product extension of the probability space that leaves the law of (Oi+ , ζi , Zi0 , Yi ) unchanged. Define Yi′ from Yi by Yis′A (1̄) := YisA (1̄) + ∆

(s = 0, . . . , h − 1),

all other coordinates unchanged.

The variables (Oi+ , ζi , Zi0 ) are not changed, so their law is the actual one. Consistency still holds on {Ri ̸= 1̄}, a set of probability one. The law of (Oi+ , ζi , Zi0 , Yi′ ) therefore lies in P, and the target it defines is τi′ = τi + ∆. Conditional mean. With ∆ := g(Oi+ ) − E[τi | Oi+ ], we get E[τi′ | Oi+ ] = E[τi | Oi+ ] + ∆ = g(Oi+ ) almost surely. Accuracy. • Take ∆ := f (Oi+ ) + δξ − τi . Then τi′ = f + δξ, so cPEHE(f ) = SD(δξ) = δ, and Var(τi′ ) = Var f + δ 2 > 0. • Take ∆ := ϱf c / SD(f ) + (1 − ϱ2 )1/2 ξ − τi . Then Var(τi′ ) = 1 and Cov(f, τi′ ) = ϱ SD(f ), so corr(f, τi′ ) = ϱ. Several patients and learned rules. Suppose that under P the patients are independent and identically distributed. Fix one measurable map F and set ∆i := F (Oi+ , Yi , ξi ) for every patient, with ξ1 , ξ2 , . . . independent standard normals that are independent of everything else. The modified patient vectors are one fixed function of independent and identically distributed vectors, so they are again independent and identically distributed. Every patient’s factual data are unchanged pointwise. A learner trained on factual data therefore returns the same rule under the modified law. Given a realized training set, that rule is a fixed function f . Using the F of the previous step for this f gives a law in P, independent and identically distributed across patients and with the actual factual law, under which a new patient has cPEHE(f ) = δ or corr(f, τi ) = ϱ. Assumptions. T2 and T3(i)–(ii) restrict only the law of (Oi , ζi , Zi0 ), which the construction leaves unchanged. Every constructed law therefore satisfies them if the actual law does. T1 and T3(iii) restrict the query regime directly: they fix E[τi | ζi , Oi ] as a known function. A construction with E[∆ | ζi , Oi ] ̸= 0 violates them. Structural zero at s = 0. If one also requires the s = 0 term of τi to vanish (Proposition 4(a)), modify only YisA (1̄) for s = 1, . . . , h − 1, each by h∆/(h − 1). This again gives τi′ = τi + ∆, and nothing else changes. Conclusion. The factual law thus fixes neither E[τi | Oi+ ], nor cPEHE(f ), nor corr(f, τi ). A guarantee for any estimator must restrict how the query-regime outcomes relate to the factual variables. The usual route by overlap, E[Y (1̄) | X] = E[Y | R = 1̄, X], needs P (R = 1̄ | X) > 0 and is unavailable. The zero-propensity fact is needed only when factual post-decision outcomes are part of the data. For data that contain no postdecision row, the construction applies directly. Since σ(Oi ) ⊆ σ(Oi+ ), the non-identification of E[τi | Oi ] is the case where g is a function of Oi . □ Proof of (c2). Joint law. Let PS be the law of Si . It has finite support (T3(i)), so integrals over s are finite sums over designs with P (Si = s) > 0. By T2, ζi is independent of Si , and Zi0 is independent of ζi given Si , with conditional law P 0 (· | s). By T3(i), given (Si , ζi ) = (s, ζ), the record Zi = M (Zi0 , s, ζ) has the law of M (·, s, ζ) under P 0 (· | s), with density k(· | s, ζ) with respect to µs . Hence the joint law of (Si , ζi , Zi ) is PS (ds) πζ (dζ) k(z | s, ζ) µs (dz). Null sets. Let N := {(s, z) : m(z | s) = 0}. Then P (Oi ∈ N ) =

X

Z PS (s)

1N (s, z) m(z | s) µs (dz) = 0,

s

and on N , k(z | s, ζ) = 0 for πζ -almost every ζ. Now fix a jointly measurable h(ζ, s) with E|h(ζi , Si )| < ∞, and let Z n o Nh := (s, z) : |h(ζ, s)|k(z | s, ζ)πζ (dζ) = ∞ . By Tonelli’s theorem, E|h(ζi , Si )| =

X

Z PS (s)

Z µs (dz)

|h(ζ, s)|k(z | s, ζ)πζ (dζ) < ∞,

s

so µs ({z : (s, z) ∈ Nh }) = 0 for every s in the support, and P (Oi ∈ Nh ) = 0. Posterior formula. Set R h(ζ, s)k(z | s, ζ)πζ (dζ) off N ∪ Nh , H(s, z) := m(z | s)

74

and H := 0 on N ∪ Nh . On N the inner integral is 0 = H m, and Nh is µs -null. So for bounded measurable φ, Fubini’s theorem gives Z hZ i X E[h(ζi , Si )φ(Oi )] = PS (s) φ(s, z) h(ζ, s)k(z | s, ζ)πζ (dζ) µs (dz) s

=

X

Z PS (s)

φ(s, z)H(s, z)m(z | s)µs (dz) = E[H(Oi )φ(Oi )].

s

R Hence H(Oi ) = E[h(ζi , Si ) | Oi ] almost surely. Off the null set N ∪ Nh , H(Oi ) = h(ζ, Si ) Πζ (dζ | Oi ). Taking h = 1B (ζ), which is bounded and has Nh = ∅, shows that Πζ is a version of the conditional law of ζi given Oi ; on N , any choice, such as πζ , is allowed. With h = τ̄ we obtain the displayed formula. Πζ is computed from Oi , k and πζ , none of which involves an outcome under the query regime. The query regime enters only through τ̄ , which T3(iii) assumes known. With T1. The tower property (σ(Oi ) ⊆ σ(ζi , Oi )) gives E[τi | Oi ] = E[E[τi | ζi , Oi ] | Oi ] = E[τ̄ (ζi , Si ) | Oi ]. Without T1. Take τ̄ (ζ, s) := E[τi | ζi = ζ, Si = s], with τi ∈ L2 . Then E[τi | Oi ] − E[τ̄ (ζi , Si ) | Oi ] = E[τi − τ̄ (ζi , Si ) | Oi ]. Conditional expectation is a contraction in L2 , so the norm of this difference is at most ∥τi − τ̄ (ζi , Si )∥ = {E Var(τi | ζi , Si )}1/2 . This is in turn at most {E Var(τi | ζi )}1/2 = ∥τi − E[τi | ζi ]∥: E[τi | ζi , Si ] is the σ(ζi , Si )measurable variable closest to τi in L2 , and E[τi | ζi ] is one such variable. □ Proof of (c3). In the factual record only j ∗ is dosed (C0). So at a draw d of channel k the tanh argument is Γik (d) = Λk ζi qs (d). Under the hypothesis it is zero at every draw of every channel; for channels with Λk = 0 this is trivial. The shift c α tanh(0) = 0 at every draw, whatever the attenuation, and the hold copies draw values, so M (z 0 , s, ζ) = z 0 for every ζ. (In code, a zero shift returns the base value unchanged.) Hence: • k(· | s, ζ) is the density of P 0 (· | s) for every ζ, so k(z | s, ζ) = m(z | s); • Πζ (dζ | Oi ) = πζ (dζ) on {Si = s}, both off N and, by the convention of (c2), on N ; R • the prediction of (c2) is τ̄ (ζ, s)πζ (dζ). Proof of (c4).

□

Fix s and (k, d) as in the hypothesis, and write a := Λk qs (d) ̸= 0.

0 • Almost surely on {Si = s}, the attenuation is inactive and no clip binds. So there Zik (d) = Zik (d) + Wi , with Wi := c tanh(aζi ). 0 • By T2, Zik (d) and ζi are independent given Si = s, and ζi has law R πζ given Si = s. Hence the conditional characteristic functions satisfy ϕZ (t) = ϕ0 (t) ψ(t), with ψ(t) := exp{itc tanh(aζ)}πζ (dζ).

• Because P (Si = s) > 0, ϕZ is determined by the law of Oi , and ϕ0 is known by T3(i). So ψ = ϕZ /ϕ0 is determined on {ϕ0 ̸= 0}. This set has full Lebesgue measure and is therefore dense. ψ is continuous, so it is determined everywhere, and by the uniqueness theorem the law of Wi is determined. • The map ζ 7→ c tanh(aζ) is a continuous, strictly monotone bijection of R onto (−c, c). So πζ is the image of the law of Wi under its inverse. 0 It remains to show that a bounded record satisfies the characteristic-function condition. If |Zik (d)| ≤ B almost 0 0 itZik (d) iwZik (d) surely given Si = s, then ϕ0 (t) = E[e | Si = s] extends to ϕ0 (w) = E[e | Si = s] for complex w. By dominated differentiation, using |eiwx | ≤ e|w|B for |x| ≤ B, this extension is entire. Since ϕ0 (0) = 1, it is not identically zero, so its zeros are isolated. Its real zeros are therefore countable and form a Lebesgue-null set. □ 1/2 Proof of (d). (d1) Fix i and an orthogonal matrix Ui whose first row is p̃⊤ . Then Ui r̃i = (Si , Wi⊤ )⊤ with i /A 1/2 Si = Bi /A . Under the working model,

Ui r̃i = (A1/2 βi , 0, . . . , 0)⊤ + Ui ei ,

Ui ei ∼ N (0, I).

So Si = A1/2 βi + (Ui ei )1 , and Wi = (Ui ei )2: ∼ N (0, I) is independent of (βi , Si ) and free of (µ, v). The Jacobian of Ui is one, so the working density of r̃i is N (Si ; A1/2 µ, 1 + Av) · N (Wi ; 0, I). The working likelihood Q of n independent patients is therefore i N (Si ; A1/2 µ, 1 + Av) times a factor P free of (µ, v). Write η := A1/2 µ and x := 1 + Av ∈ [1, ∞). The log-likelihood is − 21 i {(Si − η)2 /x + log x} + const. 75

• For every x it is maximized over η at S̄. • The profile − n2 {V̂n /x + log x} has derivative 2xn2 (V̂n − x). It increases for x < V̂n and decreases for x > V̂n , so its maximizer on [1, ∞) is max(V̂n , 1). • The Si are independent and identically distributed with finite variance, so V̂n → V almost surely by the strong law of large numbers. Since max(·, 1) is continuous, 1 + Av̂ → max(V, 1) almost surely. Finally, Wi is independent of (βi , Si ) under the working model, and Covw (βi , Si ) = A1/2 v, Varw (Si ) = 1 + Av. Gaussian regression gives Ew [βi | r̃i ] = Ew [βi | Si ] = µ +

A1/2 v µ + vBi (Si − A1/2 µ) = . 1 + Av 1 + Av

In ICare-RH’s parametrization (βi = γbi , bi ∼ N (m, 1), v = γ 2 , µ = γm), this is γ(m + hi )/(1 + ai ) with hi = γBi and ai = γ 2 A. For the Laird–Ware learners (p̃i = ei /σ̂, v = σb2 , λ = σb2 /σ̂ 2 ), the deviation from µ is ⊤ λe⊤ i (ri − ei µ)/(1 + λei ei ), which is the coded BLUP. (d2) Fix (µ̂, v̂) and let patient i be independent of the fitting sample. The moments below are over (b∗i , εi ) with (µ̂, v̂) fixed. The prediction is τ̂i = Dµ̂ + Dŵ(Si − A1/2 µ̂), so τ̂ic = DŵSic and τic = Db∗c i . • If v̂ = 0, then ŵ = 0, the prediction is constant, u = 0 and cPEHE = στ . • If v̂ > 0, then ŵ > 0. Since Cov(Si , b∗i ) = A1/2 σ∗2 + Cov(εi , b∗i ) = A1/2 σ∗2 and Var(Si ) = Aσ∗2 + ω0 = V , r = sign(D2 ŵ) corr(Si , b∗i ) =

 SNR 1/2 A1/2 σ∗2 , = V V 1/2 σ∗

u=

|D|ŵV 1/2 ŵV 1/2 = . |D|σ∗ σ∗

Case V > 1. By (d1), 1+Av̂ → V almost surely. Hence ŵ → w† := A1/2 (V −1)/(AV ) = (V −1)/(A1/2 V ) > 0, and v̂ > 0 for all large n. Then u→

V −1 A1/2 σ∗ V 1/2

=

V −1 , (SNR V )1/2

u V −1 ω0 − 1 → =1+ . r SNR SNR

• Shrinkage weight. The best linear predictor of b∗i from Si has weight wBL := Cov(b∗i , Si )/ Var(Si ) = A1/2 σ∗2 /V = SNR/(A1/2 V ), so w† /wBL = (V − 1)/SNR, the limit of u/r. • Calibration. Since SNR > 0, the limit has u = r if and only if ω0 = 1. • Beating zero. The limiting values have u > 0 and r > 0, so by (a2) the limiting predictor beats zero if and only if u/r < 2, that is ω0 − 1 < SNR. r is constant once v̂ > 0 and u converges almost surely. So if the limit of u is below 2r, then 0 < u < 2r for all large n; if it is above, then u > 2r for all large n. • cPEHE. By (a1), the limiting cPEHE2 / Var(τ ) is 1 − r2 + r2 (u/r − 1)2 =

ω0 SNR (ω0 − 1)2 ω0 SNR + (ω0 − 1)2 + = . V SNR V V SNR2

• Mean error. Eτ̂i = Dµ̂ (because ESi − A1/2 µ̂ is multiplied by Dŵ and µ̂ → ESi /A1/2 ), so it tends to D ESi /A1/2 = D Eb∗i + D Eεi /A1/2 , while Eτi = D Eb∗i . The mean error enters neither r nor u, which are centered quantities. Case V < 1. V̂n → V < 1 almost surely, so almost surely V̂n < 1 for all large n. Then v̂ = 0 and the prediction is the constant Dµ̂, which does not beat zero. Case V = 1. 1 + Av̂ = max(V̂n , 1) → 1 almost surely, so ŵ → 0 and u = ŵ/σ∗ → 0 almost surely. • On {v̂ = 0} the prediction is constant and does not beat zero. • On {v̂ > 0}, r = SNR1/2 > 0 is fixed. Almost surely u < 2r for all large n, and then (a2) says the predictor beats zero. By (a4) the gain is at most uστ → 0.

76

P • Write V̂n = n−1 i (Si − ESi )2 − (S̄ − ESi )2 . If ESi4 < ∞ and |Si − ESi | is not almost surely constant, P then (Si − ESi )2 has mean 1 and positive finite variance. By the central limit theorem, n1/2 {n−1 i (Si − ESi )2 − 1} converges to a centered normal law, while n1/2 (S̄ − ESi )2 → 0 in probability. Hence P (v̂ > 0) = P (V̂n > 1) → 1/2. c 2 2 2 ∗ (d3) Now τic = Dτ b∗c i + νi and Var(τi ) = Dτ σ∗ /ρlin . Because νi is uncorrelated with bi and εi , Cov(Si , νi ) = 1/2 2 0 and Cov(τ̂i , τi ) = DŵDτ A σ∗ . Hence, for a fitted v̂ > 0 (so ŵ > 0) and DDτ > 0,

r=

 SNR 1/2 DŵDτ A1/2 σ∗2 , = ρ lin V |D|ŵV 1/2 |Dτ |σ∗ /ρlin

u=

|D|ŵV 1/2 ρlin . |Dτ |σ∗

If V > 1, then ŵ → w† almost surely as in (d2). So u → ρlin (D/Dτ )(V −1)/(SNR V )1/2 and u/r → (D/Dτ )(V − 1)/SNR. Both limits are positive, and by (a2) the limiting predictor beats zero if and only if this ratio is below 2. □ Numerical checks. The identities of (a) were checked on all 57 scored learner-by-panel rows of the two shipped benchmark metrics tables (13 static learners, the Causal Transformer, RMSN and CRN, the structureinformed linear response model, the oracle-structure learner and a prior-only reference, in three panels): the identity (a1) holds with residual at most 5.3×10−15 , the criterion (a2) agrees with the point-estimate comparison of centered PEHE with SD(τ ) in 57 of 57 rows, the cap (a4) holds in 57 of 57, and every centered PEHE is at or above the rescaling bound στ (1 − r2 )1/2 of (a3). Table S16 gives the decomposition for the learners discussed in the main text. Parts (b) and (d) were checked by exact computation on a discrete σ-field and by Monte Carlo with non-Gaussian slopes and noise and a nonzero noise mean (7 of 7 settings of (d2), and an independent Monte Carlo with 4 × 106 patients to three or four decimals). Further facts: • Generic learners. Excluding the random-slope models, the two structure-informed references and the prior-only reference, the 42 generic rows of the static and sequence benchmarks (11 static learners, the Causal Transformer, RMSN and CRN, which the main text excludes from scoring, in three panels) have |r| ≤ 0.124 (information term at least 0.985), so optimal rescaling would reduce their centered PEHE by at most 0.8% of SD(τ ). In 3 rows (all the S-learner) u < 0.1; the other 39 have u ≥ 2r; no row has neither. The only rows that beat zero at point estimates are the S-learner in all three panels, by 0.051%, 0.379% and 0.033% of SD(τ ), within its label-free cap uστ of (a4) (0.66%, 4.86% and 1.52%), and the prior-only reference on the fixed schedule, by less than 0.001%. The g-computation row of G-Net (Supplementary Table S4) was scored after these checks and is not included in them. • Random-slope models. In the natural-dosing panels their information terms are 0.90 to 0.94 and their dispersion terms 4.60 to 5.86; at u = r their centered PEHE would be 0.949 to 0.969 of SD(τ ). Inverting (d2) under D = Dτ and ρlin = 1 gives ω0 of 1.46 to 4.84 and SNR of 0.064 to 0.532 across the six rows; with the scale ratios of a synthetic replica under (d3) it gives ω0 of 2.75 to 3.63 and SNR of 0.125 to 0.419. The inversion is a model-based reading and tests nothing. A replica of rs_dlm_mech with unequal Ai gives (d3) predictions of r = 0.809 (observed 0.773) and u/r = 1.60 (1.64) under NATMIN and u/r = 2.20 (2.19) under FIX19, but its SNR (5 to 10) is far above the inverted values, so it checks the formulas rather than the parameters. • Structure-informed references. The oracle-structure learner has u/r = 0.950, 0.818 and 0.955 in the discovery, replication and fixed-schedule panels (calibration slopes r/u of 1.05, 1.22 and 1.05), so its centered PEHE is within 1.2% of the rescaling bound στ (1 − r2 )1/2 of (a3). By (b2), its r2 (0.336, 0.416, 0.303) estimates a lower bound, up to sampling error, on the information about τ in the record, and is at or above the supervised readers’ R2 (0.33, 0.31, 0.24). By (b3), u close to r is necessary but not sufficient for a conditional expectation, so it does not show that the learner attains that information. • ICare-RH on synthetic panels. On the synthetic scenarios of Supplementary Table S5, ICare-RH’s arm U (Ornstein–Uhlenbeck baseline) is within 1% of the rescaling bound in 14 of 16 scenarios (the exceptions are autoregressive drift, 3.2%, and drifting gain, 7.8%) and has u/r of 0.82 to 1.09 in 15 of 16 (1.71 under drifting gain). The kernel-informed random-slope model inside the same code with independent working errors is over-dispersed in 16 of 16 scenarios (u/r from 1.94 to 6.77) and fails to beat zero in 15 of 16, as (d2) predicts when the working covariance omits variance along the exposure direction. • Transport. Doses reconstructed from the shipped inputs have no dose at or after row 54 in any panel, and a fuzz of 6,000 calls of natmin_actions gives none either. The query multiplier of the generator kernel is 0.6838 to 0.6842. On a synthetic replica, a label-free transport estimator built from (c2) reaches correlation 0.841 (NATMIN) and 0.873 (FIX19) with no treated patient at the query, and with no pre-decision dosing its prediction is constant, as (c3) states. Part (c) shows that identification by overlap is unavailable at the 77

query; it does not show that lack of overlap caused the generic learners’ failure: in an in-support synthetic control with the query at row 30, no generic learner was informative either. By (c2), the error of ignoring T1 is at most {E Var(τ | ζ)}1/2 , about 0.20 SD(τ ) from a cross-fitted trait-only R2 of 0.959; a cross-fitted R2 estimates R∗2 (σ(ζ)) from below, so this is an upper estimate of the bound up to sampling error. Supplementary Table S16: Calibration geometry of selected learners on the washout benchmark (Proposition 6). Spread ratio u = SD(τ̂ )/SD(τ ), correlation r with the true effect, information term 1 − r2 and dispersion term (u − r)2 of Proposition 6(a1), whose sum is cPEHE2 /Var(τ ); a predictor beats zero exactly when 0 < u < 2r. Values are from the shipped metrics tables of the static and sequence benchmarks (population moments over the 400 scored patients); rs_dlm_mech is the kernel-informed random-slope model. Disc., discovery; Repl., replication; Fixed, fixed schedule. Learner

Panel

u

r

1 − r2

(u − r)2

Beats zero

Oracle-structure learner

Disc. Repl. Fixed Disc. Repl. Fixed Disc. Repl. Fixed Disc. Repl. Fixed Disc. Repl. Fixed

0.550 0.528 0.526 0.556 0.536 0.531 2.405 2.528 1.682 2.588 2.701 3.490 0.007 0.049 0.015

0.580 0.645 0.550 0.581 0.645 0.556 0.260 0.249 0.205 0.315 0.281 0.209 0.081 0.102 0.029

0.664 0.584 0.697 0.662 0.584 0.691 0.932 0.938 0.958 0.901 0.921 0.956 0.994 0.990 0.999

0.001 0.014 0.001 0.001 0.012 0.001 4.599 5.192 2.183 5.169 5.858 10.767 0.005 0.003 0.000

yes yes yes yes yes yes no no no no no no by 0.05% by 0.38% by 0.03%

Linear response model (structure given)

Random-slope lag model

Random-slope lag model, kernel

S-learner (boosting)

5.7 Interpreting ICare-RH’s placebo failure Proposition 7 (ICare-RH’s placebo gate: the likelihood favors kernels along which the baseline has unmodeled variance). Standard and specific parts. Parts (a)(i) to (a)(iii) are the profile likelihood of a oneway random-slope variance component at a fixed baseline (Harville), and the mixture reference named in (a)(vii) is that of Self and Liang for a variance on the boundary of its parameter space. Parts (a)(iv) to (a)(vii) and (b) to (e) are specific to ICare-RH and its placebo gate. Gate clauses K1, K2 and K3 are defined at the start of this note. Setting. Fix a panel copy with n patients, a kernel K = (ℓ, λ, κ) ∈ K := {0, . . . , 6} × [0.02, 0.99] × [0.05, 50], a baseline and a unit loading direction u = (u1 , . . . , u12P ) ∈ R12 , ∥u∥ = 1 (for one channel k, u is the kth unit vector; the slope variance v := γ 2 below then equals k γk2 , as in Methods). The baseline consists of fixed effects βk and working covariances Ωik : Ωik is the principal submatrix, on the draw rows Dik ⊆ {0, . . . , 89}, of |t−t′ | 2 2 2 the 90 × 90 matrix Ωfull with entries σa,k + σo,k ρk + σe,k 1{t = t′ } (random intercept, Ornstein–Uhlenbeck, k 1/2

2 white noise; the code keeps ρk ≤ 0.9975 and σe,k ≥ e−12 , so Ωfull is positive definite). Write Ωik for the k ′ ⊤ −1 ′ lower Cholesky factor of Ωik , ⟨w, w ⟩Ω := w Ω w and cos2Ω (w, w′ ) := ⟨w, w′ ⟩2Ω /(⟨w, w⟩Ω ⟨w′ , w′ ⟩Ω ). Let pik := (ϕκ (eiK (t)))t∈Dik be the exposure profile at the draw rows, rik := (yik (t) − x⊤ it βk )t∈Dik the baseline residual at the draw rows (there yik (t) = Zik (t), the held record), and X X −1 −1 Ai := u2k p⊤ Bi := uk p⊤ ik Ωik pik , ik Ωik rik , k

k

P

and assume i Ai > 0. In arm U the loadings are γk = γuk (γ ̸= 0) and bi ∼ N (m, 1); write µ := γm for the mean slope and v := γ 2 for the slope variance. Put n i v(Bi − µAi )2 1 Xh − 2µBi + µ2 Ai − + log(1 + vAi ) Nn (µ, v) := 2 i=1 1 + vAi

(v ≥ 0),

P P P P Pn (v) := minµ∈R Nn (µ, v), µ̂ := i Bi / i Ai , Ri := Bi − µ̂Ai , ω = ω(K) := i R2i / i Ai (Definition 10) and the variance gain Gn := Pn (0) − inf v≥0 Pn (v) ≥ 0. Write G(ω) := (ω − 1 − log ω)1{ω > 1} and Υω (ξ) := log(1 + ξ) − ωξ/(1 + ξ) for ξ > −1. (a) Objective, score identity and profile. 78

(i) For v > 0, Nn (µ, v) is the constant-free stage-2 objective of the code (_stage2 with the intercept-only, unpenalized prior map) at loadings γu and prior mean m = µ/γ; it differs from ICare-RH’s negative marginal log-likelihood at the fixed baseline by a term that depends neither on (µ, v) nor on K. Pn (0) = P P −( i Bi )2 /(2 i Ai ) is the fixed-slope fit at the same baseline; in the code’s parametrization it is reached only as γ → 0 with γm → µ̂. (ii) Pn is analytic on (−1/ maxi Ai , ∞) and Pn (v) = Pn (0) +

v2 vX  Ai (1 − ω) + Pn′′ (0) + O(v 3 ), 2 i 2

Pn′′ (0) =

X

Ai R2i −

i

P 1 X 2 ( i Ai Ri )2 . Ai − P 2 i i Ai

Hence if ω > 1 then Pn (v) < Pn (0) for all sufficiently small v > 0 and Gn > 0, and if ω < 1 then Pn (v) > Pn (0) for all sufficiently small v > 0. In every case P (ω − 1)2+ ( i Ai )2 P 2Gn ≥ . 4 i Ai R2i (iii) Equal information (Ai ≡ A > 0): Gn = n2 G(ω), attained at v̂A = (ω − 1)+ , and G is strictly increasing on [1, ∞). With unequal information Gn > 0 can occur with ω < 1, so “a positive slope variance is fitted iff ω > 1” holds locally in general and globally only under equal information. P P (iv) Kernel comparison: minµ, v≥0 Nn = −( i Bi )2 /(2 i Ai ) − Gn , a first-moment term plus the variance gain. (v) Population. Let patients be i.i.d. with design Si , E[Bi | Si ] = 0, ωi := E[Bi2 | Si ]/Ai on {Ai > 0}, 0 < EAi < ∞, E[Ai ωi ] < ∞, ϖ := E[Ai ωi ]/EAi , N̄ (µ, v) := E N1 (µ, v) and N̄ (v) := minµ N̄ (µ, v). Then the minimizing µ is 0 for every v and N̄ (0) = 0. If ϖ > 1, inf v≥0 N̄ < 0, so every pseudo-true slope variance (Definition 11), if one exists, is positive. A pseudo-true value exists when ωi is bounded, and need not exist otherwise. If ωi ≤ 1 almost surely, v = 0 is the unique minimizer. Under equal information the minimizer is v ∗ A = (ϖ − 1)+ . (vi) Kernel selection. If for K, K ′ the conditions of (v) hold with equal information and ϖK >  two kernels ′ K max(ϖK ′ , 1), then Pr minµ,v Nn < minµ,v NnK → 1. By (i) this is a comparison of profile likelihoods at the fixed baseline: in large samples the likelihood prefers the kernel with more unmodeled variance along its exposure direction. The statement is pairwise; by a union bound it extends to any finite set of kernels, not to the continuum K. (vii) K1. At a fixed baseline the slope-variance likelihood ratio equals 2Gn . For the implemented statistic N N N (slope_variance_lrt), whose null and alternative refit their baselines, LRK1 ≥ 2GN n , where Gn , ω , Ri are computed at the null’s own fitted baseline, provided the alternative’s sweep-0 loading fit attains the stage-2 infimum there (up to the values below 10−8 ). Consequently, if with probability P code’s truncation of P N −1 −1 N 2 2 2 tending to 1 ω ≥ 1 + ε, n i Ai ≥ A > 0 and n i Ai (Ri ) ≤ R, then LRK1 ≥ nε A /(4R) → ∞, 2 the K1 test rejects at every fixed level with probability tending to 1, and the 50:50 χ̄ (0, 1) law cannot be the limit law of LRK1 . Under equal information LRK1 ≥ n G(ω N ); the value n G(ω) itself is the fixedbaseline ratio under equal information. (b) What the excess contains. One channel, baseline held, patients independent given their designs, E[ri | −1 −1 ⊤ −1 Si ] = 0, Cov(ri | Si ) = Σi . Then E[Bi2 | Si ] = Ai ωi (K) with ωi (K) = p⊤ i Ωi Σi Ωi pi /pi Ωi pi on {Ai > 0}. (i) Decomposition. Write Σi − Ωi = vR qi qi⊤ + vδ gi gi⊤ + Mi , with vR := c2 Λ2k σζ2 , qi = piK0 the generator profile at the draw rows, ti := (t)t∈Di the linear-drift direction, vδ ≥ 0 and Mi a symmetric remainder of either sign. Then ωi (K) − 1 = vR ⟨qi , qi ⟩Ωi cos2Ωi (piK , qi ) + vδ ⟨ti , ti ⟩Ωi cos2Ωi (piK , ti ) +

−1 ⊤ (Ω−1 i piK ) Mi Ωi piK . ⟨piK , piK ⟩Ωi

The first two terms are nonnegative. The third is negative along exposure directions in which Ωi overcovers the residual covariance, so ωi (K) < 1 is possible.
P P P P P (ii) Pooled value. Given the designs, E ω = ω̄ − 2 i A2i ωi − ω̄ i A2i /( i Ai )2 with ω̄ := i Ai ωi / i Ai . For i.i.d. patients with EBi2 , EA2i < ∞, almost surely   vR E⟨p, q⟩2Ω + vδ E⟨p, t⟩2Ω + E (Ω−1 p)⊤ M Ω−1 p . ω → ϖ(K) = 1 + E⟨p, p⟩Ω 79

(iii) Zero-effect copy without remainder (vR = 0, Mi = 0). ωi (K) > 1 iff vδ > 0 and ⟨piK , ti ⟩Ωi ̸= 0. ωi (K) is bounded, so a pseudo-true slope variance exists at K; it is 0 if vδ = 0 or ⟨piK , ti ⟩Ωi = 0 almost surely, and positive if vδ > 0 and Pr(⟨piK , ti ⟩Ωi ̸= 0) > 0. (iv) Homogeneous designs, Mi = 0. If the per-patient quantities in (i) do not depend on i for the kernels compared, (a)(vi) applies with ϖK = ωi (K). On the zero-effect copy, if vδ > 0 and t ̸= 0 (some draw after row 0), the fit prefers, with probability tending to 1, the kernel with the larger cos2Ω (pK , t). With a response, a kernel K is preferred to K0 if     vδ ⟨t, t⟩Ω cos2Ω (pK , t) − cos2Ω (q, t) > vR ⟨q, q⟩Ω 1 − cos2Ω (pK , q) , and K0 is preferred to K if the reverse strict inequality holds and vR ⟨q, q⟩Ω > 0. (c) Telescoping and the random-walk limit. For draws 0 = d0P< · · · < dn (n ≥ 1) with gaps gj := dj+1 − dj (so gj = ∆j t for the drift direction t := (t)t ), let ⟨w, w′ ⟩RW := j<n ∆j w ∆j w′ /gj , a semi-inner product that vanishes on constant vectors. (i) For every draw pattern and kernel, ⟨pK , t⟩RW = pK (dn ) − pK (d0 ) = pK (dn ),

⟨t, t⟩RW = dn ,

and, when ⟨pK , pK ⟩RW > 0, cos2RW (pK , t) = pK (dn )2 /(dn ⟨pK , pK ⟩RW ). A profile that is back at its starting value 0 at the last pre-decision draw is exactly orthogonal to linear drift in this metric. (ii) If σe2 = 0, σa2 is fixed and ρ ↑ 1 with σo2 (1 − ρ2 ) = σb2 fixed, then w⊤ Ω−1 w′ → σb−2 ⟨w, w′ ⟩RW for all w, w′ . In that limit, on the zero-effect copy of (b)(iii) and when ⟨piK , piK ⟩RW > 0, ωi (K) − 1 → (vδ /σb2 ) piK (dn )2 /⟨piK , piK ⟩RW . (iii) Under NATMIN (binary doses on rows 6 to 53 only) the generator kernel satisfies q(t) < 5 · 0.8t−55 for t ≥ 55, so q(89) < 5 · 0.834 < 0.0026 (FIX19: q(t) < 5 · 0.8t−26 for t ≥ 26). For any kernel, with tlast ≤ 53 i last t−tlast −ℓ last i the last dosed row and t ≥ ti + ℓ, pK (t) ≥ λ pK (ti + ℓ); at t = 89 and λ = 0.99 the factor is 0.9936−ℓ (0.696 at ℓ = 0) when tlast = 53 and at least 0.9983 = 0.434 for every tlast ≥ 6. By (i) the i i random-walk inner product of such a profile with drift equals its retained exposure pK (dn ). (d) Query contrast. ϕκ (e+∆)−ϕκ (e) = κ∆/{(κ+e)(κ+e+∆)}. Under NATMIN or FIX19, with Ei := eiK (89), the step s = 0, . . . , 11 (row 91 + s) and the query increment min(101, 91+s−ℓ)

X

∆qs :=

λ91+s−ℓ−k (s ≥ ℓ − 1),

∆qs := 0 (s < ℓ − 1),

k=90

which equals (1 − λs−ℓ+2 )/(1 − λ) for s ≥ ℓ − 1 except at (ℓ, s) = (0, 11), where it is λ(1 − λ12 )/(1 − λ) (sustained dosing covers rows 90 to 101 only), 11

Di (K) =

 κ 1 X ϕκ (λs+2 Ei + ∆qs ) − ϕκ (λs+2 Ei ) < , 12 s=0 κ + λ13 Ei

Di (K) ∈ [0, 1) is strictly decreasing in Ei , and |τ̂i | ≤ |γk∗ Ei | κ/(κ + λ13 Ei ) with Ei the posterior mean of bi . Di (K) depends only on the patient’s doses and K, so for a given K it is the same on the active and zero-effect copies. (f ) Pooled spread. The outer folds f = 1, . . . , F carry fits θ̂(f ) (kernel K̂ (f ) , loading γ̂k∗ , prior mean, baseline), each computed without the patients of fold f . Let M be the law that draws f with probability πf := nf /n (f ) and then a fresh patient independent of the fits, and predicts τ̂ = γ̂k∗ E (f ) D(K̂ (f ) ), the fits held fixed. If (f ) 2 (f ) (f ) EM (E ) < ∞ and Ē (S) := EM [E | S, f ], then h i
2
(f ) (f ) VarM (τ̂ ) = EM γ̂k∗ D(K̂ (f ) ) VarM (E (f ) | S, f ) + VarM γ̂k∗ D(K̂ (f ) ) Ē (f ) (S) . The second term is the variance of a function of (S, f ) alone, which under Definition 5 is independent of P the trait. It splits into a within-fold part f πf Var(· | f ) and a between-fold part Varf E(· | f ) that comes from differences between the fold fits; when Ē (f ) ≡ ē(f ) is constant within each fold, the within-fold part is P (f ) (f ) 2 ) Var(D(K̂ (f ) )). Given the fit of fold f , the query patients of fold f are i.i.d. from the population, f πf (γ̂k∗ ē so the pooled out-of-fold variance is a sample analog of VarM (τ̂ ). (e) Validity of the placebo spread. In each panel K2 passes iff σ0 ≤ 0.05 (absolute clause) and σ0 ≤ 0.25 σA (ratio clause), where σ0 , σA are the population standard deviations of the pooled out-of-fold predictions τ̂ on the zero-effect and active copies. 80

(i) If, given the designs, patients are independent with E[BiP| Si ] = P 0 and Var(B (working i ) = Ai P P i | S P covariance correct along pK at the true baseline), then E i R2i = i Ai − i A2i / i Ai < i Ai for every K. (ii) Under (S), (I) and (E0) to (E5) on the zero-effect copy, the fitted mean slope and slope variance tend to 0 in probability in every outer fold (the kernel and direction need not converge), so σ0 → 0 in probability and the absolute clause passes with probability tending to 1. If in addition (A1) to (A3) hold on the active copy, σA tends in probability to a positive limit and K2 passes in each panel with probability tending to 1. Passing K2 is therefore asymptotically necessary for the conjunction of a baseline correctly specified along the exposure direction (E1) and an in-family response that the fit captures (A1), given the regularity conditions (E0), (E2) to (E5), (A2) and (A3). The absolute clause concerns the zero-effect copy alone. A failure of the ratio clause can also come from σA → 0 (no response captured), from a nonzero first moment or from a non-global optimum, so a K2 failure does not by itself locate the misspecification in the baseline. (iii) In the setting of P6(d) (assumption (P6d)), at the population pseudo-true values of the working model, 1/2 fitted separately on each copy: σ0 = DA−1/2 (ω0 − 1)+ /ω0 , σA = DA−1/2 (SNR + ω0 − 1)/(SNR + ω0 )1/2 when SNR + ω0 > 1, and for ω0 ≥ 1 ω0 − 1 (SNR + ω0 )1/2 σ0 = 1/2 · ∈ [0, 1). σA SNR + ω0 − 1 ω 0

The ratio is 0 when ω0 ≤ 1 < SNR + ω0 ; it tends to 0 as ω0 ↓ 1, tends to 1 as SNR/(ω0 − 1) → 0, and does not depend on D or A, while σ0 is proportional to D. An informative but misspecified estimator (correlation (SNR/(SNR + ω0 ))1/2 > 0 with the target) can therefore fail the ratio clause while passing the absolute clause. The range [0, 1) holds for these population values only; the ratio of two finite-sample fits, fitted separately on the two copies, can exceed 1. Assumptions.

Each part uses only the assumptions listed for it.

(S) Setting (all parts). One channel or a fixed loading direction u; kernel K ∈ K and baseline (fixed effects and working covariances Ωik ) held P fixed; arm U, whose prior map is an unpenalized intercept, profiled in closed form (_profile_prior); i Ai > 0. The identity in (a) holds at any fixed baseline, in particular at the stage-1 baseline and at the converged baseline of the joint fit, whose response block (fit_response inside fit_joint) attempts to solve the stage-2 problem at the current baseline. The implemented fit also optimizes the direction over 12 channels, the lag by profiling and (λ, κ) by local search. Statements about the kernel the likelihood prefers refer to exact minimizers of the stage-2 objective, which the code’s local optimizers do not guarantee. (I) Independence. Patients are independent given their designs Si (population statements: i.i.d.); outer folds and the K1 halves are assigned by seeded person-level permutations that use no patient data. Used in (a)(v)–(vii), the pooled statements of (b), the pooled spread in (d), and (e). (M) Mean-zero projections. E[Bi | Si ] = 0, i.e. the fixed effects carry the mean of the residual at the draw rows; in (b) this means that the trait and the drift slope are centered. A nonzero mean adds a kernel-dependent first-moment term (a)(iv), which is not treated. (EI) Equal information (idealization). Where stated, Ai (K) and the conditional overdispersion ωi (K) do not depend on i. The real and synthetic panels have heterogeneous designs. (a)(ii), (a)(v) and (b)(i)–(iii) hold without (EI); the kernel preferences (a)(vi) and (b)(iv) use it. (O) Optimizer (K1 corollary only). When GN n > 0, the alternative’s sweep-0 loading fit (fit_loadings, three fixed starts, L-BFGS-B) attains inf v>0 PnN (v) at the null’s baseline. The half-A kernel is treated as fixed on half B (the halves are disjoint). (L) Linear regime (part (b)). Attenuation inactive (α ≡ 1) and tanh x ≈ x for the response, so the trait contributes cΛk ζi qi with Var(ζi ) = σζ2 ; the drift is δi gi with Eδi = 0, Var δi = vδ , independent of ζi and of the modeled baseline. (b)(i) and (b)(ii) allow any symmetric remainder Mi ; (b)(iii) and (b)(iv) assume Mi = 0, i.e. that Ωi is exactly the covariance of the remaining residual. That idealization fails for the fitted covariance on the benchmark generator’s synthetic zero-effect copies: ω < 1 at 33 of 49 kernels (NATMIN) and 42 of 49 (FIX19), with minima 0.68 and 0.61 below every correctly specified placebo’s minimum (0.82 to 0.94), so the fitted Ω over-covers some exposure directions and only the ordering at the persistent end of the kernel family carries over. Because Ω is fitted with the drift present, (b) gives the direction of the effect, not its size. 81

(RW) Random-walk metric (part (c)). The identity (c)(i) is exact in the random-walk metric. That metric is the limit (c)(ii) of the implemented intercept + OU + white-noise metric only when the white-noise variance is 0 and the OU persistence tends to 1 at fixed per-row innovation variance. The implementation bounds persistence by 0.9975 and keeps σe2 ≥ e−12 . On the benchmark generator’s synthetic tapes (a reproduction of job 10692) the fitted heart-rate baseline had persistence 0.9938 and white-noise variance about 3.2 times the per-row innovation variance, so the limit is not reached and (c) describes a mechanism, not the fitted metric. (D) Dosing support (parts (c)(iii), (d)). Doses of j ∗ are binary (build_series refuses other values); NATMIN doses only on rows 6 to 53, FIX19 only on rows 6 to 24; no dose of j ∗ on rows 54 to 89; the query contrast (sustained_contrast) appends doses 1 (resp. 0) on rows 90 to 101, treats rows from 102 on as undosed, and averages rows 91 to 102; lags ℓ ≤ 6, 0 < λ ≤ 0.99. (E0) Compactness. With probability tending to 1 the fitted parameter θ̂ = (K̂, û, µ̂, v̂) lies in Θ := K × U × [−µmax , µmax ] × [0, vmax ], where U is a compact set of unit directions (or the fit is restricted to Θ). The code’s loadings and prior mean are unbounded, so this is an assumption. (E1) Correct specification on the zero-effect copy. At the true baseline, given Si , the whitened residuals of the channels in the direction have mean 0 and identity covariance and are uncorrelated across channels (the generator shares latent noise across channels, so this is an idealization). (E2) Baseline consistency and uniform law of large numbers. The fitted baseline converges in probability to the true baseline, and supΘ |n−1 N̂n − N̄ | → 0 in probability, where N̂n (θ) is the stage-2 objective at the fitted baseline and N̄ (θ) the population objective at the true baseline; Θ contains the population minimizers. (E3) Approximate minimization. n−1 N̂n (θ̂) ≤ inf Θ n−1 N̂n + op (1). The code’s block coordinate ascent with local optimizers does not guarantee this. (E4) Non-degenerate exposure. Pr(Ai (K, u) > 0) > 0 for every (K, u) ∈ K × U, E supK×U Ai < ∞, and Ai (K, u) is continuous in (λ, κ, u) for each lag. P 2 (E5) Query moments. In each fold the out-of-fold query patients satisfy n−1 q q supK×U Bq = Op (1) at the fitted baseline; the training and query sizes of every fold tend to infinity, with the number of folds fixed. (A1) In-family response on the active copy. At the true baseline the working model holds at θ0 = (K0 , u0 , µ0 , v0 ) with v0 > 0 and u0,k∗ ̸= 0. (A2) Identification. On the active copy the population objective N̄ is continuous on Θ and has the unique minimizer θ0 up to the sign flip (u, µ) 7→ (−u, −µ), and (E0) and (E2) to (E5) hold there. (A3) Continuity of the spread. The query-population mean and standard deviation of τ̂ are continuous functions of (parameter, baseline) at (θ0 , true baseline), and in each fold the query-sample mean and standard deviation converge to them in probability uniformly over a neighborhood of that point. (P6d) Setting of P6(d) (part (e)(iii)). Equal information A; standardized projection Bi /A1/2 = A1/2 b∗i + εi on the active copy and εi on the zero-effect copy (shared noise), εi ⊥ b∗i , Var b∗ = σ∗2 , SNR := Aσ∗2 > 0, Var ε = ω0 ; constant D > 0; the working model b ∼ N (µ, v), Var ε = 1 at its pseudo-true value on each copy; τ̂i = D Ew [bi | Bi ]. −1/2

Proof. Notation. For patient i, kalman_whiten returns Ωik w for any column w at the draw rows, where −1/2 −1/2 −1/2 Ωik is the inverse of the lower Cholesky factor. Let p̃i := (uk Ωik pik )k and r̃i := (Ωik rik )k , stacked over channels. Then Ai = ∥p̃i ∥2 and Bi = p̃⊤ i r̃i . In the code, response_stats returns the per-channel values −1 ⊤ −1 p⊤ Ω p and p Ω r , and A , B are their u-weighted sums. If Ai = 0 then p̃i = 0 and Bi = 0. Write i i ik ik ik ik ik P ik ξi := vAi and Nn = i Ni . Lemma 1 (the objective and its standardized form). Under the working model rik = γuk bi pik + εik , εik ∼ N (0, Ωik ) independent over k and of bi ∼ N (m, 1), h i P − log p(ri ) = 21 ∥r̃i ∥2 + k log |Ωik | + Ni log 2π + Ni (µ, v), with Ni the number of draw values. For Ai > 0, 2Ni (µ, v) =

(Bi − µAi )2 B2 + log(1 + ξi ) − i , Ai (1 + ξi ) Ai 82

(⋆)

and Ni ≡ 0 if Ai = 0. Proof. Whitening gives r̃i = γbi p̃i + ε̃i with ε̃i ∼ N (0, I) independent of bi , so r̃i ∼ N (µp̃i , I + v p̃i p̃⊤ i ). By ⊤ −1 the matrix determinant lemma det(I + v p̃i p̃⊤ ) = 1 + ξ , and by the Sherman–Morrison formula (I + v p̃ p̃ = i i i ) i /(1 + ξ ). Hence the quadratic form is I − v p̃i p̃⊤ i i 2 v{p̃⊤ v(Bi − µAi )2 i (r̃i − µp̃i )} = ∥r̃i ∥2 − 2µBi + µ2 Ai − . 1 + ξi 1 + ξi P The Jacobian of ri 7→ r̃i contributes 12 k log |Ωik |. For (⋆), use (Bi − µAi )2 /Ai = Bi2 /Ai − 2µBi + µ2 Ai and 1/(1 + ξi ) = 1 − ξi /(1 + ξi ), which turn the right side of (⋆) into −2µBi + µ2 Ai − v(Bi − µAi )2 /(1 + ξi ) + log(1 + ξi ). □ P P −1 ⊤ −1 (a)(i). In _stage2 the code forms the arrays a = k γk2 p⊤ ik Ωik pik = vAi and h = k γk pik Ωik rik = γBi and P 1 2 2 returns 2 i [−2m h+m a−(h−ma) /(1+a)+log(1+a)] with m profiled over R (_profile_prior; the ridge acts only on non-intercept coefficients, which arm U does not have). With m = µ/γ: m h = µBi , m2 a = µ2 Ai and (h−ma)2 = v(Bi −µAi )2 , so the value is Nn (µ, v), and profilingPm over R is Pprofiling µ over R because γ ̸= 0. By Lemma 1 the full negative marginal log-likelihood differs by 21 i [∥r̃i ∥2 + k log |Ωik | + Ni log 2π]; the residuals rik and covariances Ωik are fixed by the baseline, so this term involves neither (µ, v) norP K nor u. At v = 0 1 2 the model is r̃i ∼ N (µp̃i , I), the fixed-slope model at the same baseline, and N (µ, 0) = n i (µ Ai − 2µBi ) is 2 P P 2 minimized at µ̂ with value −( i Bi ) /(2 i Ai ). In the code’s parametrization γ = 0 gives a = h = 0 and the value 0 ≥ Pn (0), so Pn (0) is approached only along γ → 0 with γm → µ̂. P (a)(ii). By (⋆), for fixed v > −1/ objective is a quadratic in µ with leading coefficient 12 i Ai /(1 + P maxi Ai the P ξi ) > 0, minimized at µ̂(v) := i Bi (1 + ξi )−1 / i Ai (1 + ξi )−1 , which gives

∥r̃i − µp̃i ∥2 −

2 P i X v B2 X i Bi /(1 + ξi ) i P − + log(1 + ξi ) , 1 + ξi i Ai /(1 + ξi ) i i

1h Pn (v) = − 2

a sum of rational functions with non-vanishing denominators and of logarithms of positive arguments on the interval (−1/ maxi Ai , ∞). Hence Pn is analytic there and has a Taylor expansion at 0 with an O(v 3 ) remainder. Since ∂µ Nn (µ̂(v), v) = 0, Pn′ (v) = ∂v Nn (µ̂(v), v), and differentiating (⋆), Pn′ (v) =

1 X h Ai (Bi − µ̂(v)Ai )2 i − , 2 i 1 + ξi (1 + ξi )2

Differentiating once more, with µ̂′ (0) = − Pn′′ (0) =

P

i Ai Ri /

Pn′ (0) =

X  1 X  1 X Ai − R2i = Ai (1 − ω). 2 i 2 i i

P

i Ai (quotient rule on µ̂(v)),

i X X 1h X 2 − Ai + 2 Ai R2i + 2µ̂′ (0) Ai Ri , 2 i i i

which is the stated expression. If ω > 1 then Pn′ (0) < 0, so Pn (v) < Pn (0) on some interval (0, v1 ) and Gn > 0; if ω < 1 then Pn′ (0) > 0 and Pn (v) > Pn (0) on some (0, v1 ). For the lower bound, Gn ≥ Pn (0) − Pn (v) ≥ Nn (µ̂, 0) − Nn (µ̂, v) for every v ≥ 0, since Pn (0) = Nn (µ̂, 0) and Pn (v) ≤ Nn (µ̂, v). By (⋆), 2[Ni (µ̂, 0) − Ni (µ̂, v)] = (R2i /Ai ) ξi /(1 + ξi ) − log(1 + ξi ) when Ai > 0, and 0 otherwise. Using ξ/(1 + ξ) ≥ ξ − ξ 2 and log(1 + ξ) ≤ ξ for ξ ≥ 0, 2Gn ≥

X h R2 i

i

Ai

i X X (ξi − ξi2 ) − ξi = v(ω − 1) Ai − v 2 Ai R2i . i

i

2 If ω P> 1 then P i Ai Ri > 0 (some Ri ̸= 0 has Ai > 0, since Ai = 0 forces Ri = 0); taking v = (ω − 1) i Ai /(2 i Ai R2i ) gives the bound. If ω ≤ 1 the bound is 0 ≤ 2Gn . P (a)(iii). With Ai ≡ A the weights in µ̂(v) are equal, so µ̂(v) = µ̂ and i R2i /A = nω. By (⋆), with ξ = vA,

P

Pn (v) − Pn (0) =

i n 1 X h R2i R2 − i + log(1 + ξ) = Υω (ξ). 2 i A(1 + ξ) A 2

Υ′ω (ξ) = (1 + ξ − ω)/(1 + ξ)2 . If ω ≤ 1, Υω is nondecreasing on [0, ∞) and its minimum there is Υω (0) = 0. If ω > 1, Υω decreases on [0, ω − 1] and increases afterward, so its minimum is Υω (ω − 1) = log ω − (ω − 1) < 0. Hence Gn = n2 G(ω) at v̂A = (ω − 1)+ , and G ′ (ω) = 1 − 1/ω > 0 on (1, ∞). Since A = ⟨p, p⟩Ω in the one-channel case, the maximizer satisfies v̂⟨p, p⟩Ω = ω − 1 when ω > 1. −3 Unequal information: Gn > 0 with ω < 1. Take 1000 P patients with Ai = 10 and Bi = ±0.1 (signs alternating) and 10 patients with Ai = 1, Bi = 0. Then i Bi = 0, µ̂(v) = 0 for all v by symmetry, Pn (0) = 0, 83

2 i Ri = 10,

P

P

i Ai = 11 and ω = 10/11 < 1. At v = 9000, ξi = 9 in the first group and (⋆) gives Pn (9000) = 1 [1000(log 10 − 9) + 10 log 9001] = −3303.2 < 0, so Gn ≥ 3303. ICare-RH’s fit_loadings on these per-channel 2 2

values returns γ̂ = 8890 with objective −3303.2.

(a)(iv). minµ,v≥0 Nn = inf v≥0 Pn (v) = Pn (0) − Gn by definition of Gn , and Pn (0) was computed in (a)(i). (a)(v). Conditioning on Si in (⋆) and using E[Bi | Si ] = 0, E[Bi2 | Si ] = Ai ωi , E[2Ni (µ, v) | Si ] =

µ2 Ai ωi + µ2 Ai − ωi + log(1 + ξi ) = + Υωi (ξi ) 1 + ξi 1 + ξi

on {Ai > 0} (and 0 otherwise). Each term is integrable because 0 ≤ log(1 + ξi ) ≤ vAi and 0 ≤ ξi ωi /(1 + ξi ) ≤ vAi ωi . The µ term is nonnegative and vanishes only at µ = 0 (as Pr(Ai > 0) > 0), so the minimizing µ is 0, N̄ (v) = 21 E[Υωi (vAi )] and N̄ (0) = 0. The difference quotient Υωi (vAi )/v tends to Ai (1 − ωi ) as v ↓ 0 and is bounded in absolute value by Ai + Ai ωi , so by dominated convergence N̄ ′ (0+) = 21 E[Ai (1 − ωi )] = 1 2 (EAi )(1 − ϖ). If ϖ > 1 this is negative, N̄ (v) < 0 for small v > 0, and v = 0 does not minimize the expected negative log-likelihood, which differs from N̄ by a term free of (µ, v) (Lemma 1). Every minimizer, when one exists, therefore has v > 0. Existence. N̄ is continuous on [0, ∞) by dominated convergence (on [0, V ] the integrand is bounded by V Ai (1 + ωi )). If ωi ≤ ωmax almost surely, then 2N̄ (v) ≥ E log(1 + vAi ) − ωmax → ∞ as v → ∞ by monotone convergence and Pr(Ai > 0) > 0, so the minimum over [0, ∞) is attained. Without a bound it need not be: let Pr(Ai = 2−k ) = c k −2 for k ≥ 1 (c the normalizing constant), ωi = 2k on that event and Bi | Si ∼ N (0, Ai ωi ) = N (0, 1). Then E[Ai ωi ] = EBi2 = 1, EA2i < ∞, and at v = 2K the atoms k ≥ K have ξi ≤ 1 and contribute Υωi (ξi ) = log(1 + ξi )P − 2K /(1 + ξi ) ≤ log 2 − 2K−1 , while the atoms k < K contribute at most K log(1 + 2 ) ≤ (K + 1) log 2. Since k≥K ck −2 ≥ c/K, 2N̄ (2K ) ≤ (K + 2) log 2 − c 2K−1 /K → −∞, and no minimizer exists. If ωi ≤ 1 almost surely, Υωi (ξ) ≥ Υ1 (ξ) = log(1 + ξ) − ξ/(1 + ξ) > 0 for ξ > 0, so N̄ (v) > 0 = N̄ (0) for v > 0 (using Pr(Ai > 0) > 0). Under equal information E[Aωi ]/A = Eωi = ϖ and N̄ (v) = 21 Υϖ (vA), minimized at v ∗ A = (ϖ − 1)+ by the analysis of Υ in (a)(iii). P P P P P (a)(vi). Under equal information, i R2i = i Bi2 − ( i Bi )2 /n, so ω K = (nAK )−1 [ i Bi2 − ( i Bi )2 /n] → EBi2 /AK = ϖK almost surely by the strong law (the BP i are i.i.d. with mean 0 and finite variance AK ϖK ), and likewise for K ′ . The first-moment term satisfies E( i Bi )2 /(2nAK ) = ϖK /2, so it is Op (1). By (a)(iii)– (iv), n−1 min NnK = − 12 G(ω K ) + Op (n−1 ) → − 12 G(ϖK ) in probability, by continuity of G. Because G is 0 on (0, 1] and strictly increasing on [1, ∞), ϖK > max(ϖK ′ , 1) implies G(ϖK ) > G(ϖK ′ ), so n−1 (min NnK − ′ min NnK ) converges in probability to a negative constant and the stated probability tends to 1. For a finite set of competitors K ′ , the union of the finitely many exceptional events has probability tending to 0. By (a)(i) the term that separates Nn from the negative marginal log-likelihood is the same for K and K ′ , so the comparison is one of profile likelihoods. (a)(vii). At a fixed baseline the null (fixed slope) is the model with v = 0 and µ free, the alternative has v ≥ 0 and µ free, and by (a)(i) the log-likelihood ratio is 2[Pn (0) − inf v≥0 Pn (v)] = 2Gn . For the implemented statistic, the null (fit_common_baseline on heart rate with design [X, ϕ]) fits the N , µN ) by exact generalized least variance parametersP and tail decay and, at those values, computes (βroute N −1/2 2 N squares, minimizing i ∥(Ωi ) (yi − Xi β) − µp̃i ∥ jointly, with null’s working covariance. With r̃i := P Ωi theP −1/2 N ⊤ N (ΩN (yi − Xi βroute ), the normal equation for µ gives µN = i p̃⊤ i ) i r̃i / i p̃i p̃i = µ̂ , so i X Xh  N N N N N 2 N 1 (µ̂N )2 AN − 2µ̂ B = H + P (0), H := ∥r̃ ∥ + log |Ω | + N log 2π , nllnull = HN + 12 i i n i i i 2 i

i

using ∥r̃ − µp̃∥2 = ∥r̃∥2 − 2µB + µ2 A. The alternative (fit_joint with the kernel fixed) starts from the null’s variance parameters, tail decay and route coefficients with the slope column removed; its sweep-0 objective is HN + PnN (γ̂ 2 ) by (a)(i), with γ̂ from fit_loadings, and a later sweep is kept only if it lowers the objective, so N N nllalt ≤ HN + PnN (γ̂ 2 ). Under (O), PnN (γ̂ 2 ) = PnN (0) − GN n when Gn > 0, hence LRK1 = 2(nllnull − nllalt ) ≥ 2Gn ; N when Gn = 0 the inequality holds because the code sets negative or tiny ratios to 0. The divergence follows 1 2 2 2 2 2 from (a)(ii): on the stated event, 2GN n ≥ ε (nA) /(4nR) = nε A /(4R). The reported p-value 2 Pr(χ1 > LRK1 ) then tends to 0 in probability, so the test rejects at every fixed level with probability tending to 1; and a sequence that diverges in probability cannot converge in law to the tight law χ̄2 (0, 1). Under equal information (a)(iii) gives N LRK1 ≥ 2GN n = nG(ω ). −1 −1 −1 ⊤ ⊤ −1 (b)(i). With E[ri | Si ] = 0 and Cov(ri | Si ) = Σi , E[Bi2 | Si ] = p⊤ i Ωi E[ri ri | Si ]Ωi pi = pi Ωi Σi Ωi pi , which ⊤ −1 ⊤ ⊤ −1 2 is linear in Σi . The part Ωi contributes pi Ωi pi = Ai ; a rank-one part zz contributes (pi Ωi z) = ⟨pi , z⟩2Ωi = −1 ⊤ Ai ⟨z, z⟩Ωi cos2Ωi (pi , z) (for z ̸= 0; the term is 0 if z = 0); the remainder contributes (Ω−1 i pi ) Mi Ωi pi . Dividing by Ai > 0 gives the display. Under (L) the trait term cΛk ζi qi and the drift δi gi are centered, mutually uncorrelated

84

and uncorrelated with the modeled part, so they contribute exactly vR qi qi⊤ + vδ gi gi⊤ ; Mi is whatever of Σi − Ωi is left. P 2 P 2 P P (b)(ii). Given the − 2µ̂ i Ai Bi + µ̂2 i A2i . By independence and zero means, i Bi P P designs, 2i RiP= E[µ̂Bi ] = Ai ωi / j Aj and Eµ̂ = j Aj ωj /( j Aj )2 , hence E

X

R2i =

i

X

Ai ωi −

i

2

P P P 2 ( i A2i )( i Ai ωi ) Ai ωi i P P + ; ( i Ai )2 i Ai

P 2 2 dividing by i Ai gives the stated formula. with P For i.i.d. patients−1 P EB2i , EAi 2< ∞ (so E|Ai Bi | < 2∞), the −1 strong law gives µ̂ → EBi /EAi = 0, n i Ai Bi → E[Ai Bi ], n i Ai → EAi and hence ω → EBi /EAi = −1 ⊤ E[Ai ωi ]/EAi almost surely. By (b)(i), Ai (ωi − 1) = vR ⟨pi , qi ⟩2Ωi + vδ ⟨pi , ti ⟩2Ωi + (Ω−1 i pi ) Mi Ωi pi , which gives the stated limit. (b)(iii). With vR = 0 and Mi = 0, ωi (K) − 1 = vδ ⟨piK , ti ⟩2Ωi /Ai , positive iff vδ > 0 and ⟨piK , ti ⟩Ωi ̸= 0. Boundedness. By the Cauchy–Schwarz inequality in ⟨·, ·⟩Ωi , ωi (K) − 1 ≤ vδ ⟨ti , ti ⟩Ωi ≤ vδ ∥ti ∥2 /λmin (Ωi ). Ωi P89 is a principal submatrix of Ωfull , so by eigenvalue interlacing λmin (Ωi ) ≥ λmin (Ωfull ) > 0, and ∥ti ∥2 ≤ t=0 t2 . P89 Hence ωi (K) ≤ 1 + vδ t=0 t2 /λmin (Ωfull ), a bound free of i, and (a)(v) gives existence of a pseudo-true value. If vδ = 0 or ⟨piK , ti ⟩Ωi = 0 almost surely, then ωi = 1 almost surely and (a)(v) gives the unique pseudo-true value 0. If vδ > 0 and Pr(⟨piK , ti ⟩Ωi ̸= 0) > 0, then ωi ≥ 1 with strict inequality on an event of positive probability on which Ai > 0, so ϖ > 1 and (a)(v) gives a positive pseudo-true value. (b)(iv). When the per-patient quantities do not depend on i, Ai ≡ AK and ϖK = ωi (K); the Bi are i.i.d. with mean 0 and variance AK ϖK < ∞, so (a)(vi) applies. On the zero-effect copy ϖK − 1 = vδ ⟨t, t⟩Ω cos2Ω (pK , t) with ⟨t, t⟩Ω free of K and positive when t ̸= 0. If vδ > 0 and cos2Ω (pK , t) > cos2Ω (pK ′ , t), then ϖK > ϖK ′ ≥ 1 and (a)(vi) gives the preference for K. With a response, ϖK0 − 1 = vR ⟨q, q⟩Ω + vδ ⟨t, t⟩Ω cos2Ω (q, t) ≥ 0 because cos2Ω (q, q) = 1, and ϖK > ϖK0 is the displayed inequality after rearrangement; since ϖK0 ≥ 1, (a)(vi) gives the preference for K. If the reverse strict inequality holds and vR ⟨q, q⟩Ω > 0, then ϖK0 > max(ϖK , 1) and (a)(vi) gives the preference for K0 . P (c)(i). For the drift direction t, ∆j t = gj , so ⟨pK , t⟩RW = j<n ∆j pK = pK (dn ) − pK (d0 ) by telescoping, for any draw pattern. Row 0 is always a draw and e (0) = 0 (_memory), so pK (d0 ) = ϕκ (0) = 0. Also iK P ⟨t, t⟩RW = j gj = dn > 0, which gives the cosine when ⟨pK , pK ⟩RW > 0. The form vanishes on constant vectors because their increments are 0. (c)(ii). The stationary OU process on the integer grid satisfies o(t + 1) = ρ o(t) + ϵOU with Var ϵOU = σo2 (1 − t t 2 2 2 ρ ) = σb ; sampled at the draws it is a Gaussian Markov chain with o(d0 ) ∼ N (0, σo ) and o(dj+1 ) | o(dj ) ∼ N (ρj o(dj ), σo2 (1 − ρ2j )), ρj := ρgj . Reading off the joint density, its precision defines the bilinear form ⟨w, w′ ⟩ρ :=

′ P w0 w0′ (wj+1 −ρj wj )(wj+1 −ρj wj′ ) . + j<n 2 (1−ρ2 ) σ 2 o j σo

Pgj −1 2m ρ → σb2 gj , so ⟨w, w′ ⟩ρ → As ρ ↑ 1 with σb2 fixed: σo2 = σb2 /(1 − ρ2 ) → ∞, ρj → 1 and σo2 (1 − ρ2j ) = σb2 m=0 σb−2 ⟨w, w′ ⟩RW . With the random intercept, Ω = σa2 11⊤ + ΣOU and w⊤ Ω−1 w′ = P by Sherman–Morrison ′ 2 ′ 2 2 2 ⟨w, w ⟩ρ − σa ⟨1, w⟩ρ ⟨1, w ⟩ρ /(1 + σa ⟨1, 1⟩ρ ). Since ⟨1, w⟩ρ = w0 /σo + j (wj+1 − ρj wj )/{σo (1 + ρj )} → 0 and ⟨1, 1⟩ρ ≥ 0, the correction vanishes. Substituting the limit into (b)(iii), ⟨p, t⟩2Ω /⟨p, p⟩Ω → σb−2 ⟨p, t⟩2RW /⟨p, p⟩RW = σb−2 p(dn )2 /⟨p, p⟩RW by (c)(i), provided ⟨p, p⟩RW > 0. (c)(iii). The generator kernel is (2, 0.8, 1). Under NATMIN the last dose is at a row ≤ 53 and enters memory P P47 at row ≤ 55, so e(t) = 0.8t−55 e(55) for t ≥ 55, and e(55) = k≤53 0.853−k A(k) ≤ j=0 0.8j < 5 for binary doses on rows 6 to 53. Hence q(t) = e(t)/(1 + e(t)) ≤ e(t) < 5 · 0.8t−55 (if e(55) = 0 then q(t) = 0), and P18 5 · 0.834 = 0.00254. Under FIX19 the last dose enters at row ≤ 26 and e(26) ≤ j=0 0.8j < 5, which gives last

5 · 0.8t−26 . For a general kernel, no dose enters after row tlast + ℓ, so e(t) = λt−ti −ℓ e(tlast + ℓ) for t ≥ tlast + ℓ; i i i t−tlast −ℓ last i and ϕκ (αe) = αe/(κ+αe) ≥ αe/(κ+e) = αϕκ (e) for α ∈ [0, 1], e ≥ 0, which gives pK (t) ≥ λ pK (ti +ℓ). Numerically 0.9936 = 0.696 and 0.9983 = 0.434. The last statement is (c)(i). e+∆ e κ∆ (d). The identity is algebra: κ+e+∆ − κ+e = (e+∆)(κ+e)−e(κ+e+∆) = (κ+e)(κ+e+∆) . In sustained_contrast (κ+e)(κ+e+∆) both regimes carry the factual doses on rows 0 to 89, then 1 (resp. 0) on rows 90 to 101; the dose array has 102 columns, and _memory enters no dose for an index at or beyond its length, so row 102 and later are undosed in both regimes. Di averages ϕκ (e(1) ) − ϕκ (e(0) ) over rows 91 to 102. The recursion is linear in the doses, so min(101, t−ℓ)

e

(1)

(t) − e

(0)

(t) =

X

λt−ℓ−k

k=90

85

(t − ℓ ≥ 90),

0

(t − ℓ < 90),

which at t = 91 + s is ∆qs and does not depend on i. The upper limit 101 binds only when 91 + s − ℓ > 101, P12 i.e. s − ℓ > 10, which for s ≤ 11 means (ℓ, s) = (0, 11); there the sum is j=1 λj = λ(1 − λ12 )/(1 − λ). Ps−ℓ+1 j Otherwise it is j=0 λ = (1 − λs−ℓ+2 )/(1 − λ). At rows t = 90, . . . , 102 the entering dose is A(t − ℓ) with t − ℓ ∈ [84, 102]: rows 84 to 89 carry no dose by (D), and rows 90 onward carry the regime’s dose, which is 0 in the zero regime. Hence e(0) (90) = λEi and e(0) (91 + s) = λs+2 Ei for s ≥ 0. Each summand equals κ∆qs /{(κ + λs+2 Ei )(κ + λs+2 Ei + ∆qs )} ≥ 0. For ∆qs > 0 it is strictly decreasing in Ei (λ > 0) and is < κ/(κ + λs+2 Ei ) ≤ κ/(κ + λ13 Ei ), because ∆qs /(κ + λs+2 Ei + ∆qs ) < 1 and λs+2 ≥ λ13 for s ≤ 11; for ∆qs = 0 it is 0. Since 11 ≥ ℓ − 1 for ℓ ≤ 6, ∆q11 is a nonempty sum of positive powers of λ > 0, so ∆q11 > 0 and Di is strictly decreasing in Ei , lies in [0, 1) and is below κ/(κ + λ13 Ei ). At the generator kernel with Ei = 0 the average is 0.684. Since τ̂i = γk∗ Ei Di (code: posterior times the heart-rate loading times Di ), |τ̂i | ≤ |γk∗ Ei | κ/(κ + λ13 Ei ). Di uses only the doses and K, and the two copies share the doses. (f ) Pooled spread. Under M the fits are constants, D(K̂ (f ) ) and γ̂k∗ are functions of (S, f ), and the law of total variance with respect to (S, f ) gives  
VarM (τ̂ ) = EM VarM (τ̂ | S, f ) + VarM EM [τ̂ | S, f ] , (f )

(f )

with VarM (τ̂ | S, f ) = (γ̂k∗ D(K̂ (f ) ))2 VarM (E (f ) | S, f ) and EM [τ̂ | S, f ] = γ̂k∗ D(K̂ (f ) )Ē (f ) (S); all moments are finite because |D| < 1 and EM (E (f ) )2 < ∞. Under M the fold label is drawn independently of the patient and the patient’s (S, ζ) has the population law, in which S ⊥ ζ (Definition 5); hence (S, f ) ⊥ ζ and the second term is the variance of a trait-free quantity. The law of total variance with respect to f splits it into P (f ) (f ) ≡ ē(f ) the within-fold variance is (γ̂k∗ ē(f ) )2 Var(D(K̂ (f ) )). Finally, f πf Var(· | f ) + Varf E(· | f ); with Ē θ̂(f ) is computed from the patients outside fold f , which are independent of the patients in fold f under (I), so given θ̂(f ) the fold-f query patients are i.i.d. from the population and their predictions are i.i.d. draws from the f -component of M. P P (e)(i). Given the designs, EBi2 = Ai , E[µ̂Bi ] = Ai / j Aj and Eµ̂2 = 1/ j Aj , so the pooled expectation in P 2 P P 2 P P 2 P P P 2 P (b)(ii) with Pωi ≡ 1 gives E i Ri = i Ai − 2 i Ai / i Ai + i Ai / i Ai = i Ai − i Ai / i Ai , which is below i Ai because some Ai > 0. (e)(ii). Parametrize by θ = (K, u, µ, v) with γk = γuk , µ = γm, v = γ 2 ; every loading vector is of this form with u a unit vector. Fix an outer fold; the argument is the same in each of the finitely many folds. Zero-effect copy. Under (E1), for each (K, u) the conditions of (a)(v) hold with ωi ≡ 1: whitened residuals with identity covariance, uncorrelated across channels, give E[Bi | Si ] = 0 and E[Bi2 | Si ] = P 2 −1/2 pik ∥2 = Ai . Hence k uk ∥Ωik h µ2 A i i + Υ1 (vAi ) , N̄ (θ) = 21 E 1 + vAi

Υ1 (ξ) = log(1 + ξ) −

ξ ≥ 0, 1+ξ

with Υ1 (ξ) > 0 for ξ > 0 and N̄ (K, u, 0, 0) = 0. By (E4) and dominated convergence (the integrand is at most (µ2max + vmax ) supK×U Ai ), N̄ is continuous on Θ. On Θε := {θ ∈ Θ : |µ| + v ≥ ε} it is positive, because Pr(Ai > 0) > 0 makes the first term positive when µ ̸= 0 and the second positive when v > 0. Θε is compact, so N ε := minΘε N̄ > 0. The value N̂n (K, u, 0, 0) is 0 at every baseline. On the event {θ̂ ∈ Θ}, whose probability tends to 1 by (E0), (E2) and (E3) give N̄ (θ̂) ≤ n−1 N̂n (θ̂) + op (1) ≤ inf n−1 N̂n + op (1) ≤ 0 + op (1), Θ

so Pr(|µ̂| + v̂ ≥ ε) ≤ Pr(θ̂ ∈ / Θ) + Pr(N̄ (θ̂) ≥ N ε ) → 0. The kernel and direction are not identified on this copy (N̄ (K, u, 0, 0) = 0 for all (K, u)) and need not converge. For a query patient, τ̂q = γk∗ Eq Dq = uk∗ Dq (µ̂ + v̂Bq )/(1 + v̂Aq ), because the posterior mean is Eq = (m + γBq )/(1 + vAq ). As 0 ≤ Dq < 1, |uk∗ | ≤ 1 and 1 + v̂Aq ≥ 1, Minkowski’s inequality gives, in each fold, 

n−1 q

X

τ̂q2

1/2

 1/2 X ≤ |µ̂| + v̂ n−1 sup Bq2 →0 q

q

q

K×U

in probability by (E5). The pooled population standard deviation σ0 is at most the root mean square of the pooled predictions, which is at most the largest of the fold root mean squares, so σ0 → 0 in probability and Pr(σ0 ≤ 0.05) → 1. Active copy. By (A1) and Lemma 1, at θ0 and the true baseline the working model is the true law of the residuals given the design, and the stage-2 objective differs from the negative log-likelihood by a term free of θ (a)(i). By the information inequality (the expected negative log-likelihood of any working law exceeds that of the true law by their Kullback–Leibler divergence) θ0 minimizes N̄ , and by (A2) the minimizer is unique up to 86

the sign flip θ0− . The argument above with {θ ∈ Θ : dist(θ, {θ0 , θ0− }) ≥ ε} in place of Θε and N̄ (θ0 ) in place of 0 (continuity from (A2)) gives θ̂ → {θ0 , θ0− } in probability in every fold, and the fitted baseline converges by (E2). The prediction is invariant under the flip, since B(−u) = −B(u) and A(−u) = A(u). By (A3), in every fold the query-sample mean and standard deviation converge in probability to the mean τ̄0 and the standard deviation P sA := SD(τ̂ (θ0 )) at (θ0 , true baseline); the pooled variance is f (nf /n)[Varf +(meanf −mean)2 ] over the folds, so σA → sA . At θ0 the true slope βi = γ0 bi ∼ N (µ0 , v0 ) and Bi = βi Ai + p̃⊤ i ε̃i , so Var(Bi | Si ) = Ai (1 + v0 Ai ) and h   v02 Ai i > 0, s2A ≥ E Var(τ̂i (θ0 ) | Si ) = u20,k∗ E Di (K0 )2 1 + v0 Ai because Di (K0 ) > 0 (by (d), ∆q11 > 0 for ℓ ≤ 6) and Pr(Ai > 0) > 0. Hence, in each of the three panels, Pr(σ0 ≤ 0.05) → 1 and Pr(σ0 ≤ 0.25σA ) → 1, and the intersection over the finitely many clauses and panels has probability tending to 1. The contrapositive is the stated necessity: if K2 fails with probability bounded away from 0, at least one of (E1), (A1) or the regularity conditions fails. The mechanism analyzed in (a) and (b) is overdispersion ω0 (K̂) > 1 along the fitted exposure direction on the zero-effect copy, which makes v̂ positive there and so keeps σ0 away from 0. The ratio clause fails also when σA → 0 with σ0 small, which happens when the fit captures no response, and a nonzero first moment or an optimum far from the global one can produce a failure as well. (e)(iii). Let Bis := Bi /A1/2 . The working model states Bis ∼ N (A1/2 µ, 1 + Av). For a true law of Bis with mean b̄ and variance V, the expected negative log-likelihood is 12 [log(1 + Av) + {V + (b̄ − A1/2 µ)2 }/(1 + Av)] plus a constant, minimized at A1/2 µ∗ = b̄ and 1+Av ∗ = max(V, 1) (the function y 7→ log y+V/y on y ≥ 1 is minimized at max(V, 1)). The working posterior mean is Ew [b | B] = (µ + vB)/(1 + vA) = (µ + vA1/2 B s )/(1 + vA), so SD(τ̂ ) = D v ∗ A1/2 V1/2 /(1 + v ∗ A). On the active copy V = SNR + ω0 (independence of b∗ and ε); on the zero-effect copy V = ω0 . Substituting Av ∗ = (V − 1)+ gives SD = DA−1/2 (V − 1)+ /V1/2 , which is the formula for σA (when SNR + ω0 > 1) and for σ0 . With f0 (y) := y 1/2 − y −1/2 = (y − 1)/y 1/2 , strictly increasing on (0, ∞), the ratio for ω0 ≥ 1 is f0 (ω0 )/f0 (SNR+ω0 ), which lies in [0, 1) because SNR > 0, equals 0 at ω0 = 1, and is 0 for ω0 < 1 < SNR + ω0 because then v ∗ = 0 on the zero-effect copy. Continuity gives the limit 0 as ω0 ↓ 1. Writing f0 (SNR + ω0 )/f0 (ω0 ) = {1 + SNR/(ω0 − 1)}{1 + SNR/ω0 }−1/2 and noting SNR/ω0 ≤ SNR/(ω0 − 1), the ratio tends to 1 as SNR/(ω0 − 1) → 0. D and A cancel in the ratio, while σ0 is proportional to D. In this setting the correlation of τ̂ with τ = Db∗ is corr(Bs , b∗ ) = (SNR/(SNR + ω0 ))1/2 > 0 (Proposition 6(d)), so the predictor is informative while the ratio can be arbitrarily close to 1; for example SNR = 0.25, ω0 = 3 gives correlation 0.277 and ratio 0.925 > 0.25. These are population values at each copy’s pseudo-true parameters. With finitely many patients the two copies’ fitted (µ̂, v̂) are random, and nothing confines the ratio of the two sample standard deviations to [0, 1); in a five-fold out-of-fold simulation of this setting with n = 400, SNR = 0.005 and ω0 = 1.2 (population ratio 0.978), the ratio of the pooled standard deviations was at least 1 in 26% and at least 1.045 in 4.1% of 20,000 replicates. ■ Numerical checks. All checks use label-free outputs of the placebo gate, synthetic panels or simulation; no target of a real panel was read. • The statements. The query increment of (d) matches the implemented sustained_contrast to 3.3 × 10−16 at every lag 0 to 6 over 3,000 random kernels and dosings, the closed forms hold in all of them, the bound of (d) is never violated, and the generator kernel with empty memory gives D = 0.6842. The second derivative Pn′′ (0) of (a)(ii) agrees with finite differences to 1.2 × 10−5 relative, and the unequal-information example of (a)(iii) gives Pn (9000) = −3303.2. In the non-existence example of (a)(v), 2N̄ (230 ) ≈ −1.06 × 107 . The decomposition of (b)(i) with an indefinite remainder gives 1.0643 by Monte Carlo against 1.0624 from the formula. The fold-mixture variance identity of (d) agrees to 0.1% (1.7259 × 10−3 against 1.7278 × 10−3 ). • (a) against ICare-RH’s own code. On the benchmark generator’s synthetic registered cell, the finitedifference score of the stage-2 objective at zero slope variance is −502.41 against −502.43 from (a)(ii); the sign rule (γ̂ 2 > 0 exactly when ω > 1) held in 294 of 294 kernel fits; the code objective equals −(first-moment term + Gn ) to 10−10 ; in all six copies the kernel with the largest total gain, the largest Gn and the largest ω coincide; Gn /[(n/2)G(ω)] lies between 0.88 and 1.01; and the active copies select the generator kernel (2, 0.80), with ω of 3.97 and 3.09. • Correctly specified placebos on the same cell. Over lags 0 to 6 and seven decays (κ = 1, 49 kernels), eight placebo copies per arm drawn from ICare-RH’s own fitted heart-rate baseline selected decay 0.99 in 5 of 16 draws (2 of 8 under NATMIN, 3 of 8 under FIX19). Their largest variance gains were 0.54 to 3.83 nats, the best gain at decay 0.99 reached 3.55 (NATMIN) and 2.87 (FIX19) nats, and their largest ω at decay 0.99 was 0.995 to 1.197. The zero-effect copies gained 3.35 nats at (lag 1, decay 0.99)

87

under NATMIN and 2.53 at (lag 0, 0.99) under FIX19, inside that range. In the heart-rate channel of this cell, whose fitted baseline has persistence 0.9938 and white-noise variance about 3.2 times the per-row innovation variance, a selection of decay 0.99 therefore cannot be distinguished from a correct baseline. Under the fitted Ω the zero-effect copies have ω < 1 at 33 of 49 kernels (NATMIN) and 42 of 49 (FIX19), with minima 0.684 and 0.605 below every placebo’s minimum (0.82 to 0.94), and ω > 1 at all seven kernels with decay 0.99 in both arms: the fitted working covariance over-covers some exposure directions, so only the ordering at the persistent end of the kernel family carries over from (b). • The ICare-RH synthetic generator under the implemented joint fit. On the generator of Supplementary Table S5, the fitted decay sat at its bound in 200 of 200 fits on out-of-family null panels (autoregressive baseline with linear drift of slope standard deviation 0.005 per row, and level shifts), against 3 of 120 size-check fits (2.5%) and 0 of 5 folds of the in-family null scenario. On real records it sat at the bound in 5 of 5 folds on both copies of both natural-dosing panels (Table S17). This pattern is far outside the model’s in-family behavior and is consistent with unmodeled slow variance along the exposure direction. Proposition 7 does not identify which slow component it is. • Drift simulations. With a white-noise baseline and cap 0.75, the kernel with decay 0.80 was selected at drift slope standard deviations of 0 and 0.005 per row and displaced to decay 0.99 at 0.015 and 0.03; the shipped autoregressive-drift variant, with slope standard deviation 0.005, fitted decays of 0.756 to 0.781 in 5 of 5 folds. With a random-walk baseline, decays 0.80 to 0.85 were selected at drift standard deviations 0 to 0.06 and beat the bound by 22 to 66 nats, as (c) leads one to expect for a baseline metric close to the random-walk limit. • Lag. On real records lag 0 was preferred over lag 1 in 4 of 5 discovery folds (by 1.45 to 9.81 nats; the fifth fold prefers lag 4). In the replication panel lag 1 beat lag 0 by 6.4 to 10.2 nats in four active folds and by 5.7 to 9.2 nats in all five zero-effect folds, and the fixed-schedule panel is mixed. These fits show that decay 0.99 is selected and that the lag is not identified. • K1. On in-family synthetic nulls the test rejected in 5 of 120 cases (4.2%), 47% of the likelihood ratios were below 10−6 and their 90% quantile was 1.49, against 1.64 for the χ̄2 (0, 1) reference. On 40 out-of-family null panels with baseline drift or level shifts it rejected in 199 of 200 folds (median ratio 28.6). Part (a)(vii) is consistent with these rejections but was not used to explain them, because ω N was not computed on those nulls, and it does not predict the in-family size. • K2. Table S17 gives the gate’s values. The absolute clause passed in all three panels and the ratio clause failed in all three. The in-family synthetic pairs pass both clauses (ratios 0.125, 0.030 and 0.024), as (e)(ii) predicts. The discovery ratio of 1.045 lies outside the population range of (e)(iii); a five-fold simulation of that setting shows that separately fitted finite samples can exceed 1 (proof of (e)(iii)), and the small discovery σA (0.0068) leaves σA → 0 as a live explanation. The proposition does not decide between them. • Query contrast and design spread (d). In the discovery panel the fitted kernels of the active copy give median query contrasts of 0.012 to 0.048 across folds, 14 to 58 times smaller than the generator kernel’s 0.684 (means 11 to 40 times smaller; coefficient of variation 0.71 to 0.79). Pooled over folds, 43% to 69% of the trait-free design variance at unit prior mean is between folds (43% and 45% on the active and zero-effect copies of the discovery panel, 69% and 51% in replication), because the fitted loadings and kernels differ across folds. A design-only calculation with the fitted kernels and heart-rate loadings reproduces the placebo-to-active ratios (1.031 against 1.045; 0.465 against 0.428) if the mean posterior slope has magnitude 1.14 to 1.15 (discovery) or 0.80 to 0.87 (replication). This is descriptive and not a test.

5.8 What the theory does not show • The twin is not shown to be an ideal forecaster or a Bayes filter. Propositions 1 and 2 use only that the twin’s state at hour t is a function of the record through hour t. The twin’s arms are not those of an ideal forecaster in the idealized class of Proposition 2 (Table S12, last column). • The hour-6 arm is not shown to be the model’s best hour-6 read. In a disclosed post hoc read of the eICU-CRD test panel, the model’s hour-6 forecast of the earlier window exceeded S1 by 0.021 and 0.050 AUROC in the two saturation settings at (30, 24), although neither excess survived Holm correction over the 12 settings (Section 5.1). In those two settings, therefore, S2 − S1 cannot be assumed to be at most the ideal information gain.

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Supplementary Table S17: ICare-RH’s placebo gate on real records (Proposition 7). Label-free outputs of the registered gate, read before any target. σA and σ0 : population standard deviations of the pooled out-of-fold predicted effects on the active and zero-effect copies. The gate required σ0 ≤ 0.05 and σ0 /σA ≤ 0.25 in every panel. Decay at bound: folds in which the fitted exposure decay equals its upper bound of 0.99. Query contrast: median over patients of the fitted kernel’s query contrast Di (K̂), range over folds (generator kernel: 0.684). Design-only ratio: placebo-to-active spread ratio predicted from the fitted kernels, heart-rate loadings and doses alone (Proposition 7(d)). A dash marks a value that was not computed.

σA σ0 (limit 0.05) σ0 /σA (limit 0.25) Decay at bound, active copy (folds) Decay at bound, zero-effect copy (folds) Median query contrast, active copy (range over folds) Design-only ratio

Discovery

Replication

Fixed schedule

0.0068 0.0071 1.045 5/5 5/5 0.012–0.048 1.031

0.0419 0.0179 0.428 5/5 5/5 – 0.465

0.0703 0.0450 0.640 1/5 3/5 – –

• How much of the twin’s gain passes through the last measured value is not determined. Exact CVR is false a priori for a model that reads every channel. The four AUROCs of the difference-in-differences cannot reject HCS , and they cannot confirm it when the two attainability conditions of Proposition 2(b) hold strictly. The label-free quantities that would bear on it, corr(V, V ′ ) and the own-value splice of Proposition 2(c), were not computed, because the per-stay values of the validation isolation run are not in the shipped aggregates and the splice needs a new model read. Personalization beyond the patient’s current state is not shown. • Supervised readers are not upper bounds. Readers fitted with the true effects as labels estimate lower bounds on the information content of their feature class (Definition 6); an estimator using the same record can exceed them. • The realism audits bound distances from below only. The audit of the generated dataset bounds the distance between the real and generated summaries from below and certifies no upper bound (Proposition 3(a)); the audit of the washout panels’ generated outcome values is a separate application of the same instrument, carries the same limits, and cannot see that those values respond to simulated rather than recorded dosing (Methods). • The cause of ICare-RH’s placebo failure is not located. Proposition 7(e) covers only the spread clauses K2, not the lag clause K1 or the final clause K3. The absolute clause passed in all three panels, and the ratio clause failed in all three. A failure of the ratio clause is consistent with a baseline misspecified along the exposure direction, with a response the fit did not capture (the spread of predicted effects on the discovery panel’s active copy was 0.0068), with a non-global optimum or with finite-sample variation. The data do not decide between these explanations. The kernel preference of Proposition 7(a)(vi) requires equal information, no mean response along the exposure direction and exact minimizers at a fixed baseline, none of which the implementation guarantees. The slow component that the likelihood absorbs on real records is also not identified. • The response and baseline models are idealizations. The linear regime, a single carrier, and whitenoise, AR(1) or random-walk baselines are simplified models that do not describe real records. Three disagreements between the theory’s magnitudes and the record are reported but not explained. In the hourly norepinephrine cell the readers tied, although the single-carrier model predicts a cadence-aware advantage. On heart rate, sampling of the scored channel alone accounts for a factor of about 1.4 in the plain reader’s loss at real cadence, against observed factors of 5.9 and 7.2 in C1 , and for NaCl on creatinine a twelve-channel model under-predicts that loss. Finally, Proposition 5(e) does not cover the cadence-aware reader’s lack of separation on the fixed schedule. • Open items. Four questions remain open. The correlation corr(V, V ′ ) and the own-value splice test would bear on the current-state hypothesis (Proposition 2). A per-channel refit of the registered readers would test the attribution of the plain reader’s loss at real cadence (Propositions 4 and 5). The advantage of a 1/gap-weighted reader depends on how jump variance grows with the gap, which the shipped aggregates cannot check (Proposition 5). ICare-RH’s lag on real records and the slow component its likelihood absorbs are not yet known (Proposition 7).

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Supplementary Note 6: Cohort flow, forecasts and clinical-endpoint probes on eICU-CRD This note gives the details behind the eICU-CRD forecast and endpoint results of the main text. It traces the cohort to the stays scored in the registered read of the test set and then reports the calibration of the forecasts, the death forecast and the provenance of the read. The last part describes the two clinical-endpoint probes in full. Cohort flow. Supplementary Table S18 follows the ICU stays from the three source databases to the populations evaluated in the forecast read. Harmonization removed 34 of the 200,859 eICU-CRD stays, and MIMIC-III was restricted to CareVue stays of patients who do not appear in MIMIC-IV. Each eICU-CRD patient, with all of their stays, belongs to one split, so the patients of the three splits sum to the 139,362 eICU-CRD patients of the cohort. Baseline characteristics were not tabulated by cohort. Discrimination by age, sex and unit type is given in Supplementary Table S7, and by ethnicity in its legend. Evaluated populations. Each crossing setting was scored in a population selected by measurement rather than drawn at random from the test set. All forecasts were read at one landmark, hour 48 (Methods, Forecast evaluation). A stay entered a crossing setting only if its target was measured at hour 48 without having crossed and its outcome was then resolved by a crossing, by an exit from the unit or by measurement of the target in every hour of the horizon. The evaluated populations therefore shrink as the horizon lengthens and are smaller for SpO2 (227 to 352 stays) than for heart rate (801 to 1,449 stays). Longer horizons also have higher crossing rates, from 0.066 to 0.444 for heart rate and from 0.173 to 0.815 for SpO2 , and AUPRC, which depends on that rate, rises with them (Supplementary Table S19). Calibration. The forecasts were not recalibrated on test data, and at short horizons their mean came close to the observed crossing rate. At 6 and 12 hours the mean forecast was 91% to 105% of the observed rate for every vital sign. At 24 and 48 hours it was lower, 68% to 89% of the observed rate, with the largest shortfall for heart rate at 48 hours (0.300 against 0.444). Calibration slopes of the crossing forecasts ranged from 0.65 to 1.27 across settings. The lowest, 0.65 to 0.73 for SpO2 at 6 to 24 hours and for respiratory rate at 48 hours, show that these forecasts were too extreme. Reliability tables for every seed and setting are in joint_factual_metrics_by_seed.csv, a source file of Supplementary Table S19. Death forecast. The model has no separate mortality head, so the death forecast, fixed in the analysis plan as a descriptive setting, is the largest of the four target-specific death cumulative incidences at 48 hours. It discriminated death in the unit within 48 hours with an AUROC of 0.798 (95% interval across seeds 0.768 to 0.828) and an AUPRC of 0.195 against an observed rate of 0.0256. Its mean forecast, 0.0023, was far below that rate, in part by construction, because each target-specific head counts a death only if it precedes that target’s crossing. In each seed, all but two to five of the 11,498 forecasts were below 0.1, and the logistic calibration fit diverged. At an alert threshold of 0.03 the forecast flagged 41 to 83 stays across seeds (0.36 to 0.72 per 100), with net benefit of 0.0012 to 0.0026, and at 0.15 it flagged at most three. Its net benefit over alerting on every patient or on none (Supplementary Table S8) therefore rests on very few alerts. Recalibration was not part of the registered read. Registration and reproduction. The read followed an analysis plan (digest ecf365f4) that was locked before the test set was opened for this model. The plan fixed the landmark, the settings, the metrics, both kinds of interval (across seeds and from a patient bootstrap within each seed) and the rule that every setting is reported. Before this read, much of the test set had been scored by models trained with an earlier objective, as recorded in the test-read registry (Supplementary Note 2). The read ran as jobs 5276 (predictions), 5293 (metrics) and 5312 (verification). The verifier bundled with job 5293 failed after the metrics had been computed, because it carried an outdated copy of the plan digest. Job 5312 then reproduced all 85 seed-by-setting results exactly from the sealed prediction files, without a new model run. The same plan registered two further reads of this test set with the seed-29 checkpoint. The first, an earlier version of the synchronization contrast (jobs 5264 and 5275), is superseded because its hour-6 control differed from the synchronized forecast in estimand and calibration and because its panel required survival to the last landmark. The second (job 5265) compared a logistic model on the frozen state at hour 48 with LightGBM on summaries of the same 48 hours, for respiratory-rate crossings within the next 48 hours. The log-loss difference (logistic model minus LightGBM) was −0.013 (95% interval −0.031 to 0.004; 697 stays of 674 patients). The subgroup and decision-curve analyses (Supplementary Tables S7 and S8, job 10545) re-read the same prediction files and are descriptive.

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Clinical-endpoint probes. A threshold crossing does not by itself define a clinical event, so two registered development probes, P0 and P1, asked whether the patient state also carries information on clinical endpoints (Supplementary Fig. S3; Methods of the main text, Clinical endpoints). The probes scored four endpoints within 24 hours of landmarks at 12, 24 and 48 hours. These were death in the unit, hypotension (mean arterial pressure below 65 mmHg in two consecutive measured hours), acute kidney injury by the KDIGO creatinine criterion, and hypoxemia (SpO2 below 90% in two consecutive measured hours). Both probes used the tuning set of the eICU-CRD training split. The pretrained encoder had seen every stay of the set’s 14,607 patients in training, and the 14,122 patients with at least one landmark row were scored. On this set the boosting comparator, a fixed CatBoost recipe on 1,136 summary features of the record, reached AUROCs of 0.941, 0.816, 0.824 and 0.868 for the four endpoints in the order above (Supplementary Fig. S3a). Frozen state (probe P0). The frozen state of the vital-sign model, which was trained on other targets, trailed boosting by 0.020 to 0.086 AUROC (Supplementary Fig. S3b). The probe’s protocol, written before the run, had forecast deficits of about 0.01 to 0.02. The deficit was smallest for death, which the forecasting heads model as a competing event. Added to the boosting features, the frozen state raised AUROC by at most 0.0014, and every interval included zero. Trained encoders (probe P1). Training the same encoder architecture from random initialization on the four endpoints closed 42% to 94% of the frozen-state deficit (Supplementary Fig. S3c). This encoder still trailed boosting on every endpoint, by 0.004 to 0.021 AUROC, with every interval below zero. Adding the fine-tuned state of the pretrained encoder to the boosting features (the hybrid) improved discrimination of hypotension by 0.0042 and of acute kidney injury by 0.0075 (95% CIs 0.0015 to 0.0067 and 0.0035 to 0.0116). The hybrid also lowered log-loss on these two endpoints, and its AUROC intervals for death and hypoxemia included zero (Supplementary Fig. S3b,d). Used alone, the encoder fine-tuned from the pretrained twin was worse than the encoder from random initialization on hypotension, acute kidney injury and hypoxemia, by 0.006 to 0.014 AUROC. This comparison is confounded, because fine-tuning used a lower learning rate for the encoder. Status of the probes. Neither probe met its registered gate, and P1 missed it narrowly (Table 2 of the main text). No family-wise correction was applied, so the intervals are unadjusted. P1 was designed after P0 had read the tuning labels, and the pretrained encoder had been trained on every tuning-set patient (Supplementary Note 2). We therefore report the two P1 results above as development observations on eICU-CRD. The hybrid improved discrimination of hypotension and acute kidney injury, and the encoder trained from random initialization trailed boosting on every endpoint. The boosting comparator, both encoders and the hybrid were later refitted within the MIMIC-IV training split and read once on the previously analyzed MIMIC-IV test split under a registered protocol (main text and Supplementary Note 7). Supplementary Table S18: Cohort inclusion flow. ICU stays and patients from the source databases to the populations evaluated in the forecast read on the eICU-CRD test set; –, not tabulated. Counts are from inclusion_flow.csv, cohort_characteristics.csv and joint_factual_metrics_by_seed.csv in source_tables/revision_v3/; the training and validation patient counts are from the landmark event-time reports in reports/revision_v3/. Stage

Stays

Patients

Raw ICU stays

356,849

–

Harmonized stays

356,815

–

Experimental cohort

323,677

–

200,825 140,473 30,213 30,139 30,099

139,362 97,553 20,904 20,905 20,886

227 to 1,449

220 to 1,349

11,498

9,506

eICU-CRD (208 hospitals) Training split Validation split Test split Test set Evaluable for a crossing Evaluable for death

91

Restriction or note eICU-CRD 200,859; MIMIC-III 61,532; MIMIC-IV 94,458 BlendedICU harmonization; 34 eICU-CRD stays removed eICU-CRD 200,825; MIMIC-IV 94,458; MIMIC-III 28,394 CareVue stays of 23,692 patients not in MIMIC-IV (from 61,532 stays of 46,476 patients) Split by patient

All 40 stays of the 19 patients opened in an earlier analysis removed Per setting at hour 48: target measured and not crossed, outcome resolved In the unit after hour 48 with a valid discharge record

Supplementary Table S19: Discrimination and calibration of the joint competing-risks forecasts on the eICUCRD test set. Forecasts issued at hour 48 by the five checkpoints in the registered read (Methods, Forecast evaluation). Stays, patients and events (threshold crossings, or deaths in the unit for the death forecast) describe the evaluable population of each setting and are identical across seeds. Metrics are five-seed means. The AUROC interval is a 95% t interval across seeds (4 degrees of freedom), and the next column gives the AUROC of seed 29, the checkpoint with the lowest validation loss, with a 95% interval from 2,000 bootstrap replicates that resample patients. Log-loss and Brier score are those of the raw forecasts. Intercept and slope: logistic regression of the outcome on the logit of the forecast (0 and 1 under perfect calibration). The death forecast is the largest of the four target-specific death cumulative incidences at 48 hours; its calibration fit diverged in every seed (n.e., not estimable). Sources: source_tables/revision_v3/joint_factual_seed_summary.csv, joint_factual_metrics_by_seed.csv and joint_factual_person_cluster_bootstrap.csv. AUROC Mean Observed rate forecast [95% CI]

AUROC, seed 29 [95% CI]

Forecast

H (h)

Stays

Patients

Events

Heart rate

6 12 24 48

1,449 1,309 1,029 801

1,349 1,224 967 758

95 154 252 356

0.066 0.118 0.245 0.444

0.061 0.820 [0.813, 0.826] 0.823 [0.784, 0.860] 0.111 0.780 [0.774, 0.786] 0.781 [0.740, 0.818] 0.198 0.729 [0.725, 0.734] 0.730 [0.695, 0.764] 0.300 0.709 [0.704, 0.715] 0.708 [0.673, 0.744]

0.253 0.333 0.458 0.655

0.200 0.055 0.308 0.091 0.503 0.165 0.674 0.237

0.12 0.10 0.19 0.65

1.01 1.01 0.90 0.91

Respiratory rate

6 12 24 48

1,102 1,016 838 697

1,058 979 809 674

108 201 298 407

0.098 0.198 0.356 0.584

0.102 0.784 [0.771, 0.796] 0.791 [0.745, 0.834] 0.185 0.733 [0.720, 0.747] 0.740 [0.702, 0.779] 0.316 0.693 [0.681, 0.705] 0.701 [0.664, 0.738] 0.468 0.660 [0.649, 0.672] 0.665 [0.621, 0.707]

0.341 0.415 0.534 0.703

0.272 0.078 0.443 0.141 0.603 0.208 0.682 0.243

0.46 0.22 0.13 0.47

1.27 1.09 0.89 0.67

SpO2

6 12 24 48

352 307 257 227

339 296 250 220

61 105 150 185

0.173 0.342 0.584 0.815

0.182 0.656 [0.645, 0.667] 0.657 [0.584, 0.726] 0.313 0.656 [0.640, 0.671] 0.649 [0.586, 0.713] 0.502 0.648 [0.631, 0.664] 0.645 [0.577, 0.717] 0.688 0.725 [0.697, 0.753] 0.725 [0.636, 0.810]

0.286 0.492 0.690 0.912

0.448 0.140 0.618 0.215 0.675 0.240 0.480 0.154

−0.54 −0.05 0.36 0.63

0.65 0.73 0.68 1.27

SaO2

6 12 24 48

962 897 788 701

910 851 751 673

143 235 375 481

0.149 0.262 0.476 0.686

0.139 0.703 [0.697, 0.710] 0.710 [0.668, 0.752] 0.248 0.710 [0.707, 0.712] 0.709 [0.672, 0.744] 0.410 0.686 [0.680, 0.693] 0.686 [0.651, 0.724] 0.588 0.646 [0.625, 0.667] 0.646 [0.599, 0.691]

0.334 0.452 0.642 0.784

0.385 0.117 0.523 0.174 0.650 0.229 0.620 0.216

−0.02 0.13 0.30 0.54

0.93 1.04 1.00 0.75

Death in the unit

48

11,498

9,506

294

0.0256

0.0023

0.195

0.138

n.e.

n.e.

0.798 [0.768, 0.828]

92

0.808 [0.780, 0.835]

AUPRC

Log-loss

Brier Intercept

0.025

Slope

Supplementary Note 7: The registered MIMIC-IV read Scope and sources. This note reports every outcome and every setting of the registered read of the persondisjoint MIMIC-IV test split, as the all-results rule of the protocol requires. The read had two registered families. Family A applied the five frozen eICU-CRD checkpoints to MIMIC-IV without refitting and asked whether synchronization still improved the forecasts. Family B refitted the clinical-endpoint probes within the MIMIC-IV training split and asked whether the fine-tuned twin state still added to gradient boosting. The frozen model’s death and unit-exit forecasts were also scored at hour 48 as disclosed secondary outcomes, with no criterion. The test split had been read before. The protocol lists 34 earlier entries, 27 of them scientific reads of the split, including zero-shot transfers of earlier versions of the eICU-CRD-trained twin and outcomes that overlap those scored here (protocol section 2.3, reproduced later in this note). The read is therefore a registered re-read of a previously analyzed split under a rule fixed before the read. The protocol (MIMIC_IV_READ_PROTOCOL_ 20260930_V1.md), the result note (MIMIC_IV_READ_RESULT_20261002_V1.md) and the release review (MIMIC_ IV_READ_RELEASE_REVIEW_20261002_V1.md) are in reports/revision_v3/. The 41 shipped aggregate files are in source_tables/revision_v3/mimic4_read_6f9c4a6_13172_v1/. A script (make_si_note7_tables.py) wrote every table of this note from these files and from the committed files named in the captions, and it checks the counts and outcomes stated in this text against them. The power and precision table, the disclosure table and the lists of deviations and limitations are converted from the text of the protocol. As the protocol requires (sections 3.9 and 13), the provenance comes first, then the mechanism records, then the outcomes. Design in brief. Family A used three landmark and horizon pairs, (L, H) = (12, 6), (18, 12) and (30, 24), and three targets, heart rate, respiratory rate and SpO2 , which gave 9 settings. The arterial-saturation target was not scored, because in MIMIC-IV that channel holds sparse blood-gas values. The synchronized forecast (S2) read hours 1 to L and forecast a first crossing of the target’s threshold in the next H hours. The primary comparator (S1) was the forecast of the same window that the same frozen model made at hour 6. The registered secondary comparator was the patient’s last measured value of the target at hour L (LOCF). A descriptive comparator (S1-adm) was the hour-6 forecast of a crossing within H hours of hour 6. Inputs were standardized with MIMIC-IV training statistics, so each threshold was the MIMIC-IV training mean of its target. Rosters required a record reaching hour 30. The estimate was the five-seed mean AUROC difference, with 10,000 whole-person bootstrap replicates, two-sided add-one P values and Holm correction over the 9 settings, separately for S2 − S1 (A1) and S2 − LOCF (A2). A setting survived when its Holm-adjusted P was below 0.05, its estimate was positive and most replicates lay on the side of its estimate. Each comparison received one of four outcome classes (protocol section 3.7): criterion met in all 9 settings; criterion met in 5 to 8 settings with none significantly worse before correction (A_MAJORITY); criterion not met (A_NOT_SHOWN); and criterion not met with at least one setting worse after Holm correction (A_CONTRADICTED). A2 used the same classes with the prefix A2. Family B refitted four arms within the 66,007 stays of the MIMIC-IV training split. The comparator b was the fixed CatBoost recipe on summary features of the record, fitted once with tree seed 13. The encoder f was initialized from the pretrained twin and fine-tuned on the endpoints, and g was the same architecture trained from random initialization, each with seeds 13, 17, 23, 29 and 31. The hybrid h was the CatBoost recipe on the features of b and the fine-tuned state of f . Endpoints were scored within 24 hours of landmarks at 12, 24 and 48 hours: hypotension (E2) and acute kidney injury by the KDIGO creatinine criterion (E3) as primary endpoints, and hypoxemia (E4) as a registered secondary endpoint. Death in the unit was not scored as an endpoint, because the harmonized MIMIC-IV records keep a hospital-death flag without its time. The primary test was one-sided for h − b, the five-seed mean of h minus b, on E2 and E3, with 10,000 whole-person bootstrap replicates, add-one P values and Holm correction over the two endpoints. Intervals are two-sided 95% percentile intervals, so a gain significant at the second Holm step can have a lower interval bound at or below zero. For the disclosed secondary outcomes, the frozen checkpoints were read at hour 48 in stays whose record reached hour 48 and whose unit stay lasted beyond it. The death forecast PD was the largest of the four target-specific death cumulative incidences at 48 hours. The exit forecast PX (H) was the largest, over the four heads, of the cumulative incidence of leaving the unit, alive or dead, before the head’s target crossed its threshold, for H = 6, 12, 24 and 48 hours. The outcomes were hospital death in a hospitalization whose unit stay ended within 48 hours of hour 48 (MORT-W), hospital death at any time in the hospitalization (MORT-H) and leaving the unit within H hours of hour 48 (LOS-XH). Their intervals are 95% whole-person bootstrap intervals of the five-seed mean (10,000 replicates) and Student t intervals across the five seeds (4 degrees of freedom). Provenance and execution. The protocol was committed alone as commit P (3ad11ead, tag mimic4-readprotocol-v1). The scholar then gave a dated acknowledgment of it (decision item 20 of reports/revision_ 93

v3/ROUTE_B_DECISION_20260926_V1.md). The code, tests and launchers were committed next as package commit A, a sequence of seven commits ending at 80c86ec7. Manifest job 13123 ran from A and listed 14,211 test stays of 9,805 persons. No test person had a training or validation stay, and the two split columns agreed in every MIMIC-IV row. Commit C2 (6f9c4a62, tag mimic4-read-freeze-v1) then froze the protocol by pinning these manifest values. Registry commit B (3c58d87d) entered the two rows of the test-read registry and the run row before any read was submitted. The first chain of jobs from B (13138 to 13148) stayed pending in the queue and was cancelled before any stage started, so nothing was read. The read then ran from relaunch commit D1 (0e8b093b) and registry commit B′ 1 (52b5eef7) as run 52b5eef7a7f8_20261002t120818z_mimic4_ read (Supplementary Table S20a). Every job and array task ended with state COMPLETED and exit code 0:0. Each stage was written to run its native tests before touching data and to stop on a failure. The stage logs were not shipped, so this rests on the exit codes and on the launcher. A shipping script written after the protocol (scripts/revision_v3_mimic4_read_ship_v1.py, commit a29db2a3) copied the 41 allowlisted aggregate files after scanning them against every stay and person identifier of the test manifest and of the MIMIC-IV training split. No stay-level file left the cluster or was committed. A release review by a separate agent with read-only access re-applied the decision rules of protocol sections 3.6, 3.7, 4.4 and 4.5 to the shipped contrasts, recomputed the Holm adjustments with a maximum difference of 0.0 and found no correctness error. The reviewing agent belongs to the same model family as the agent that built and ran the read, so the review is independent in context only. Release, freeze, labels and verification. Stage R released the occurrence of measurements in the test split, without outcomes, predictions or metrics, under an authorization that binds the 34 disclosed earlier entries, 27 of them scientific reads of the split, for the full split of 14,211 stays of 9,805 persons. Stage FZ recorded the SHA-256 digest of every prediction file before any label existed (freeze digest beginning 2f480e66). Stage LB verified those digests without parsing a prediction value and then built the labels; its receipt records neither counts nor metrics. No test stay was a rescued zero-measurement stay. Stage SC wrote one read-once marker and scored the labels. Stage VF read the labels under that marker, wrote no marker and reproduced 185,380 values of the scored outputs at a tolerance of 10−12 . VF calls the same scoring function as SC, so its agreement shows that the scoring is deterministic; it does not test an independent implementation.

Mechanism records The protocol required three descriptive records to be reported before the outcomes. They have no decision role and are stated here as recorded. Family A. The roster of every setting held 0.986 to 1.055 times the number of stays projected from the MIMIC-IV training census, inside the registered bounds of 0.5 to 2 (record M-A1; Supplementary Table S20b). No prediction was exactly 0 or 1, and the target was unmeasured at hour 6 in 2.7% to 6.3% of roster stays (MA4). Between 66.7% and 84.8% of roster stays had rows after the terminal clock (M-A5). This is the disclosed deviation D15: labels stop at the terminal clock, and a record can reach hour 30 through such rows. The per-seed AUROCs of S2, S1 and LOCF (M-A2) are given with the other per-seed values in Supplementary Table S21. S2 exceeded S1 in every seed of every setting. Record M-A3 sets the observed-hour fractions of 12 input channels in test hours 1 to 30 beside the training-census fractions over whole records, as registered (Supplementary Table S20c). The two windows differ, so these ratios do not estimate a shift between the test and training splits. The input shift that the frozen model meets in MIMIC-IV was described before the read by the MIMIC-IV training census (job 13062), which compared the observed-hour fractions of the 41 physiological channels in seeded samples of MIMIC-IV and eICU-CRD training stays (Supplementary Table S20d; protocol deviation D16). The arterial-saturation channel, which in eICU-CRD combines the bedside monitor’s periodic saturation readings with blood-gas values, was observed in 0.012 of MIMIC-IV hours against 0.492 of eICU-CRD hours. Heart rate and non-invasive blood pressure were observed less often in MIMIC-IV, and most laboratory values more often. No arm of the read varied the input mapping, so neither this record nor any outcome attributes a result to the shift. Family B. The numbers of test rows with known status and of events at each endpoint and landmark lay between 0.986 and 1.012 and between 0.919 and 1.114 times their projections from the training census (Supplementary Table S20e). The bootstrap population was the 9,623 test persons with a test row. The training fit sets of b and h are given in Supplementary Table S20f. The physical-unit guard passed on the training and test splits, with channel medians that differed by at most 0.5 units between the splits (Supplementary Table S20g). All ten encoder runs clipped the gradient of every batch, 212 of 212 batches in every epoch at a clip norm of

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1.0 (Supplementary Table S20h). In the four eICU-CRD encoder runs of probe P1, every batch was clipped as well, 360 of 360 in every epoch (source_tables/revision_v3/route_b_p1_result_13032_v1/train/). The pretrained encoder f stopped at the 15-epoch cap with its best epoch at the cap for seeds 13, 17 and 23, and on the patience rule for seeds 29 and 31. The scratch encoder g stopped on the patience rule for four seeds and at the cap for seed 13, whose best epoch was 14. In probe P1 on eICU-CRD, one of the two pretrainedencoder runs (seed 29) also ended at the cap. No training receipt records access to the test or validation split. Disclosed secondary outcomes. Of the 14,211 manifest stays, 6,951 stays of 5,396 persons entered the population (Supplementary Table S20i). The main-text Results describe them as stays still in the unit after hour 48. The registered entry rule also requires a record reaching hour 48, so it excludes 65 stays still in the unit whose record ends before hour 48; the main-text Methods (Forecast evaluation) state the rule. One third of the hospital deaths (MORT-H events) occurred in a hospitalization whose unit stay ended within 48 hours of hour 48 (MORT-W events). No exit forecast needed the sum correction of protocol section 3.11. The head that attains the maximum of the death forecast varies by seed (Supplementary Table S20j). Supplementary Table S20: Execution and mechanism records of the registered MIMIC-IV read. Panel a gives the jobs of the read. Panels b to j give the descriptive mechanism records that the protocol required before the outcomes: family A and the input shift that the frozen model meets (b to d), family B (e to h) and the disclosed secondary outcomes (i and j). Settings (L, H): read to hour L, forecast the next H hours. Files named without a path are in source_tables/ revision_v3/mimic4_read_6f9c4a6_13172_v1/. a Jobs of the read. Run 52b5eef7a7f8_20261002t120818z_mimic4_read, launched from registry commit B′ 1 (52b5eef7); the manifest stage M ran earlier from package commit A. Stages with several tasks ran as Slurm job arrays, and their elapsed time is the range over tasks. State: Slurm accounting state and exit code. Source: job_log_summary.json.

Stage

Role

M

TEST manifest (identity columns only; run from package commit A) Family B extract (MIMIC-IV training split only) Family B encoder training, arms f and g Family B tree fits, arms b and h Release of the TEST observation occurrence Family A predictions (hours 1 to 30) Secondary-outcome predictions (hours 1 to 48) Family B predictions Freeze of every prediction file Labels, built after the freeze Scoring under the read-once marker Verification of the scored outputs

BX BT BF R AP AS BP FZ LB SC VF

Job

Tasks

Elapsed (h:min:s)

State

13123

1

00:00:33

COMPLETED, exit 0:0

13163

1

00:06:26

COMPLETED, exit 0:0

13164 13165 13166

10 15 1

00:03:19 to 00:03:36 00:03:57 to 00:06:19 00:04:41

COMPLETED, exit 0:0 COMPLETED, exit 0:0 COMPLETED, exit 0:0

13167 13168

5 5

00:02:10 to 00:02:36 00:01:52 to 00:02:15

COMPLETED, exit 0:0 COMPLETED, exit 0:0

13169 13170 13171 13172 13173

1 1 1 1 1

00:02:52 00:00:32 00:04:43 00:11:14 00:12:05

COMPLETED, exit 0:0 COMPLETED, exit 0:0 COMPLETED, exit 0:0 COMPLETED, exit 0:0 COMPLETED, exit 0:0

b Family A: rosters, predictions and the terminal clock. Roster stays against the projection from the MIMIC-IV training census (record M-A1; registered bounds 0.5 to 2); share of roster stays whose target was unmeasured at hour 6 (M-A4); share of roster stays with rows after the terminal clock (M-A5); share of prediction values equal to exactly 0 or 1 (M-A4). Source: mechanism_a.json.

Setting SpO2 (12, 6) SpO2 (18, 12) SpO2 (30, 24) Heart rate (12, 6) Heart rate (18, 12) Heart rate (30, 24) Respiratory rate (12, 6) Respiratory rate (18, 12) Respiratory rate (30, 24)

Roster stays

Projected stays

Ratio

Target unmeasured at hour 6 (%)

Rows after the terminal clock (%)

Predictions at 0 or 1 (share)

2,485 1,493 559 3,277 2,066 876 2,341 1,227 401

2,520 1,466 566 3,232 2,019 850 2,321 1,226 380

0.986 1.018 0.988 1.014 1.023 1.031 1.009 1.001 1.055

5.8 6.0 6.3 2.7 2.9 3.4 3.6 4.1 4.2

81.6 77.9 66.7 83.8 79.8 78.0 84.8 82.3 76.1

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

95

c Family A, record M-A3: observed-hour fractions. Fraction of hours in which each input channel was observed in test hours 1 to 30, identical across seeds, beside the fraction over whole records in the MIMIC-IV training census, as registered. The two windows differ, so the ratio does not estimate a shift between the test and training splits. Source: mechanism_a.json.

Channel

Test, hours 1 to 30

Training census, whole records

Ratio

0.865 0.857 0.850 0.025 0.600 0.279 0.261 0.479 0.076 0.052 0.077 0.054

0.466 0.461 0.456 0.012 0.287 0.180 0.139 0.271 0.060 0.021 0.036 0.036

1.85 1.86 1.86 2.18 2.09 1.55 1.88 1.77 1.27 2.42 2.15 1.52

Heart rate Respiratory rate SpO2 SaO2 Non-invasive mean arterial pressure Invasive mean arterial pressure Temperature Urine output Creatinine Lactate pH Partial thromboplastin time Valid test hours in hours 1 to 30: 411,460

d Input shift seen by the frozen model. Fraction of hours in which each of the 41 physiological input channels was observed, and its median standardized value with each database standardized by its own training statistics, in seeded samples of MIMIC-IV and eICU-CRD training stays, computed before the read by the MIMIC-IV training census (job 13062). In eICU-CRD the SpO2 channel holds nurse-charted pulse oximetry and the SaO2 channel combines the bedside monitor’s periodic saturation readings with blood-gas values; in MIMIC-IV the SpO2 channel holds hourly bedside pulse oximetry and the SaO2 channel holds blood-gas values. No channel was remapped for the frozen model (protocol choice C9). Source: source_tables/revision_v3/mimic4_train_census_32de53f_13062_v1/c3_input_ shift_channels.csv. Channel Heart rate Invasive systolic pressure Invasive diastolic pressure Invasive mean arterial pressure Non-invasive systolic pressure Non-invasive diastolic pressure Non-invasive mean arterial pressure SpO2 SaO2 Temperature Respiratory rate Expiratory tidal volume Plateau pressure Lactate Blood glucose Magnesium Sodium Creatinine Calcium Chloride Potassium Partial thromboplastin time Bilirubin Alanine aminotransferase Aspartate aminotransferase Alkaline phosphatase Albumin Phosphate Bicarbonate Blood urea nitrogen pH PaO2 PaCO2 Hemoglobin White blood cells Platelets Urine output Glasgow Coma Scale, total Glasgow Coma Scale, eye Glasgow Coma Scale, motor Glasgow Coma Scale, verbal

Observed hours, MIMIC-IV

Observed hours, eICU-CRD

Ratio

Median z, MIMIC-IV / eICU-CRD

0.466 0.163 0.163 0.180 0.287 0.287 0.287 0.456 0.012 0.139 0.461 0.025 0.018 0.021 0.123 0.058 0.066 0.060 0.053 0.064 0.071 0.036 0.016 0.016 0.016 0.016 0.008 0.053 0.060 0.060 0.036 0.033 0.033 0.055 0.054 0.055 0.271 0.128 0.129 0.128 0.129

0.587 0.103 0.103 0.103 0.503 0.503 0.467 0.365 0.492 0.217 0.567 0.015 0.015 0.006 0.158 0.023 0.046 0.043 0.042 0.044 0.050 0.008 0.012 0.012 0.013 0.012 0.015 0.015 0.040 0.043 0.012 0.013 0.012 0.043 0.038 0.038 0.146 0.125 0.107 0.107 0.106

0.79 1.58 1.58 1.74 0.57 0.57 0.62 1.25 0.02 0.64 0.81 1.69 1.23 3.57 0.78 2.46 1.42 1.38 1.26 1.46 1.42 4.37 1.27 1.26 1.26 1.26 0.52 3.43 1.48 1.38 2.90 2.63 2.67 1.27 1.41 1.43 1.85 1.02 1.21 1.20 1.21

−0.155 / −0.095 −0.075 / 0.035 −0.175 / −0.155 −0.155 / −0.075 −0.095 / −0.085 −0.125 / −0.065 −0.115 / −0.065 0.175 / 0.165 0.585 / 0.145 −0.065 / −0.045 −0.215 / −0.225 0.195 / −0.055 −0.215 / −0.095 −0.145 / −0.195 −0.225 / −0.265 −0.205 / 0.035 −0.085 / 0.045 −0.365 / −0.345 −0.075 / −0.105 −0.015 / −0.005 −0.185 / −0.195 −0.365 / −0.315 −0.275 / −0.225 −0.205 / −0.205 −0.205 / −0.215 −0.355 / −0.325 0.055 / −0.025 −0.215 / −0.195 −0.095 / 0.005 −0.335 / −0.345 −0.025 / −0.065 −0.125 / −0.225 −0.115 / −0.175 0.005 / −0.115 −0.145 / −0.235 −0.195 / −0.255 −0.285 / −0.255 0.565 / 0.565 0.435 / 0.415 0.335 / 0.345 0.635 / 0.625

Seeded training samples: MIMIC-IV 5,000 stays (794,043 valid hours); eICU-CRD 5,000 stays (543,162 valid hours)

96

e Family B: test rows against the projection. Test rows with known status and events per endpoint and landmark against the projection from the MIMIC-IV training census, and the persons contributing rows. Source: mechanism_b.json. Endpoint

Landmark (h)

Known rows

Projected

Ratio

Events

Projected

Ratio

Persons

12 24 48 12 24 48 12 24 48

9,586 6,592 3,234 10,023 8,893 4,777 12,648 10,348 6,252

9,560 6,669 3,279 9,998 8,916 4,722 12,669 10,362 6,267

1.003 0.988 0.986 1.003 0.997 1.012 0.998 0.999 0.998

2,113 989 315 1,214 892 301 342 248 125

1,949 952 311 1,199 901 291 307 232 136

1.084 1.039 1.013 1.013 0.990 1.034 1.114 1.069 0.919

7,278 5,283 2,777 7,558 6,776 3,935 8,922 7,612 4,929

Hypotension (E2) Hypotension (E2) Hypotension (E2) Acute kidney injury (E3) Acute kidney injury (E3) Acute kidney injury (E3) Hypoxemia (E4, secondary) Hypoxemia (E4, secondary) Hypoxemia (E4, secondary)

Bootstrap population: 9,623 test persons with a test row; rescued zero-measurement stays excluded: 0 (training split), 0 (test split) f Family B: training fit sets. Rows, persons and events of the training fit sets of b and h, identical for every tree fit of an endpoint. Source: fit_receipt_*.json. Endpoint

Training fit rows

Persons

Events

90,610 109,782 136,085

33,878 38,575 41,721

14,925 11,109 3,132

Hypotension (E2) Acute kidney injury (E3) Hypoxemia (E4, secondary)

g Family B: physical-unit guard. Median of each channel in physical units and fraction of invalid measured hours on the training and test splits (family B stops if more than 1% of measured hours are invalid). Sources: mechanism_b.json and label_receipt.json. Channel

Training median

Training invalid

Test median

Test invalid

76.0 78.0 97.0 88.4

0

76.5 78.0 97.0 88.4

0 3.1×10−5 0 8.2×10−5

Invasive mean arterial pressure (mmHg) Non-invasive mean arterial pressure (mmHg) SpO2 (%) Creatinine (µmol/L)

1.6×10−5 0 5.9×10−5

h Family B: training histories of the encoders. Ten runs on the MIMIC-IV training split: f , initialized from the pretrained eICU-CRD twin, and g, random initialization, each with seeds 13, 17, 23, 29 and 31. AdamW, at most 15 epochs, early stopping with a patience of 2 epochs on the mean binary cross-entropy of the inner split; the rows by epoch give that value, and ∗ marks the best epoch, whose weights were kept. Cap, stopped at the 15-epoch cap; patience, stopped by the patience rule. The 54,102 fitting and 5,936 early-stopping stays are the 60,038 of the 66,007 training-split stays that have at least one landmark row at risk with known status for hypotension, acute kidney injury or hypoxemia (train_stays_with_labels in each receipt). Sources: mechanism_b.json and train_receipt_*.json. f (pretrained), seed

Stopped by Epochs run Best epoch Batches clipped per epoch Epoch 1 Epoch 2 Epoch 3 Epoch 4 Epoch 5 Epoch 6 Epoch 7 Epoch 8 Epoch 9 Epoch 10 Epoch 11 Epoch 12 Epoch 13 Epoch 14 Epoch 15

g (random initialization), seed

13

17

23

29

31

13

17

23

29

31

cap 15 15 212 of 212

cap 15 15 212 of 212

cap 15 15 212 of 212

patience 14 12 212 of 212

patience 13 11 212 of 212

cap 15 14 212 of 212

patience 7 5 212 of 212

patience 8 6 212 of 212

patience 7 5 212 of 212

patience 14 12 212 of 212

0.2468 0.2427 0.2413 0.2402 0.2383 0.2371 0.2351 0.2347 0.2352 0.2341 0.2340 0.2329 0.2322 0.2333 0.2321∗

0.2478 0.2419 0.2395 0.2368 0.2361 0.2348 0.2347 0.2348 0.2339 0.2345 0.2334 0.2332 0.2340 0.2319 0.2314∗

0.2493 0.2437 0.2407 0.2378 0.2365 0.2359 0.2354 0.2361 0.2337 0.2333 0.2324 0.2314 0.2332 0.2312 0.2304∗

0.2494 0.2443 0.2412 0.2404 0.2372 0.2368 0.2361 0.2347 0.2338 0.2347 0.2336 0.2325∗ 0.2326 0.2338 –

0.2476 0.2430 0.2399 0.2385 0.2373 0.2375 0.2348 0.2342 0.2343 0.2332 0.2322∗ 0.2327 0.2326 – –

0.2418 0.2347 0.2368 0.2322 0.2303 0.2332 0.2299 0.2306 0.2290 0.2291 0.2281 0.2281 0.2275 0.2274∗ 0.2294

0.2447 0.2367 0.2370 0.2337 0.2318∗ 0.2318 0.2354 – – – – – – – –

0.2450 0.2356 0.2338 0.2343 0.2328 0.2305∗ 0.2313 0.2336 – – – – – – –

0.2410 0.2354 0.2342 0.2321 0.2308∗ 0.2312 0.2329 – – – – – – – –

0.2452 0.2361 0.2373 0.2328 0.2319 0.2331 0.2314 0.2293 0.2313 0.2289 0.2294 0.2281∗ 0.2282 0.2293 –

Every run: 54,102 training stays and an inner early-stopping split of 5,936 stays of 4,213 persons

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i Disclosed secondary outcomes: population and events, identical across seeds. Source: mechanism_sec.json. Quantity

Value

Included: record reaches hour 48 and unit stay lasts beyond it of which persons Excluded: terminal clock at or before hour 48 Excluded: record shorter than 48 h, terminal clock after hour 48 Excluded: unresolved terminal Manifest stays (sum)

6,951 5,396 7,193 65 2 14,211

Events: hospital death, unit exit within 48 h (MORT-W) Events: hospital death (MORT-H) Events: leaving the unit within 6 h (LOS-X6) Events: leaving the unit within 12 h (LOS-X12) Events: leaving the unit within 24 h (LOS-X24) Events: leaving the unit within 48 h (LOS-X48)

324 972 784 1,318 2,237 3,548

Share of MORT-H events that are MORT-W events

0.333

j Disclosed secondary outcomes: records by seed. Exit-forecast sums above 1 by at most 10−6 that were set to 1; percentiles of the death forecast PD and of the 48-hour exit forecast PX (48); and the share of stays in which each target-specific head attains the maximum that defines each forecast. Source: mechanism_sec.json. Seed Quantity

13

Exit-forecast sums set to 1

17

23

29

31

0

0

0

0

0

Death forecast PD , 5th percentile Death forecast PD , 25th percentile Death forecast PD , 50th percentile Death forecast PD , 75th percentile Death forecast PD , 95th percentile

0.00033 0.00065 0.0012 0.0030 0.019

0.00029 0.00057 0.0010 0.0022 0.012

0.00044 0.00092 0.0015 0.0028 0.021

0.00040 0.00079 0.0014 0.0030 0.017

0.00047 0.00087 0.0014 0.0026 0.020

Exit forecast PX (48), 5th percentile Exit forecast PX (48), 25th percentile Exit forecast PX (48), 50th percentile Exit forecast PX (48), 75th percentile Exit forecast PX (48), 95th percentile

0.024 0.062 0.24 0.44 0.61

0.020 0.051 0.22 0.43 0.62

0.023 0.072 0.26 0.47 0.65

0.025 0.071 0.27 0.47 0.62

0.022 0.054 0.19 0.40 0.58

PD attained by the heart-rate head (%) PD attained by the respiratory-rate head (%) PD attained by the SpO2 head (%) PD attained by the SaO2 head (%)

7.8 11.9 79.4 0.9

17.3 54.6 27.9 0.1

1.6 36.4 61.9 0.0

21.7 21.3 56.9 0.0

31.0 4.7 63.2 1.1

PX (48) attained by the heart-rate head (%) PX (48) attained by the respiratory-rate head (%) PX (48) attained by the SpO2 head (%) PX (48) attained by the SaO2 head (%)

50.9 17.9 10.0 21.2

46.1 31.7 6.1 16.1

49.3 30.6 10.4 9.8

50.7 23.8 14.0 11.5

52.5 29.7 2.1 15.7

Supplementary Table S21: Family A AUROCs by seed. S2, synchronized forecast; S1, the model’s hour-6 forecast of the same window (primary comparator); S1-adm, the hour-6 forecast of a crossing within H hours of hour 6 (descriptive); LOCF, the patient’s last measured value of the target at hour L, which does not depend on the seed. The last two columns are the per-seed differences whose five-seed means are the estimates of A1 and A2. Hour-30 rosters and MIMIC-IV training-mean thresholds. Source: metrics_a.csv; the per-seed AUROCs of S2, S1 and LOCF equal record M-A2 of mechanism_a.json. Setting

Seed

S2

S1

S1-adm

LOCF

S2 − S1

S2 − LOCF

SpO2 (12, 6)

13 17 23 29 31 Five-seed mean

0.604 0.576 0.594 0.593 0.590 0.592

0.574 0.562 0.564 0.569 0.562 0.566

0.556 0.542 0.551 0.560 0.537 0.549

0.675 0.675 0.675 0.675 0.675 0.675

+0.030 +0.014 +0.031 +0.024 +0.028 +0.025

−0.071 −0.099 −0.081 −0.083 −0.085 −0.084

SpO2 (18, 12)

13 17 23 29 31 Five-seed mean

0.643 0.610 0.626 0.628 0.608 0.623

0.598 0.569 0.572 0.594 0.575 0.582

0.575 0.561 0.552 0.574 0.567 0.566

0.636 0.636 0.636 0.636 0.636 0.636

+0.045 +0.041 +0.053 +0.034 +0.033 +0.041

+0.008 −0.026 −0.010 −0.007 −0.027 −0.013

SpO2 (30, 24)

13 17 23 29 31

0.638 0.618 0.608 0.605 0.585

0.585 0.589 0.567 0.599 0.569

0.601 0.594 0.570 0.614 0.576

0.622 0.622 0.622 0.622 0.622

+0.053 +0.028 +0.041 +0.006 +0.016

+0.015 −0.005 −0.014 −0.017 −0.037

98

Supplementary Table S21, continued S2

S1

S1-adm

LOCF

S2 − S1

S2 − LOCF

Five-seed mean

0.611

0.582

0.591

0.622

+0.029

−0.011

Heart rate (12, 6)

13 17 23 29 31 Five-seed mean

0.730 0.719 0.726 0.726 0.720 0.724

0.635 0.633 0.640 0.638 0.634 0.636

0.636 0.634 0.637 0.640 0.633 0.636

0.732 0.732 0.732 0.732 0.732 0.732

+0.094 +0.086 +0.086 +0.088 +0.086 +0.088

−0.002 −0.013 −0.006 −0.006 −0.012 −0.008

Heart rate (18, 12)

13 17 23 29 31 Five-seed mean

0.731 0.730 0.735 0.736 0.732 0.733

0.617 0.615 0.620 0.626 0.616 0.619

0.616 0.615 0.613 0.619 0.611 0.615

0.732 0.732 0.732 0.732 0.732 0.732

+0.113 +0.115 +0.115 +0.110 +0.116 +0.114

−0.001 −0.001 +0.003 +0.004 0.000 +0.001

Heart rate (30, 24)

13 17 23 29 31 Five-seed mean

0.651 0.656 0.656 0.652 0.655 0.654

0.544 0.546 0.542 0.543 0.539 0.543

0.561 0.558 0.554 0.556 0.551 0.556

0.696 0.696 0.696 0.696 0.696 0.696

+0.108 +0.110 +0.113 +0.109 +0.116 +0.111

−0.044 −0.040 −0.040 −0.044 −0.041 −0.042

Respiratory rate (12, 6)

13 17 23 29 31 Five-seed mean

0.637 0.627 0.624 0.624 0.619 0.626

0.561 0.558 0.561 0.557 0.556 0.559

0.571 0.565 0.560 0.564 0.561 0.564

0.603 0.603 0.603 0.603 0.603 0.603

+0.076 +0.069 +0.062 +0.067 +0.063 +0.068

+0.034 +0.024 +0.021 +0.021 +0.016 +0.023

Respiratory rate (18, 12)

13 17 23 29 31 Five-seed mean

0.647 0.611 0.614 0.628 0.607 0.622

0.580 0.552 0.555 0.557 0.550 0.559

0.588 0.554 0.554 0.567 0.560 0.565

0.588 0.588 0.588 0.588 0.588 0.588

+0.067 +0.059 +0.059 +0.071 +0.057 +0.062

+0.059 +0.023 +0.027 +0.041 +0.019 +0.034

Respiratory rate (30, 24)

13 17 23 29 31 Five-seed mean

0.623 0.588 0.590 0.606 0.562 0.594

0.547 0.517 0.508 0.516 0.507 0.519

0.595 0.557 0.554 0.569 0.552 0.566

0.624 0.624 0.624 0.624 0.624 0.624

+0.077 +0.072 +0.081 +0.091 +0.054 +0.075

−0.001 −0.036 −0.035 −0.018 −0.063 −0.030

Setting

Seed

99

Outcomes Summary. The read returned A_MAJORITY for family A, A2_CONTRADICTED for its registered secondary comparison and B_E3_ONLY for family B, and the disclosed secondary outcomes were scored. No flag was raised, and no setting or endpoint was undefined. For this combination of outcomes, the interpretation guide fixed in protocol section 3.9 reads: “the frozen model still gains from reading the newer hours in MIMIC-IV, but its advantage over the patient’s last measured value is shown on eICU only.” The isolation arms were not repeated on MIMIC-IV, so the attribution of the gain to the patient’s newly recorded measurements rests on the eICU-CRD validation split. Family A. The synchronized forecast outperformed the hour-6 forecast in 8 of 9 settings after Holm correction (estimates +0.025 to +0.114 AUROC; not significant in SpO2 (30, 24)). This meets the registered transfer criterion of at least 5 of 9 settings with none significantly worse before correction, but unlike on eICU-CRD the gain was not significant in every setting (Supplementary Table S22a). In SpO2 (30, 24) the estimate was +0.029 (95% interval −0.021 to +0.077; Holm-adjusted P = 0.26), and no setting was significantly worse before or after correction. Eight of the nine estimates were smaller than the eICU-CRD estimate for the same setting; heart rate (18, 12) was the exception. Against the patient’s last measured value, the synchronized forecast was better in 1 of 9 settings after Holm correction, respiratory rate (12, 6), and worse in 2, SpO2 (12, 6) and heart rate (30, 24), with a median difference of −0.011 (range −0.084 to +0.034 AUROC). The three registered sensitivities gave the same outcome classes with the same numbers of surviving settings: per-cell rosters, thresholds at the eICU-CRD training means and the stay-level bootstrap of the eICU-CRD analysis (Supplementary Table S23). Under the eICU-CRD-mean thresholds, SpO2 (12, 6) remained worse than the last measured value after Holm correction, and heart rate (30, 24) was worse before correction only. Because the stay-level bootstrap gave the same outcome classes as the primary analysis, the protocol’s rule that a differing class be reported in the main-text Results did not apply. In raw probabilities, S2 had lower log-loss and Brier score than S1 in 8 of 9 settings and higher values in SpO2 (12, 6) (Supplementary Table S22b). Nothing was calibrated on MIMIC-IV, whereas the eICU-CRD comparison in the main text uses log-loss and Brier score after isotonic calibration on eICU-CRD validation data, so the two are not on the same footing. Supplementary Table S22: Family A on the MIMIC-IV test split: the registered contrasts and the descriptive contrasts. Settings (L, H): read to hour L, forecast the next H hours. S2, synchronized forecast; S1, the model’s hour-6 forecast of the same window; S1-adm, the hour-6 forecast of a crossing within H hours of hour 6; LOCF, the patient’s last measured value of the target at hour L. Files are in source_tables/revision_v3/mimic4_read_6f9c4a6_13172_v1/. a Synchronization (A1) and last-value comparison (A2). Five-seed mean AUROC differences with 95% intervals from 10,000 wholeperson bootstrap replicates (stream seed 20260930), two-sided add-one P values and Holm correction over the 9 settings within each family; the smallest attainable Holm-adjusted P is about 0.0018. ∗ Survives (Holm-adjusted P < 0.05, positive and direction-concordant). † Worse after Holm correction. Stays, persons and events: hour-30 roster. The last two columns give, for A1 and A2, the eICU-CRD test-set estimates of the same model head and, for SpO2 , of the SaO2 head, whose eICU-CRD channel combines the bedside monitor’s periodic saturation readings with blood-gas values (protocol section 3.2); neither pairing enters a decision. Source: contrasts_a.csv. Setting

Stays

Persons

Events

SpO2 (12, 6) SpO2 (18, 12) SpO2 (30, 24) Heart rate (12, 6) Heart rate (18, 12) Heart rate (30, 24) Respiratory rate (12, 6) Respiratory rate (18, 12) Respiratory rate (30, 24)

2,485 1,493 559 3,277 2,066 876 2,341 1,227 401

2,241 1,387 529 2,856 1,855 822 2,106 1,148 385

819 837 372 527 620 422 885 748 307

A1: S2 − S1 [95% CI] +0.025∗ [+0.006, +0.046] +0.041∗ [+0.011, +0.071] +0.029 [−0.021, +0.077] +0.088∗ [+0.066, +0.110] +0.114∗ [+0.092, +0.136] +0.111∗ [+0.077, +0.146] +0.068∗ [+0.045, +0.089] +0.062∗ [+0.036, +0.089] +0.075∗ [+0.023, +0.126]

Holm P 0.022 0.022 0.26 0.0018 0.0018 0.0018 0.0018 0.0018 0.022

A2: S2 − LOCF [95% CI] −0.084† [−0.114, −0.053] −0.013 [−0.048, +0.023] −0.011 [−0.071, +0.048] −0.008 [−0.021, +0.005] +0.001 [−0.013, +0.016] −0.042† [−0.070, −0.014] +0.023∗ [+0.007, +0.039] +0.034 [+0.007, +0.061] −0.030 [−0.089, +0.028]

Holm P

eICU-CRD, same head: A1 / A2

eICU-CRD, SaO2 head: A1 / A2

0.0018 1.0 1.0 1.0 1.0 0.034 0.030 0.10 1.0

+0.078 / +0.062 +0.101 / +0.060 +0.153 / +0.085 +0.116 / +0.010 +0.107 / +0.027 +0.121 / +0.042 +0.102 / +0.048 +0.083 / +0.046 +0.108 / +0.049

+0.073 / +0.023 +0.083 / +0.025 +0.136 / +0.017 – – – – – –

b Descriptive contrasts. Threshold of the target in physical units (MIMIC-IV training mean; beats or breaths per minute, SpO2 in percent) and prevalence of crossings on the roster; S2 minus S1-adm; raw log-loss (probabilities clipped at 10−3 ) and raw Brier score of S2 minus S1; and the seed spread (largest minus smallest per-seed difference) of A1 and A2. Five-seed means with 95% whole-person bootstrap intervals and no correction for multiple comparisons. Nothing was calibrated on MIMIC-IV. Sources: contrasts_a.csv and metrics_a.csv. Setting SpO2 (12, 6) SpO2 (18, 12) SpO2 (30, 24) Heart rate (12, 6) Heart rate (18, 12) Heart rate (30, 24) Respiratory rate (12, 6) Respiratory rate (18, 12) Respiratory rate (30, 24)

Threshold

Prevalence

96.526 96.526 96.526 86.776 86.776 86.776 20.259 20.259 20.259

0.330 0.561 0.665 0.161 0.300 0.482 0.378 0.610 0.766

S2 − S1-adm

Log-loss, S2 − S1

Brier, S2 − S1

+0.042 [+0.024, +0.062] +0.057 [+0.029, +0.084] +0.020 [−0.028, +0.066] +0.088 [+0.066, +0.110] +0.118 [+0.094, +0.142] +0.098 [+0.061, +0.136] +0.062 [+0.040, +0.083] +0.057 [+0.029, +0.085] +0.028 [−0.028, +0.083]

+0.029 [+0.016, +0.040] −0.031 [−0.048, −0.013] −0.012 [−0.040, +0.017] −0.024 [−0.036, −0.012] −0.064 [−0.079, −0.049] −0.033 [−0.061, −0.006] −0.020 [−0.034, −0.005] −0.036 [−0.056, −0.016] −0.029 [−0.062, +0.005]

+0.014 [+0.008, +0.019] −0.015 [−0.023, −0.007] −0.007 [−0.018, +0.005] −0.007 [−0.011, −0.004] −0.025 [−0.030, −0.019] −0.022 [−0.032, −0.012] −0.013 [−0.018, −0.008] −0.021 [−0.029, −0.013] −0.016 [−0.030, −0.002]

100

Seed spread, A1

Seed spread, A2

0.016 0.020 0.047 0.008 0.006 0.009 0.014 0.014 0.036

0.028 0.035 0.052 0.011 0.006 0.005 0.018 0.040 0.062

Supplementary Table S23: Family A: registered sensitivities. Per-cell rosters (entry when the record reaches hour L); thresholds at the eICU-CRD training means applied to raw values; and the stay-level bootstrap of the eICU-CRD analysis, with P values without the add-one correction, so that a value of 0 means that no replicate fell on the other side of zero. Each block ends with its outcome classes under the rules of protocol section 3.7; the sensitivities change no outcome. ∗ Survives; † worse after Holm correction; ‡ worse before correction only. Sources: contrasts_a.csv and outcomes.json. Holm P

A2: S2 − LOCF [95% CI]

Holm P

Per-cell rosters (entry at hour L instead of hour 30) SpO2 (12, 6) 2,550 2,298 845 +0.024∗ [+0.004, +0.043] 0.038 SpO2 (18, 12) 1,507 1,401 851 +0.042∗ [+0.013, +0.072] 0.019 SpO2 (30, 24) 559 529 372 +0.029 [−0.021, +0.077] 0.26 Heart rate (12, 6) 3,408 2,962 552 +0.090∗ [+0.069, +0.111] 0.0018 Heart rate (18, 12) 2,082 1,869 636 +0.114∗ [+0.092, +0.135] 0.0018 Heart rate (30, 24) 876 822 422 +0.111∗ [+0.077, +0.146] 0.0018 Respiratory rate (12, 6) 2,430 2,180 916 +0.066∗ [+0.044, +0.087] 0.0018 Respiratory rate (18, 12) 1,237 1,157 758 +0.061∗ [+0.035, +0.087] 0.0018 Respiratory rate (30, 24) 401 385 307 +0.075∗ [+0.023, +0.126] 0.019 Outcome: A_MAJORITY (8 of 9 better); A2_CONTRADICTED (1 of 9 better, 2 worse after correction)

−0.083† [−0.113, −0.054] −0.012 [−0.048, +0.024] −0.011 [−0.071, +0.048] −0.006 [−0.019, +0.006] +0.002 [−0.012, +0.016] −0.042† [−0.070, −0.014] +0.024∗ [+0.008, +0.040] +0.033 [+0.005, +0.059] −0.030 [−0.089, +0.028]

0.0018 1.0 1.0 1.0 1.0 0.034 0.021 0.12 1.0

Thresholds at the eICU-CRD training means (SpO2 96.520%, heart rate 85.532 per minute, respiratory rate 20.082 per minute) SpO2 (12, 6) 2,485 2,241 819 +0.025∗ [+0.006, +0.046] 0.021 −0.084† [−0.114, −0.053] SpO2 (18, 12) 1,493 1,387 837 +0.041∗ [+0.011, +0.071] 0.019 −0.013 [−0.048, +0.023] SpO2 (30, 24) 559 529 372 +0.029 [−0.021, +0.077] 0.26 −0.011 [−0.071, +0.048] Heart rate (12, 6) 3,075 2,679 520 +0.081∗ [+0.060, +0.102] 0.0018 −0.006 [−0.019, +0.007] Heart rate (18, 12) 1,922 1,731 614 +0.111∗ [+0.088, +0.132] 0.0018 +0.003 [−0.012, +0.019] Heart rate (30, 24) 798 752 390 +0.100∗ [+0.064, +0.136] 0.0018 −0.040‡ [−0.070, −0.009] Respiratory rate (12, 6) 2,335 2,100 881 +0.068∗ [+0.045, +0.090] 0.0018 +0.024∗ [+0.007, +0.039] Respiratory rate (18, 12) 1,226 1,147 748 +0.062∗ [+0.035, +0.088] 0.0018 +0.034 [+0.008, +0.062] Respiratory rate (30, 24) 399 383 305 +0.076∗ [+0.025, +0.127] 0.014 −0.032 [−0.090, +0.026] Outcome: A_MAJORITY (8 of 9 better); A2_CONTRADICTED (1 of 9 better, 1 worse after correction, 1 worse before correction only)

0.0018 1.0 1.0 1.0 1.0 0.080 0.043 0.080 1.0

Stay-level bootstrap of the eICU-CRD analysis (2,000 replicates, P without the add-one correction) SpO2 (12, 6) 2,485 2,241 819 +0.025∗ [+0.004, +0.046] 0.046 −0.084† [−0.116, −0.053] SpO2 (18, 12) 1,493 1,387 837 +0.041∗ [+0.013, +0.073] 0.0080 −0.013 [−0.048, +0.022] SpO2 (30, 24) 559 529 372 +0.029 [−0.021, +0.078] 0.25 −0.011 [−0.071, +0.051] Heart rate (12, 6) 3,277 2,856 527 +0.088∗ [+0.067, +0.109] 0 −0.008 [−0.020, +0.004] ∗ Heart rate (18, 12) 2,066 1,855 620 +0.114 [+0.092, +0.135] 0 +0.001 [−0.014, +0.017] Heart rate (30, 24) 876 822 422 +0.111∗ [+0.077, +0.147] 0 −0.042† [−0.071, −0.014] Respiratory rate (12, 6) 2,341 2,106 885 +0.068∗ [+0.045, +0.090] 0 +0.023∗ [+0.007, +0.040] Respiratory rate (18, 12) 1,227 1,148 748 +0.062∗ [+0.036, +0.089] 0 +0.034 [+0.006, +0.060] Respiratory rate (30, 24) 401 385 307 +0.075∗ [+0.024, +0.125] 0.012 −0.030 [−0.090, +0.030] Outcome: A_MAJORITY (8 of 9 better); A2_CONTRADICTED (1 of 9 better, 2 worse after correction)

0 1.0 1.0 1.0 1.0 0.0080 0.042 0.13 1.0

Setting

Stays

Persons

Events

A1: S2 − S1 [95% CI]

Family B. Adding the fine-tuned state to the boosting features improved discrimination of acute kidney injury by +0.0035 (95% CI +0.0007 to +0.0065; one-sided Holm-adjusted P = 0.018), which was significant under the registered test, but not significantly that of hypotension (+0.0010; −0.0012 to +0.0033; P = 0.18; Supplementary Table S24a). No endpoint carried the flag for a hybrid worse than boosting. For hypoxemia, a registered secondary endpoint, the change was +0.0009 (−0.0034 to +0.0054; unadjusted one-sided P = 0.35). The test conditions on one tree seed of the comparator. Refitted with four further tree seeds, the comparator’s AUROC varied by 0.0008 for hypotension, 0.0004 for acute kidney injury and 0.0014 for hypoxemia (Supplementary Table S24b). With paired tree seeds, the hybrid’s gain was +0.0010 (−0.0012 to +0.0032) for hypotension and +0.0034 (+0.0006 to +0.0062) for acute kidney injury. These seed analyses were registered as sensitivities and enter no test. The encoder trained from random initialization differed from boosting by −0.0109, −0.0294 and −0.0197 AUROC for hypotension, acute kidney injury and hypoxemia, and no non-inferiority margin was registered. The pretrained encoder was below the scratch encoder for hypotension and acute kidney injury, with intervals below zero, and differed from it by less than 0.0001 for hypoxemia (f − g). Per landmark, the hybrid’s gain over boosting for acute kidney injury was +0.0016 at 12 hours, +0.0030 at 24 hours and +0.0103 at 48 hours (Supplementary Table S24c). These are point differences of the five-seed means, without intervals. Every per-seed gain for acute kidney injury was positive, from +0.0024 to +0.0045 (Supplementary Table S24d).

101

Supplementary Table S24: Family B on the MIMIC-IV test split. Arms refitted within the MIMIC-IV training split: b, fixed CatBoost recipe (tree seed 13 unless stated); f , encoder initialized from the pretrained eICU-CRD twin and fine-tuned on the endpoints; g, the same architecture from random initialization; h, CatBoost on the features of b and the fine-tuned state of f . Five-seed means of f , g and h (seeds 13, 17, 23, 29 and 31) unless stated. Intervals are 95% two-sided percentile intervals from 10,000 whole-person bootstrap replicates (stream seed 20261001). Log-loss: raw probabilities, lower is better. Files are in source_tables/revision_v3/mimic4_read_6f9c4a6_13172_v1/. a AUROC contrasts. The primary test was one-sided for h − b on hypotension and acute kidney injury, with add-one P values and Holm correction over the two endpoints; ∗ rejected. Hypoxemia was a registered secondary endpoint with an unadjusted one-sided P . The last column gives the eICU-CRD estimates of probe P1, which used seeds 23 and 29 and the tuning set of the eICU-CRD training split. Sources: contrasts_b.csv and mechanism_b.json. Contrast

AUROC difference [95% CI]

Hypotension (E2) h − b, primary test h − b, paired tree seeds (sensitivity) g−b f −b f −g h − b, seeds 23 and 29 g − b, seeds 23 and 29 f − b, seeds 23 and 29

One-sided P

Holm P

Log-loss difference [95% CI]

eICU-CRD, probe P1 [95% CI]

+0.0010 [−0.0012, +0.0033] +0.0010 [−0.0012, +0.0032] −0.0109 [−0.0145, −0.0072] −0.0155 [−0.0197, −0.0114] −0.0046 [−0.0067, −0.0025] +0.0007 [−0.0016, +0.0030] −0.0139 [−0.0178, −0.0100] −0.0153 [−0.0196, −0.0112]

0.18 – – – – – – –

0.18 – – – – – – –

−0.0007 [−0.0023, +0.0008] −0.0009 [−0.0024, +0.0006] +0.0084 [+0.0057, +0.0111] +0.0119 [+0.0090, +0.0149] +0.0035 [+0.0018, +0.0052] −0.0005 [−0.0021, +0.0011] +0.0105 [+0.0076, +0.0134] +0.0117 [+0.0087, +0.0147]

+0.0042 [+0.0015, +0.0067] – −0.0044 [−0.0079, −0.0010] – – – – –

Acute kidney injury (E3) h − b, primary test h − b, paired tree seeds (sensitivity) g−b f −b f −g h − b, seeds 23 and 29 g − b, seeds 23 and 29 f − b, seeds 23 and 29

+0.0035∗ [+0.0007, +0.0065] +0.0034 [+0.0006, +0.0062] −0.0294 [−0.0367, −0.0226] −0.0376 [−0.0452, −0.0304] −0.0082 [−0.0116, −0.0047] +0.0034 [+0.0003, +0.0064] −0.0318 [−0.0393, −0.0249] −0.0355 [−0.0430, −0.0284]

0.0088 – – – – – – –

0.018 – – – – – – –

−0.0018 [−0.0032, −0.0006] −0.0020 [−0.0033, −0.0007] +0.0159 [+0.0129, +0.0191] +0.0205 [+0.0172, +0.0240] +0.0046 [+0.0031, +0.0062] −0.0018 [−0.0032, −0.0004] +0.0168 [+0.0137, +0.0201] +0.0193 [+0.0160, +0.0229]

+0.0075 [+0.0035, +0.0116] – −0.0209 [−0.0291, −0.0128] – – – – –

Hypoxemia (E4, secondary) h − b, primary test h − b, paired tree seeds (sensitivity) g−b f −b f −g h − b, seeds 23 and 29 g − b, seeds 23 and 29 f − b, seeds 23 and 29

+0.0009 [−0.0034, +0.0054] +0.0017 [−0.0026, +0.0059] −0.0197 [−0.0275, −0.0118] −0.0198 [−0.0283, −0.0112] 0.0000 [−0.0051, +0.0050] +0.0016 [−0.0030, +0.0066] −0.0225 [−0.0309, −0.0139] −0.0197 [−0.0287, −0.0106]

0.35 – – – – – – –

– – – – – – – –

+0.0001 [−0.0005, +0.0007] −0.0001 [−0.0007, +0.0004] +0.0037 [+0.0025, +0.0049] +0.0036 [+0.0022, +0.0050] −0.0001 [−0.0010, +0.0008] +0.0002 [−0.0004, +0.0008] +0.0040 [+0.0027, +0.0053] +0.0038 [+0.0024, +0.0052]

+0.0013 [−0.0032, +0.0059] – −0.0125 [−0.0202, −0.0054] – – – – –

b AUROC and log-loss of every fitted arm. AUROC with 95% whole-person bootstrap interval, and raw log-loss, pooled over the three landmarks. The comparator b was refitted with tree seeds 17, 23, 29 and 31 as a registered sensitivity, and h was also fitted with the tree seed equal to its encoder seed (paired tree seeds). The means of seeds 23 and 29 follow the two-seed construction of probes P0 and P1 on eICU-CRD. Source: metrics_b.csv. Hypotension (E2)

Acute kidney injury (E3) Log-loss

AUROC [95% CI]

Hypoxemia (E4, secondary)

Arm

AUROC [95% CI]

Scored rows

19,412 rows, 7,278 persons, 3,417 events

23,693 rows, 8,295 persons, 2,407 events

29,248 rows, 8,922 persons, 715 events

b, tree seed 13 (primary comparator) b, tree seed 17 b, tree seed 23 b, tree seed 29 b, tree seed 31 b, mean over the five tree seeds

0.8162 [0.8072, 0.8251] 0.8165 [0.8076, 0.8253] 0.8159 [0.8071, 0.8247] 0.8167 [0.8078, 0.8255] 0.8161 [0.8072, 0.8248] 0.8163 [0.8073, 0.8251]

0.3655 0.3655 0.3657 0.3653 0.3659 0.3656

0.8321 [0.8219, 0.8421] 0.8325 [0.8224, 0.8426] 0.8325 [0.8224, 0.8426] 0.8325 [0.8223, 0.8427] 0.8322 [0.8221, 0.8423] 0.8324 [0.8222, 0.8425]

0.2528 0.2527 0.2528 0.2527 0.2529 0.2528

0.8499 [0.8326, 0.8660] 0.8498 [0.8327, 0.8659] 0.8485 [0.8313, 0.8647] 0.8491 [0.8319, 0.8650] 0.8495 [0.8323, 0.8657] 0.8494 [0.8322, 0.8654]

0.0895 0.0896 0.0898 0.0897 0.0896 0.0896

h, encoder seed 13, tree seed 13 h, encoder seed 17, tree seed 13 h, encoder seed 23, tree seed 13 h, encoder seed 29, tree seed 13 h, encoder seed 31, tree seed 13 h, five-seed mean (primary)

0.8179 [0.8091, 0.8268] 0.8167 [0.8078, 0.8255] 0.8166 [0.8078, 0.8254] 0.8172 [0.8085, 0.8261] 0.8178 [0.8090, 0.8265] 0.8172 [0.8085, 0.8261]

0.3645 0.3650 0.3651 0.3648 0.3643 0.3647

0.8360 [0.8261, 0.8460] 0.8345 [0.8244, 0.8444] 0.8362 [0.8262, 0.8460] 0.8348 [0.8248, 0.8447] 0.8366 [0.8268, 0.8465] 0.8356 [0.8258, 0.8454]

0.2505 0.2514 0.2508 0.2513 0.2510 0.2510

0.8493 [0.8326, 0.8652] 0.8534 [0.8366, 0.8690] 0.8507 [0.8339, 0.8665] 0.8524 [0.8360, 0.8681] 0.8481 [0.8312, 0.8640] 0.8508 [0.8343, 0.8663]

0.0896 0.0891 0.0897 0.0897 0.0897 0.0896

h, encoder and tree seed 17 h, encoder and tree seed 23 h, encoder and tree seed 29 h, encoder and tree seed 31 h, paired tree seeds, mean

0.8169 [0.8080, 0.8257] 0.8163 [0.8075, 0.8252] 0.8174 [0.8086, 0.8263] 0.8181 [0.8093, 0.8269] 0.8173 [0.8086, 0.8262]

0.3650 0.3651 0.3646 0.3641 0.3647

0.8341 [0.8241, 0.8440] 0.8366 [0.8266, 0.8465] 0.8347 [0.8247, 0.8447] 0.8372 [0.8273, 0.8471] 0.8357 [0.8258, 0.8456]

0.2517 0.2504 0.2509 0.2506 0.2508

0.8526 [0.8358, 0.8683] 0.8513 [0.8345, 0.8670] 0.8523 [0.8357, 0.8679] 0.8497 [0.8329, 0.8656] 0.8510 [0.8346, 0.8665]

0.0891 0.0897 0.0896 0.0896 0.0895

f , seed 13 f , seed 17 f , seed 23 f , seed 29 f , seed 31 f , five-seed mean

0.8015 [0.7923, 0.8108] 0.8016 [0.7925, 0.8108] 0.8020 [0.7928, 0.8112] 0.7998 [0.7905, 0.8092] 0.7987 [0.7895, 0.8079] 0.8007 [0.7916, 0.8099]

0.3768 0.3779 0.3768 0.3775 0.3779 0.3774

0.7935 [0.7824, 0.8044] 0.7955 [0.7844, 0.8064] 0.7982 [0.7870, 0.8090] 0.7950 [0.7837, 0.8059] 0.7903 [0.7792, 0.8012] 0.7945 [0.7835, 0.8051]

0.2737 0.2727 0.2715 0.2728 0.2759 0.2733

0.8291 [0.8114, 0.8465] 0.8345 [0.8171, 0.8517] 0.8327 [0.8157, 0.8494] 0.8277 [0.8103, 0.8448] 0.8264 [0.8089, 0.8436] 0.8301 [0.8131, 0.8467]

0.0930 0.0921 0.0926 0.0939 0.0938 0.0931

g, seed 13 g, seed 17 g, seed 23 g, seed 29 g, seed 31 g, five-seed mean

0.8127 [0.8037, 0.8215] 0.8022 [0.7930, 0.8114] 0.8020 [0.7928, 0.8110] 0.8027 [0.7934, 0.8118] 0.8071 [0.7982, 0.8160] 0.8053 [0.7963, 0.8141]

0.3685 0.3757 0.3764 0.3755 0.3732 0.3739

0.8067 [0.7956, 0.8174] 0.7984 [0.7871, 0.8093] 0.8007 [0.7895, 0.8116] 0.7998 [0.7888, 0.8107] 0.8076 [0.7965, 0.8183] 0.8027 [0.7917, 0.8133]

0.2666 0.2711 0.2691 0.2701 0.2665 0.2687

0.8365 [0.8191, 0.8534] 0.8222 [0.8041, 0.8399] 0.8319 [0.8141, 0.8493] 0.8229 [0.8045, 0.8409] 0.8371 [0.8195, 0.8540] 0.8301 [0.8127, 0.8471]

0.0916 0.0959 0.0928 0.0941 0.0916 0.0932

h, mean of seeds 23 and 29 f , mean of seeds 23 and 29 g, mean of seeds 23 and 29

0.8169 [0.8082, 0.8258] 0.8009 [0.7917, 0.8102] 0.8023 [0.7931, 0.8114]

0.3649 0.3771 0.3760

0.8355 [0.8255, 0.8453] 0.7966 [0.7854, 0.8073] 0.8003 [0.7892, 0.8112]

0.2510 0.2721 0.2696

0.8515 [0.8350, 0.8671] 0.8302 [0.8133, 0.8468] 0.8274 [0.8095, 0.8448]

0.0897 0.0932 0.0935

102

Log-loss

AUROC [95% CI]

Log-loss

c AUROC by landmark. Pooled AUROC, AUROC at the landmarks of 12, 24 and 48 hours, and pooled raw log-loss with its 95% wholeperson interval. In the h − b rows, the pooled columns give the registered contrast and its log-loss difference, and the landmark columns give point differences of the five-seed mean of h and b, without intervals (descriptive). Sources: metrics_b.csv and contrasts_b.csv. Arm

Pooled AUROC

12 h

24 h

48 h

Hypotension (E2) b (tree seed 13) b, mean over tree seeds h, five-seed mean h, paired tree seeds f , five-seed mean g, five-seed mean h−b

Pooled log-loss [95% CI]

0.8162 0.8163 0.8172 0.8173 0.8007 0.8053 +0.0010

0.8061 0.8059 0.8061 0.8060 0.7901 0.7951 0.0000

0.8068 0.8078 0.8104 0.8106 0.7899 0.7954 +0.0036

0.8179 0.8160 0.8154 0.8158 0.7977 0.8055 −0.0025

0.3655 [0.3548, 0.3759] 0.3656 [0.3549, 0.3759] 0.3647 [0.3540, 0.3753] 0.3647 [0.3540, 0.3752] 0.3774 [0.3665, 0.3882] 0.3739 [0.3639, 0.3839] −0.0007 [−0.0023, +0.0008]

Acute kidney injury (E3) b (tree seed 13) b, mean over tree seeds h, five-seed mean h, paired tree seeds f , five-seed mean g, five-seed mean h−b

0.8321 0.8324 0.8356 0.8357 0.7945 0.8027 +0.0035

0.8402 0.8406 0.8418 0.8422 0.8024 0.8107 +0.0016

0.8285 0.8286 0.8315 0.8315 0.7925 0.8035 +0.0030

0.7888 0.7888 0.7992 0.7985 0.7348 0.7375 +0.0103

0.2528 [0.2437, 0.2620] 0.2528 [0.2437, 0.2620] 0.2510 [0.2419, 0.2603] 0.2508 [0.2417, 0.2601] 0.2733 [0.2638, 0.2833] 0.2687 [0.2592, 0.2784] −0.0018 [−0.0032, −0.0006]

Hypoxemia (E4, secondary) b (tree seed 13) b, mean over tree seeds h, five-seed mean h, paired tree seeds f , five-seed mean g, five-seed mean h−b

0.8499 0.8494 0.8508 0.8510 0.8301 0.8301 +0.0009

0.8504 0.8507 0.8532 0.8534 0.8346 0.8344 +0.0028

0.8640 0.8631 0.8644 0.8648 0.8455 0.8467 +0.0004

0.8193 0.8175 0.8149 0.8148 0.7842 0.7832 −0.0044

0.0895 [0.0829, 0.0963] 0.0896 [0.0830, 0.0965] 0.0896 [0.0829, 0.0964] 0.0895 [0.0829, 0.0964] 0.0931 [0.0861, 0.1002] 0.0932 [0.0868, 0.0998] +0.0001 [−0.0005, +0.0007]

d Contrasts by encoder seed. The arm of one seed minus b (tree seed 13), with 95% whole-person bootstrap intervals; descriptive. Source: contrasts_b.csv. hs − b [95% CI]

gs − b [95% CI]

fs − b [95% CI]

Hypotension (E2)

13 17 23 29 31

+0.0017 [−0.0009, +0.0044] +0.0005 [−0.0022, +0.0030] +0.0004 [−0.0021, +0.0029] +0.0010 [−0.0014, +0.0036] +0.0016 [−0.0010, +0.0041]

−0.0035 [−0.0072, +0.0002] −0.0140 [−0.0182, −0.0099] −0.0142 [−0.0182, −0.0102] −0.0135 [−0.0176, −0.0094] −0.0091 [−0.0134, −0.0049]

−0.0147 [−0.0192, −0.0103] −0.0146 [−0.0190, −0.0103] −0.0142 [−0.0185, −0.0101] −0.0164 [−0.0209, −0.0121] −0.0175 [−0.0219, −0.0131]

Acute kidney injury (E3)

13 17 23 29 31

+0.0039 [+0.0006, +0.0073] +0.0024 [−0.0007, +0.0056] +0.0041 [+0.0007, +0.0075] +0.0027 [−0.0005, +0.0059] +0.0045 [+0.0015, +0.0076]

−0.0254 [−0.0330, −0.0182] −0.0337 [−0.0414, −0.0262] −0.0314 [−0.0387, −0.0244] −0.0322 [−0.0399, −0.0250] −0.0244 [−0.0318, −0.0174]

−0.0386 [−0.0463, −0.0312] −0.0366 [−0.0446, −0.0290] −0.0339 [−0.0415, −0.0266] −0.0371 [−0.0447, −0.0299] −0.0418 [−0.0497, −0.0340]

Hypoxemia (E4, secondary)

13 17 23 29 31

−0.0005 [−0.0057, +0.0048] +0.0035 [−0.0015, +0.0087] +0.0008 [−0.0050, +0.0068] +0.0025 [−0.0022, +0.0073] −0.0017 [−0.0069, +0.0035]

−0.0133 [−0.0213, −0.0052] −0.0277 [−0.0376, −0.0179] −0.0179 [−0.0264, −0.0092] −0.0270 [−0.0365, −0.0174] −0.0128 [−0.0207, −0.0048]

−0.0207 [−0.0299, −0.0115] −0.0153 [−0.0247, −0.0057] −0.0172 [−0.0266, −0.0079] −0.0221 [−0.0321, −0.0121] −0.0234 [−0.0336, −0.0133]

Endpoint

Seed

103

Disclosed secondary outcomes. The death forecast discriminated hospital death with unit exit within 48 hours (MORT-W) with an AUROC of 0.687 (95% whole-person interval 0.657 to 0.718) and hospital death at any time in the hospitalization (MORT-H) with 0.726 (0.708 to 0.744). The exit forecast discriminated leaving the unit within 6 to 48 hours with AUROCs of 0.765 to 0.808 (Supplementary Table S25a). Every AUROC interval lay above 0.5. Every mean forecast lay below its observed rate, and every whole-person interval of the difference lay below zero. These shortfalls arise in part by construction: each target-specific head counts a death or a unit exit only if it precedes that target’s crossing, whereas the outcomes count every death or exit in their windows, and both death outcomes also count deaths after a live unit exit. The shipped files do not separate the parts. Supplementary Table S25b gives every metric for every seed, with the five-seed means and their Student t intervals. The logistic calibration intercept and slope of MORT-W and MORT-H are recorded as NON_ESTIMABLE_SEPARATION in every seed and as NON_ESTIMABLE_IN_ONE_OR_MORE_SEEDS for the five-seed mean. Why these fits failed has not been examined, and we state the codes without interpretation. Leaving out the arterial-saturation head, whose input channel is nearly empty in MIMIC-IV, changed no fiveseed mean AUROC by more than 0.002 (Supplementary Table S25c). The reliability tables of every seed are in Supplementary Table S25d. Earlier reads of the same outcomes. Hospital death and length of stay had been scored on this split in many earlier reads (protocol section 2.3, reproduced later in this note). As the protocol requires, the committed results at a landmark of 48 hours of the two corrected-clock reads TRB-M4-20 and TRB-M4-21, whose models were trained on MIMIC-IV, are given beside this read (Supplementary Table S25e,f). TRB-M4-20 scored hospital death with nine tabular models among stays with an ICU length of stay of at least 48 hours on a corrected clock. This read scored the same outcome with the frozen eICU-CRD model on the legacy clock, among stays whose terminal clock lies after hour 48 and whose record reaches hour 48 (protocol deviation D29). The size of this population is given above. Protocol deviation D29 also stated that the scorer would count its overlap with the TRB-M4-20 stays, and the shipped aggregates contain no such count. TRB-M4-21 regressed the remaining ICU length of stay and reports mean absolute errors in hours. The unit-exit outcomes of this read are thresholds of the remaining stay, so they cannot be compared with those errors (protocol deviation D30). The earlier results are context, and no comparison between the reads was made or tested. Supplementary Table S25: Disclosed secondary outcomes of the MIMIC-IV read and earlier reads of the same outcomes. Panels a to d: frozen eICU-CRD checkpoints read at hour 48 in 6,951 stays of 5,396 persons; death forecast PD for the two death outcomes and exit forecast PX (H) for leaving the unit within H hours of hour 48. No criterion was registered. Files of panels a to d are in source_tables/revision_v3/mimic4_read_6f9c4a6_13172_v1/. Panels e and f: committed results at a landmark of 48 hours of two earlier disclosed reads of the MIMIC-IV test split by models trained on MIMIC-IV (corrected clock; stays with an ICU length of stay of at least 48 hours). Their populations, clocks and models differ from those of this read, and no comparison is made or tested. a Discrimination and mean forecast. AUROC: five-seed mean with 95% whole-person bootstrap interval (10,000 replicates of the five-seed mean) and 95% Student t interval across seeds (4 degrees of freedom). Mean forecast and AUPRC: five-seed means with t intervals. The difference between the mean forecast and the observed rate has a whole-person interval. Sources: metrics_sec.csv and outcomes.json. Outcome

Events

Observed rate

AUROC

Whole-person 95% interval

t interval across seeds

Mean forecast [t interval]

Mean forecast minus observed rate [95% CI]

AUPRC [t interval]

Hospital death, unit exit within 48 h (MORT-W) Hospital death (MORT-H) Leaving the unit within 6 h (LOS-X6) Leaving the unit within 12 h (LOS-X12) Leaving the unit within 24 h (LOS-X24) Leaving the unit within 48 h (LOS-X48)

324 972 784 1,318 2,237 3,548

0.047 0.140 0.113 0.190 0.322 0.510

0.687 0.726 0.808 0.791 0.767 0.765

[0.657, 0.718] [0.708, 0.744] [0.793, 0.822] [0.779, 0.803] [0.756, 0.778] [0.754, 0.776]

[0.666, 0.708] [0.695, 0.757] [0.800, 0.816] [0.783, 0.799] [0.765, 0.768] [0.764, 0.766]

0.0039 [0.0031, 0.0048] 0.0039 [0.0031, 0.0048] 0.091 [0.077, 0.105] 0.130 [0.114, 0.146] 0.195 [0.174, 0.215] 0.267 [0.242, 0.291]

−0.043 [−0.048, −0.038] −0.136 [−0.145, −0.127] −0.022 [−0.029, −0.015] −0.059 [−0.068, −0.051] −0.127 [−0.137, −0.117] −0.244 [−0.255, −0.233]

0.155 [0.130, 0.179] 0.377 [0.354, 0.400] 0.372 [0.348, 0.395] 0.460 [0.439, 0.480] 0.573 [0.566, 0.580] 0.740 [0.735, 0.745]

104

b Every metric by seed. Per-seed AUROC, AUPRC, log-loss (probabilities clipped at 10−7 ), Brier score, mean forecast, and the intercept and slope of a logistic regression of the outcome on the logit of the forecast (0 and 1 under perfect calibration), with five-seed means and 95% Student t intervals across seeds. a Calibration status recorded as NON_ESTIMABLE_SEPARATION. b Recorded as NON_ESTIMABLE_IN_ ONE_OR_MORE_SEEDS. The codes are stated without interpretation. Source: metrics_sec.csv. Log-loss

Brier

Mean forecast

Intercept

Slope

Hospital death, unit exit within 48 h (MORT-W) Seed 13 0.699 0.172 Seed 17 0.704 0.137 Seed 23 0.666 0.153 Seed 29 0.693 0.176 Seed 31 0.672 0.135 Five-seed mean 0.687 0.155 95% t interval [0.666, 0.708] [0.130, 0.179]

AUROC

AUPRC

0.256 0.271 0.256 0.254 0.260 0.259 [0.251, 0.268]

0.046 0.046 0.045 0.045 0.045 0.046 [0.045, 0.046]

0.0039 0.0027 0.0044 0.0042 0.0043 0.0039 [0.0031, 0.0048]

not estimablea not estimablea not estimablea not estimablea not estimablea not estimableb not estimableb

not estimablea not estimablea not estimablea not estimablea not estimablea not estimableb not estimableb

Hospital death (MORT-H) Seed 13 0.750 Seed 17 0.744 Seed 23 0.692 Seed 29 0.736 Seed 31 0.708 Five-seed mean 0.726 95% t interval [0.695, 0.757]

0.402 0.373 0.358 0.388 0.362 0.377 [0.354, 0.400]

0.756 0.806 0.766 0.756 0.767 0.770 [0.745, 0.796]

0.137 0.138 0.136 0.137 0.137 0.137 [0.136, 0.138]

0.0039 0.0027 0.0044 0.0042 0.0043 0.0039 [0.0031, 0.0048]

not estimablea not estimablea not estimablea not estimablea not estimablea not estimableb not estimableb

not estimablea not estimablea not estimablea not estimablea not estimablea not estimableb not estimableb

Leaving the unit within 6 h (LOS-X6) Seed 13 0.807 0.373 Seed 17 0.802 0.347 Seed 23 0.806 0.361 Seed 29 0.819 0.396 Seed 31 0.807 0.381 Five-seed mean 0.808 0.372 95% t interval [0.800, 0.816] [0.348, 0.395]

0.289 0.296 0.287 0.280 0.295 0.289 [0.281, 0.297]

0.085 0.087 0.085 0.083 0.087 0.086 [0.084, 0.087]

0.089 0.085 0.106 0.098 0.078 0.091 [0.077, 0.105]

0.12 0.07 −0.06 0.17 0.37 0.13 [−0.06, 0.33]

0.89 0.84 0.91 0.99 0.95 0.92 [0.84, 0.99]

Leaving the unit within 12 h (LOS-X12) Seed 13 0.795 0.466 Seed 17 0.786 0.444 Seed 23 0.787 0.443 Seed 29 0.801 0.482 Seed 31 0.787 0.464 Five-seed mean 0.791 0.460 95% t interval [0.783, 0.799] [0.439, 0.480]

0.413 0.424 0.410 0.401 0.429 0.415 [0.401, 0.430]

0.132 0.134 0.131 0.128 0.136 0.132 [0.128, 0.136]

0.128 0.123 0.144 0.142 0.113 0.130 [0.114, 0.146]

0.43 0.33 0.23 0.36 0.53 0.38 [0.23, 0.52]

0.92 0.83 0.88 0.96 0.89 0.90 [0.83, 0.96]

Leaving the unit within 24 h (LOS-X24) Seed 13 0.768 0.572 Seed 17 0.765 0.569 Seed 23 0.766 0.566 Seed 29 0.768 0.576 Seed 31 0.768 0.581 Five-seed mean 0.767 0.573 95% t interval [0.765, 0.768] [0.566, 0.580]

0.581 0.601 0.569 0.567 0.610 0.586 [0.562, 0.609]

0.196 0.199 0.192 0.191 0.204 0.196 [0.190, 0.203]

0.195 0.186 0.211 0.210 0.172 0.195 [0.174, 0.215]

0.61 0.58 0.49 0.52 0.74 0.59 [0.47, 0.71]

0.81 0.74 0.80 0.82 0.80 0.79 [0.75, 0.83]

Leaving the unit within 48 h (LOS-X48) Seed 13 0.766 0.739 Seed 17 0.764 0.739 Seed 23 0.764 0.735 Seed 29 0.766 0.742 Seed 31 0.765 0.745 Five-seed mean 0.765 0.740 95% t interval [0.764, 0.766] [0.735, 0.745]

0.757 0.787 0.734 0.730 0.815 0.765 [0.720, 0.810]

0.260 0.266 0.252 0.251 0.278 0.261 [0.248, 0.275]

0.269 0.258 0.285 0.284 0.238 0.267 [0.242, 0.291]

1.13 1.15 1.04 1.07 1.30 1.14 [1.01, 1.26]

0.78 0.73 0.77 0.80 0.78 0.77 [0.74, 0.80]

c Three-head sensitivity. Both summaries recomputed over the heart-rate, respiratory-rate and SpO2 heads, leaving out the arterialsaturation head because its input channel is nearly empty in MIMIC-IV (registered sensitivity, descriptive). Five-seed means with 95% Student t intervals; the second column repeats the four-head AUROC. b As in panel b. Source: metrics_sec.csv. Outcome Hospital death, unit exit within 48 h (MORT-W) Hospital death (MORT-H) Leaving the unit within 6 h (LOS-X6) Leaving the unit within 12 h (LOS-X12) Leaving the unit within 24 h (LOS-X24) Leaving the unit within 48 h (LOS-X48)

AUROC, four heads 0.687 0.726 0.808 0.791 0.767 0.765

AUROC, three heads

AUPRC

Mean forecast

Intercept

Slope

0.687 [0.666, 0.708] 0.726 [0.695, 0.757] 0.808 [0.800, 0.816] 0.790 [0.782, 0.799] 0.766 [0.764, 0.767] 0.763 [0.762, 0.765]

0.155 [0.130, 0.179] 0.377 [0.354, 0.400] 0.371 [0.347, 0.395] 0.457 [0.436, 0.479] 0.571 [0.564, 0.578] 0.739 [0.734, 0.743]

0.0039 [0.0030, 0.0048] 0.0039 [0.0030, 0.0048] 0.089 [0.075, 0.104] 0.127 [0.110, 0.144] 0.191 [0.170, 0.213] 0.263 [0.238, 0.288]

not estimableb not estimableb 0.13 [−0.06, 0.32] 0.38 [0.24, 0.53] 0.60 [0.47, 0.72] 1.15 [1.02, 1.27]

not estimableb not estimableb 0.90 [0.82, 0.98] 0.88 [0.81, 0.95] 0.78 [0.74, 0.83] 0.76 [0.73, 0.80]

105

d Reliability tables. Ten equal-width bins of the four-head forecast in each seed; bins empty in every seed are omitted. Each cell gives the number of stays, the mean forecast and the observed rate; –, empty bin. The reliability tables of the three-head sensitivity are in the source file. Source: metrics_sec.csv. Seed: stays; mean forecast; observed rate Outcome

Forecast bin

13

17

23

29

31

Hospital death, unit exit within 48 h (MORT-W)

0.0 to 0.1 0.1 to 0.2

6,949; 0.004; 0.046 2; 0.110; 0.500

6,951; 0.003; 0.047 –

6,948; 0.004; 0.046 3; 0.123; 0.333

6,944; 0.004; 0.046 7; 0.136; 0.857

6,941; 0.004; 0.046 10; 0.117; 0.500

Hospital death (MORT-H)

0.0 to 0.1 0.1 to 0.2

6,949; 0.004; 0.140 2; 0.110; 0.500

6,951; 0.003; 0.140 –

6,948; 0.004; 0.140 3; 0.123; 0.333

6,944; 0.004; 0.139 7; 0.136; 0.857

6,941; 0.004; 0.139 10; 0.117; 0.800

Leaving the unit within 6 h (LOS-X6)

0.0 to 0.1 0.1 to 0.2 0.2 to 0.3 0.3 to 0.4 0.4 to 0.5 0.5 to 0.6 0.6 to 0.7 0.7 to 0.8 0.8 to 0.9 0.9 to 1.0

4,722; 0.030; 0.048 1,422; 0.144; 0.176 459; 0.240; 0.283 150; 0.351; 0.413 97; 0.443; 0.495 51; 0.546; 0.608 33; 0.655; 0.727 13; 0.734; 0.692 3; 0.835; 1.000 1; 0.901; 0.000

4,783; 0.029; 0.049 1,327; 0.143; 0.182 506; 0.241; 0.310 190; 0.343; 0.405 79; 0.444; 0.430 36; 0.543; 0.639 19; 0.631; 0.684 9; 0.743; 0.667 2; 0.871; 0.500 –

4,178; 0.029; 0.039 1,536; 0.146; 0.135 715; 0.243; 0.263 301; 0.344; 0.362 119; 0.447; 0.479 63; 0.540; 0.571 32; 0.638; 0.594 7; 0.734; 0.714 – –

4,414; 0.031; 0.040 1,492; 0.144; 0.145 621; 0.239; 0.275 219; 0.342; 0.466 98; 0.445; 0.490 58; 0.547; 0.586 29; 0.640; 0.690 13; 0.747; 0.692 7; 0.842; 0.857 –

4,902; 0.029; 0.051 1,340; 0.143; 0.178 468; 0.241; 0.344 164; 0.342; 0.494 60; 0.441; 0.683 14; 0.544; 0.857 3; 0.624; 0.333 – – –

Leaving the unit within 12 h (LOS-X12)

0.0 to 0.1 0.1 to 0.2 0.2 to 0.3 0.3 to 0.4 0.4 to 0.5 0.5 to 0.6 0.6 to 0.7 0.7 to 0.8 0.8 to 0.9 0.9 to 1.0

3,585; 0.031; 0.066 1,673; 0.148; 0.203 1,022; 0.245; 0.374 387; 0.340; 0.470 149; 0.446; 0.597 71; 0.546; 0.535 40; 0.657; 0.775 19; 0.741; 0.842 4; 0.838; 1.000 1; 0.909; 0.000

3,779; 0.029; 0.071 1,480; 0.148; 0.224 963; 0.245; 0.363 468; 0.342; 0.466 160; 0.442; 0.544 61; 0.538; 0.541 27; 0.639; 0.741 9; 0.743; 0.778 3; 0.828; 1.000 1; 0.901; 1.000

3,344; 0.030; 0.059 1,427; 0.149; 0.195 1,208; 0.247; 0.305 610; 0.343; 0.446 224; 0.439; 0.513 91; 0.546; 0.626 37; 0.646; 0.649 10; 0.743; 0.800 – –

3,304; 0.030; 0.054 1,602; 0.148; 0.185 1,122; 0.246; 0.316 574; 0.344; 0.469 199; 0.440; 0.598 83; 0.542; 0.627 43; 0.644; 0.721 15; 0.746; 0.867 9; 0.845; 0.778 –

3,904; 0.031; 0.074 1,575; 0.148; 0.244 897; 0.243; 0.363 413; 0.339; 0.487 124; 0.440; 0.694 33; 0.541; 0.848 5; 0.647; 0.600 – – –

Leaving the unit within 24 h (LOS-X24)

0.0 to 0.1 0.1 to 0.2 0.2 to 0.3 0.3 to 0.4 0.4 to 0.5 0.5 to 0.6 0.6 to 0.7 0.7 to 0.8 0.8 to 0.9 0.9 to 1.0

2,735; 0.034; 0.105 1,058; 0.149; 0.303 1,136; 0.253; 0.437 1,180; 0.350; 0.508 610; 0.437; 0.602 143; 0.540; 0.650 52; 0.645; 0.827 30; 0.741; 0.800 6; 0.839; 1.000 1; 0.916; 0.000

2,980; 0.032; 0.115 1,045; 0.149; 0.353 1,001; 0.250; 0.429 1,038; 0.347; 0.512 601; 0.441; 0.612 204; 0.537; 0.637 57; 0.643; 0.772 19; 0.742; 0.842 4; 0.826; 1.000 2; 0.920; 1.000

2,620; 0.034; 0.095 991; 0.150; 0.301 1,036; 0.250; 0.412 1,071; 0.350; 0.489 831; 0.445; 0.563 306; 0.537; 0.667 75; 0.637; 0.667 20; 0.742; 0.850 1; 0.843; 1.000 –

2,609; 0.034; 0.095 968; 0.149; 0.299 1,016; 0.253; 0.415 1,286; 0.349; 0.474 765; 0.442; 0.579 202; 0.537; 0.698 69; 0.640; 0.783 23; 0.739; 0.826 13; 0.848; 0.769 –

3,097; 0.035; 0.121 1,138; 0.149; 0.363 1,072; 0.251; 0.450 1,002; 0.349; 0.538 498; 0.439; 0.631 117; 0.542; 0.752 23; 0.633; 0.913 4; 0.721; 1.000 – –

Leaving the unit within 48 h (LOS-X48)

0.0 to 0.1 0.1 to 0.2 0.2 to 0.3 0.3 to 0.4 0.4 to 0.5 0.5 to 0.6 0.6 to 0.7 0.7 to 0.8 0.8 to 0.9 0.9 to 1.0

2,220; 0.044; 0.207 882; 0.148; 0.452 860; 0.249; 0.594 875; 0.351; 0.673 861; 0.449; 0.711 800; 0.548; 0.759 378; 0.635; 0.804 62; 0.733; 0.871 11; 0.832; 1.000 2; 0.923; 0.500

2,412; 0.041; 0.226 904; 0.148; 0.482 816; 0.248; 0.604 814; 0.351; 0.671 805; 0.449; 0.727 724; 0.547; 0.772 371; 0.640; 0.784 94; 0.735; 0.894 9; 0.831; 0.778 2; 0.941; 1.000

2,084; 0.045; 0.197 883; 0.149; 0.443 853; 0.250; 0.567 831; 0.349; 0.649 849; 0.449; 0.704 782; 0.550; 0.752 523; 0.642; 0.788 140; 0.729; 0.864 6; 0.820; 0.667 –

2,074; 0.046; 0.184 874; 0.148; 0.475 788; 0.249; 0.553 858; 0.351; 0.647 970; 0.450; 0.693 888; 0.544; 0.756 408; 0.638; 0.828 76; 0.733; 0.868 12; 0.840; 1.000 3; 0.906; 0.667

2,550; 0.045; 0.233 1,012; 0.148; 0.529 866; 0.250; 0.603 788; 0.349; 0.708 814; 0.449; 0.741 648; 0.545; 0.762 237; 0.638; 0.873 35; 0.738; 0.943 1; 0.809; 1.000 –

e Earlier read TRB-M4-20: hospital death. Nine tabular models; raw probabilities. Calibration-in-the-large: mean prediction minus outcome rate. The population, clock and models differ from those of this read (protocol deviation D29). Source: source_tables/ revision_v3/g3/g3_mortality_heldout_scoreboard_v1.csv, rows mimic4, context 48, raw surface (report reports/revision_v3/ G3_MORTALITY_HELDOUT_FRONTIER_V1.md). Model

AUROC

AUPRC

Log-loss

Brier

Intercept

Slope

Calibrationin-the-large

7,016 test stays of 5,446 persons with an ICU length of stay of at least 48 h (corrected clock); 1,006 hospital deaths (rate 0.143) lgbm_leaf31_v1 lgbm_leaf15_v1 xgb_depth4_v1 xgb_depth6_v1 catboost_depth6_v1 catboost_depth8_v1 ridge_logit_stack_all6_train_oof_ v1 nested_shrinkage_train_only_v1 tabm_0p0p3_registered_v1

0.882 0.883 0.884 0.884 0.881 0.881 0.885

0.603 0.603 0.605 0.609 0.604 0.605 0.612

0.273 0.272 0.271 0.271 0.274 0.274 0.270

0.083 0.083 0.083 0.083 0.083 0.083 0.082

0.088 0.094 0.051 0.075 0.164 0.172 −0.034

1.038 1.065 1.032 1.020 1.134 1.127 0.995

−0.0034 −0.0012 −0.0010 −0.0041 +0.0003 −0.0012 +0.0023

0.884 0.876

0.610 0.583

0.270 0.279

0.082 0.085

0.006 −0.048

1.012 0.946

+0.0006 −0.0012

106

f Earlier read TRB-M4-21: remaining ICU length of stay. Mean absolute error, in hours, of the remaining ICU length of stay predicted by 13 models. This read scored binary unit-exit outcomes instead, which are not comparable with these errors (protocol deviation D30). Source: source_tables/revision_v3/g3/g3_los_heldout_frontier_v1/scoreboard.csv, rows mimic4, context 48 (report reports/ revision_v3/G3_LOS_HELDOUT_FRONTIER_V1.md). Model

MAE (h)

Features

7,016 test stays of 5,446 persons with an ICU length of stay of at least 48 h (corrected clock) Models fitted on 32,364 training stays of 24,998 persons frozen_component_log1p_lgbm_v1 frozen_component_raw_lgbm_v1 frozen_component_sqrt1p_lgbm_v1 frozen_component_xgboost_lognormal_aft_v1 frozen_oof_simplex_cell_median_calibrated_v1 frozen_oof_simplex_stack_v1 registered_catboost_mae_seed20260826_v1 registered_constant_train_median_v1 registered_equal_weight_raw_cat_xgb_log_v1 registered_lgbm_recent_log1p_l1_seed20260826_v1 registered_lgbm_recent_raw_l1_seed13_v1 registered_lgbm_static_l1_seed13_v1 registered_xgboost_absoluteerror_seed20260826_v1

107

67.8 68.2 68.1 68.1 67.7 67.7 69.6 82.4 68.4 67.9 68.2 80.8 68.6

326 326 326 326 326 326 326 0 326 326 326 4 326

Power and precision, computed before the read The protocol computed the power of each registered outcome before the read (section 5); the scripts and their outputs are committed in reports/revision_v3/mimic4_read_power_v1/. Standard errors on eICUCRD were taken from the widths of the registered intervals. They were rescaled by the event and non-event counts projected for the MIMIC-IV test split and, for family A, by an approximate design effect for repeat stays of one person. True effects were the eICU-CRD estimates multiplied by a retained fraction. The simulation drew 200,000 runs with standard errors 1 or 1.5 times the projection (family B also 1.25 times) and applied the registered Holm procedures and outcome rules. Supplementary Table S26a gives the inputs for family A and Supplementary Table S26b its results, including a heterogeneous case in which heart and respiratory rate keep their eICU-CRD gains and the three SpO2 settings share a common true gain. Supplementary Table S26c,d gives the inputs and results for family B. The disclosed secondary outcomes had no criterion and therefore no power analysis; protocol section 5.5 stated their expected precision (Supplementary Table S26e). The protocol also recorded the author’s forecast of each outcome before the read (section 6). Supplementary Table S26: Power and precision of the registered MIMIC-IV read, computed before the read (protocol section 5). Standard errors (SE) on eICU-CRD were derived from the widths of the registered intervals and projected to the MIMIC-IV test split. Panels a, b, d and e are converted from the protocol; panel c is read from the committed power inputs. a Family A inputs (protocol section 5.2). eICU-CRD estimates with their SE, and the SE projected for the MIMIC-IV test split. Setting

eICU S2 − S1 (SE)

SpO2 (12, 6) SpO2 (18, 12) SpO2 (30, 24) Heart rate (12, 6) Heart rate (18, 12) Heart rate (30, 24) Respiratory rate (12, 6) Respiratory rate (18, 12) Respiratory rate (30, 24)

+0.078 (0.0095) +0.101 (0.0129) +0.153 (0.0243) +0.116 (0.0081) +0.107 (0.0083) +0.121 (0.0118) +0.102 (0.0078) +0.083 (0.0078) +0.108 (0.0110)

Projected MIMIC-IV SE 0.0097 0.0132 0.0262 0.0110 0.0123 0.0183 0.0109 0.0137 0.0277

eICU S2 − LOCF (SE)

Projected MIMIC-IV SE

+0.062 (0.0093) +0.060 (0.0131) +0.085 (0.0208) +0.010 (0.0034) +0.027 (0.0047) +0.042 (0.0089) +0.048 (0.0061) +0.046 (0.0075) +0.049 (0.0127)

0.0094 0.0133 0.0224 0.0046 0.0069 0.0139 0.0085 0.0133 0.0319

b Family A results (protocol section 5.3). Each cell gives the probabilities of the four outcome classes in the order: criterion met in all 9 settings / met in 5 to 8 settings with none significantly worse before correction / not met / not met with at least one setting worse after correction. Columns: multiple of the projected SE and correlation ρ between settings. Rows of the first two blocks: fraction of the eICU-CRD gain retained in every setting. Rows of the last block (heterogeneous case): the common true gain of the three SpO2 settings, with heart and respiratory rate at their full eICU-CRD gains; after the semicolon, the probability of a failed outcome with at least 6 surviving settings. 200,000 simulated runs. SE × 1.5, ρ = 0

SE × 1.5, ρ = 0.5

Synchronized versus hour-6 forecast (primary family) 1 0.97 / 0.03 / 0.00 / 0.00 0.97 / 0.03 / 0.00 / 0.00 0.75 0.82 / 0.18 / 0.00 / 0.00 0.83 / 0.17 / 0.00 / 0.00 0.5 0.28 / 0.72 / 0.01 / 0.00 0.41 / 0.52 / 0.07 / 0.00 0.25 0.00 / 0.04 / 0.96 / 0.00 0.04 / 0.14 / 0.83 / 0.00 0 0.00 / 0.00 / 0.975 / 0.025 0.00 / 0.00 / 0.980 / 0.020

0.69 / 0.31 / 0.00 / 0.00 0.27 / 0.72 / 0.01 / 0.00 0.01 / 0.40 / 0.59 / 0.00 0.00 / 0.00 / 1.00 / 0.00 0.00 / 0.00 / 0.975 / 0.025

0.72 / 0.28 / 0.00 / 0.00 0.41 / 0.52 / 0.07 / 0.00 0.11 / 0.34 / 0.55 / 0.00 0.01 / 0.03 / 0.96 / 0.00 0.00 / 0.00 / 0.980 / 0.020

Synchronized forecast versus last measured value (registered secondary) 1 0.13 / 0.86 / 0.01 / 0.00 0.25 / 0.68 / 0.07 / 0.00 0.75 0.01 / 0.69 / 0.30 / 0.00 0.10 / 0.51 / 0.38 / 0.00 0.5 0.00 / 0.06 / 0.94 / 0.00 0.03 / 0.16 / 0.81 / 0.00 0 0.00 / 0.00 / 0.98 / 0.02 0.00 / 0.00 / 0.98 / 0.02

0.00 / 0.45 / 0.54 / 0.00 0.00 / 0.06 / 0.94 / 0.00 0.00 / 0.00 / 1.00 / 0.00 0.00 / 0.00 / 0.98 / 0.02

0.07 / 0.40 / 0.53 / 0.00 0.03 / 0.16 / 0.81 / 0.00 0.01 / 0.04 / 0.95 / 0.00 0.00 / 0.00 / 0.98 / 0.02

Retained

SE × 1, ρ = 0

SE × 1, ρ = 0.5

Heterogeneous case of the primary family: heart and respiratory rate at their full eICU-CRD gains, the three SpO2 settings at a common true gain SpO2 true gain SE × 1, ρ = 0 SE × 1, ρ = 0.5 SE × 1.5, ρ = 0 SE × 1.5, ρ = 0.5 +0.02 0 −0.02

0.01 / 0.98 / 0.00 / 0.00; 0.00 0.00 / 0.93 / 0.05 / 0.02; 0.07 0.00 / 0.27 / 0.22 / 0.51; 0.68

0.06 / 0.93 / 0.00 / 0.00; 0.00 0.00 / 0.94 / 0.04 / 0.02; 0.04 0.00 / 0.37 / 0.19 / 0.44; 0.58

0.00 / 0.95 / 0.05 / 0.00; 0.01 0.00 / 0.88 / 0.10 / 0.02; 0.04 0.00 / 0.52 / 0.25 / 0.23; 0.23

0.02 / 0.89 / 0.08 / 0.00; 0.00 0.00 / 0.88 / 0.10 / 0.02; 0.01 0.00 / 0.62 / 0.20 / 0.18; 0.11

c Family B inputs (protocol section 5.4). eICU-CRD events in the tuning rows of probe P1, events projected for the MIMIC-IV test split, and the eICU-CRD estimates of h − b and g − b with their SE and the projected SE. Source: reports/revision_v3/mimic4_read_power_ v1/family_b_inputs.csv. Endpoint Hypotension (E2) Acute kidney injury (E3) Hypoxemia (E4, secondary)

eICU-CRD events

Projected events

3,136 1,762 587

3,213 2,392 674

eICU-CRD h − b (SE)

Projected SE

eICU-CRD g − b (SE)

Projected SE

+0.0042 (0.00132) +0.0075 (0.00206) +0.0013 (0.00233)

0.00134 0.00181 0.00218

−0.0044 (0.00176) −0.0209 (0.00417) −0.0125 (0.00379)

0.00179 0.00365 0.00355

108

d Family B results (protocol section 5.4). Probabilities of rejecting hypotension / rejecting acute kidney injury / rejecting both (B_BOTH) / rejecting neither (B_NONE), with correlation 0.3 between the endpoints, by the fraction of the eICU-CRD gain retained and the multiple of the projected SE. Retained (true E2, E3 gains)

SE × 1

SE × 1.25

SE × 1.5

1 (+0.0042, +0.0075) 0.75 (+0.0031, +0.0056) 0.5 (+0.0021, +0.0037) 0.25 (+0.0010, +0.0019) 0

0.93 / 0.99 / 0.93 / 0.01 0.75 / 0.90 / 0.71 / 0.07 0.42 / 0.58 / 0.34 / 0.34 0.14 / 0.19 / 0.07 / 0.74 0.03 / 0.03 / 0.01 / 0.95

0.80 / 0.93 / 0.77 / 0.04 0.56 / 0.73 / 0.49 / 0.20 0.29 / 0.41 / 0.21 / 0.51 0.10 / 0.14 / 0.05 / 0.81 0.03 / 0.03 / 0.01 / 0.95

0.64 / 0.82 / 0.60 / 0.13 0.42 / 0.58 / 0.34 / 0.34 0.22 / 0.31 / 0.14 / 0.62 0.09 / 0.11 / 0.04 / 0.84 0.03 / 0.03 / 0.01 / 0.95

e Expected precision of the disclosed secondary outcomes (protocol section 5.5). Hanley-McNeil standard error of an AUROC and binomial standard error of a rate, both multiplied by a design effect for repeat stays of one person, computed before the read from the committed counts of TRB-M4-20; 95% half-width, 1.96 standard errors. Setting

Assumed events

95% half-width of the AUROC

95% half-width of the observed rate

MORT-H MORT-W LOS-X_H

1,006 100 / 200 / 400 / 700 (unknown) event fraction 0.1 / 0.3 / 0.5 / 0.7

0.019 at AUROC 0.80; 0.022 at 0.70; 0.016 at 0.88 0.060 / 0.042 / 0.030 / 0.023 at AUROC 0.80 0.024 / 0.015 / 0.013 / 0.013 at AUROC 0.75

0.009 0.003 / 0.004 / 0.006 / 0.008 0.008 / 0.012 / 0.013 / 0.012

109

Earlier reads of the MIMIC-IV test split (protocol section 2.3) Protocol section 2.3 is reproduced here as written before the read. Its list (Supplementary Table S27) is the exact set of 34 identifiers that the release authorization of stage R bound. Section numbers in the reproduced text refer to the protocol. Registry line numbers in the reproduced text (written :66, :142-158 and similar) refer to reports/revision_v3/TEST_READ_REGISTRY.md at commit P (3ad11ead). Registry commit B (3c58d87d) later added two rows above the disclosure section of that file. From commit B on, every cited line from :66 onward is therefore two higher (TRB-M4-01 is at line 68 and TRB-M4-20 at line 87), and :14 and :24 are unchanged. The canonical list below is bound by the TEST authorization (section 7.2) as the exact sorted set of its IDs. “Scientific reads” are the IDs beginning REG- or TRB-M4- (27); the other 7 record reconciled non-reads, identity reads, a committed aggregate and mixed-split preprocessing. A record written :66 is line 66 of reports/ revision_v3/TEST_READ_REGISTRY.md, whose row gives the full access, selection and verification detail. Supplementary Table S27: Earlier reads of the MIMIC-IV test split (protocol section 2.3). The canonical disclosure list as written before the read. Record: line of reports/revision_v3/TEST_READ_REGISTRY.md at commit P (written :66; two higher from registry commit B on) or another record. Converted from the protocol. ID

Record

Jobs

Date

What was read on the MIMIC-IV test split

REG-5227

TEST_READ_REGISTRY.md: 14 TEST_READ_REGISTRY.md: 24

5227 (re-export of the job-2754 aggregate) 5896-5898

2026-08-09

TRB-M4-01

:66

not recorded

TRB-M4-02

:67

2674

TRB-M4-03 TRB-M4-04 TRB-M4-05

:68 :69 :70

2694 2731 2711, 2752, 2883, 2754

TRB-M4-06

:71

3018_[108-116]

2026-06 2026-06-10 2026-06-07 to 06-14 2026-06-15

previously opened Track B aggregate cells (mortality, LOS) mortality at 24 h of MIMIC-IV fine-tunes of the same joint lineage (parent seed 29, selection 744fe1a1); C01 preparation also read TEST features for eligibility mortality and LOS at 6/12/24/48 h; legacy onsets (AKI, hypotension, hypoxemia, lactate, vasopressor start) legacy events (SBP < 90, SpO2 < 90, HR > 130); legacy onsets; mortality; LOS as TRB-M4-02 (metrics at least) legacy events and onsets; mortality 6 and 24 h; LOS mortality and LOS; sensitivity onset cells

TRB-M4-07

:72

3321-3326

2026-06

TRB-M4-08 TRB-M4-09

:73 :74

3346-3351 3468, 3469

2026-06 2026-07-03

TRB-M4-10 TRB-M4-11 TRB-M4-12 TRB-M4-13

:75 :76 :77 :78

3499, 3500, 3574, 3595 3501-3504 3575, 3601 3534-3539

2026-07 2026-07 2026-07 2026-07

TRB-M4-14

:79

3541, 3542

2026-07

TRB-M4-15 TRB-M4-16

:80 :81

3546 3586-3591

2026-07-10 2026-07-10

TRB-M4-17 TRB-M4-18 TRB-M4-19

:82 :83 :84

TRB-M4-20

:85

3621, 3622 3647-3650, 3809, 3899 4280-4282, 4287-4291, 4300 7422-7445

2026-07-10 2026-07-11 on 2026-07-21 to 07-23 2026-08-26

TRB-M4-21

:86

7451, 7452

2026-08-26

TRB-M4-22

:87

10733 (MIMIC-IV), 10731 (MIMIC-III)

2026-09-19

TRB-M4-23

:88

10734, 10732

TRB-M4-24

:89

not recorded

TRB-M4-25

:90

not recorded

TRB-NR-01

:91

TRB-NR-02 TRB-NR-03 ID-12997

:92 :93 run_registry.csv:193

10711, 10712, 10719, 10720, 10725, 10726 10065-10067 10535 12997

2026-09-19 to 09-20 2026-05-23 to 05-25 2026-05-22 to 05-23 2026-09-19

ID-13062

run_registry row V3MIMIC4-TRAIN-CENSUS32de53f150ae-13062

13062

2026-09-30

AGG-PLAN

ROUTE_B_PREDICTION_ PLAN_20260927_V1.md: 106, :463

none

2026-09-27

REG-C01

2026-08-10

2026-05-23 to 05-27 2026-06-05

2026-09-02 2026-09-12 2026-09-28

110

raw future values of the regression targets, including heart rate and respiratory rate, at 1/6/12/24/48 h zero-shot transfer of a frozen eICU twin (A0_dyn, arm P0) and its online refinement: legacy events, onsets, mortality, LOS legacy events, onsets, mortality, LOS mortality, LOS, strict incident onsets (hypotension, hypoxemia, lactate, AKI, sepsis proxy) at 6/12/24/48 h the same 28-task grid, eICU-trained baselines legacy events, onsets, mortality, LOS the 28-task grid, target-trained baselines zero-shot transfer of a frozen eICU twin (A0_dyn_thr, arm P0) and its online refinement: legacy events, onsets, mortality, LOS a frozen eICU twin transfer (A0_dyn, arm P0) and its refinement by stratum the frozen eICU twin transfer of TRB-M4-13 by stratum zero-shot transfer of a frozen eICU twin (A0_dyn_thr, arm P0) with a threshold-calibration rescue: legacy events, onsets, mortality, LOS; closed negative mortality, LOS, lactate and sepsis onset at 24 h the same four tasks the same four tasks hospital mortality among stays with LOS ≥ C, corrected clock remaining ICU LOS; counted 14,209 finite-LOS stays of 9,804 persons and 2 non-finite zero-shot transfer of a frozen eICU twin (legacy A0_dyn): legacy events, onsets, mortality, LOS; null after Holm crossed fine-tune cell, legacy readout online personalization D1/D3: legacy events, mortality, onsets semi-synthetic MIMIC-IV root generated from TEST covariates nothing (failed at launch) nothing (fit role only) nothing on TEST (V_select only) identity columns (patient, uniquepid, split, source) of every label row, TEST included, to select eICU TRAIN the unchanged normalization verifier read patient, split and source of every label row, and the loader read the time-series index of every row (census receipt, seal. identity_projection_read_by_unchanged_ governed_code) committed TEST aggregate: 11,279 stays at risk at L = 24 and about 750 to 1,000 events per primary under legacy definitions

Supplementary Table S27, continued ID

Record

Jobs

Date

What was read on the MIMIC-IV test split

PREPROC

MIMIC_IV_READ_SCOPING_ 20260930_V1.md:86

none

2026-05

the legacy root’s clip and fill quantiles, and BlendedICU’s height and weight quantiles, were computed over all splits, TEST covariates included (label-free)

Overlap with this read (TEST_READ_REGISTRY.md:142-158): • Family A. The z = 0 crossings of heart rate, respiratory rate and SpO2 have never been scored on MIMIC-IV TEST. Overlapping surfaces were scored: raw future heart and respiratory rate (TRB-M4-06), the HR > 130 and SpO2 < 90 threshold events (the events family of TRB-M4-02 onward), and legacy hypoxemia. Frozen eICU twins of earlier versions (A0_dyn, A0_dyn_thr) were transferred zero-shot to this split in TRB-M4-07, -13, -16 and -22 (with stratified reads in -14 and -15); -16 closed negative and -22 was null after Holm, and the analysts knew these results when this zero-shot design was chosen (MIMIC_IV_READ_SCOPING_20260930_V1.md:83, icare_dt/RESULTS.md). C01 fine-tuned the same joint lineage on MIMIC-IV and read TEST mortality. • Family B. E2 (MAP below 65 mmHg in two consecutive measured hours) is word for word the legacy hypotension rule scored from TRB-M4-01 on, apart from the at-risk rule, the clock and the readout (TEST_ READ_REGISTRY.md:154). Legacy KDIGO AKI (creatinine and urine output, first-6-hour baseline), legacy hypoxemia (one hour, SpO2 or SaO2 below 90 or P/F below 300) and the SpO2 < 90 event were scored; E3 and E4 as P0 defines them were not. • Secondaries (3.11). Hospital mortality (the harmonized mortality label, MIMIC-IV’s hospital_expire_ flag copied to every ICU stay of the admission) was scored on this split at landmarks 6/12/24/48 h in the legacy reads (TRB-M4-01 to -05, -07 to -19, -22 to -24; REG-5227 re-exported a job-2754 aggregate of mortality and LOS cells), among stays with LOS ≥ C at C = 6/12/24/48 h on the corrected clock (TRB-M4-20, scoring jobs 7436-7443, merge 7444, audit 7445) and at 24 h by C01 (REG-C01, jobs 5896-5898). Length of stay was scored at landmarks in the legacy reads and as remaining ICU LOS, LOS − C among stays with LOS ≥ C (TRB-M4-21, jobs 7451/7452) (TEST_READ_REGISTRY.md:144-145, :85-86). From June 2026 every ICare-DT mortality row scored a model death CIF (ICU death with ICU discharge competing) against this hospital-death label (TEST_READ_REGISTRY.md:144, :153), including the zero-shot transfers of earlier frozen eICU twins (TRB-M4-07, -13, -16, -22). No listed read applied the five checkpoints of this read to MIMIC-IV TEST without refitting; REG-C01 read 24-hour mortality of MIMIC-IV fine-tunes of seed 29’s lineage. No binary unit-exit-within-H label was scored before, but remaining ICU LOS at C = 48 h was (TRB-M4-21). • Known TEST facts: 14,211 stays; 9,804 persons over the 14,209 finite-LOS stays; 2 stays with nonfinite LOS (they will carry an unresolved terminal here and be excluded from every label); legacy at-risk and event counts (AGG-PLAN); at C = 48 h, 7,016 stays of 5,446 persons with LOS ≥ 48 h and 1,006 hospital deaths (rate 0.143), and AUROCs of 0.876 to 0.885 for hospital death from the nine tabular models of TRB-M4-20, trained on MIMIC-IV TRAIN (corrected clock; committed scoreboard source_ tables/revision_v3/g3/g3_mortality_heldout_scoreboard_v1.csv, rows mimic4, context 48; G3_ LOS_HELDOUT_FRONTIER_V1.md:53 gives the same 7,016 / 5,446 risk set). Beyond ID-12997 and ID-13062, every governed dataset construction in the project reads the patient, split and source columns of every label row through the unchanged normalization verifier; these identity reads carry no covariate or outcome and are not listed one by one. The analysts have seen the results of every read above, every eICU result the paper reports and the MIMIC-IV TRAIN census (decision 8, ROUTE_B_DECISION_20260926_V1.md:103-106). Route B was shaped by TEST reads (TEST_READ_REGISTRY.md:138). sacct on 2026-09-30 shows no job after the census 13062, so no MIMIC-IV TEST read happened after the disclosure back-fill of 2026-09-27.

Deviations Protocol section 8 (deviations and disclosures) is reproduced in full below, as written before the read. The protocol gives it as a table of 32 rows, D1 to D32. Each item below gives the three cells of one row: the earlier decision, plan or analysis (From), what the read does (This read) and the consequence stated in the protocol (Effect). In it, C1 to C12 are the design choices of protocol section 0.3, items refer to the scholar’s decision record (reports/revision_v3/ROUTE_B_DECISION_20260926_V1.md), and other numbers refer to protocol sections.

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D1 From: item 9: corrected clock for the MIMIC-IV TEST build. This read: legacy root and clock (decision 18 supersedes for this read). Effect: native to the frozen encoder and to P1; the legacy clock is not discharge-bounded (D15). D2 From: item 9: binarized dose channels. This read: continuous; decision 18 waives for frozen weights, C7 extends to family B. Effect: recipe as run. D3 From: item 9: MIMIC-IV TRAIN normalization. This read: adopted for both families. Effect: z = 0 moves to the MIMIC-IV TRAIN mean (HR 85.53 to 86.78, RR 20.08 to 20.26, SpO2 96.52 to 96.53); eICU-mean sensitivity. D4 From: item 9: five seeds for main arms and baseline finals. This read: A: five; B: five for f, g, h; b one fit (C5, C6). Effect: tree-seed spread descriptive. D5 From: item 8: composite primaries as competing-risk CIFs, mortality and LOS as disclosed secondaries. This read: decision 18 names the P0 variants E2 and E3 as family B’s primaries, and C11 supersedes item 8’s composite CIF primaries; item 8’s mortality and LOS secondaries are kept (scholar, decision item 19) and are scored from the frozen family-A model at hour 48 (3.11), not from a composite-endpoint model. Effect: the read tests the paper’s claims, not the plan’s model; mortality and LOS are reported as descriptive secondaries with no criterion. D6 From: item 3: C01 mortality-24 excluded. This read: C01’s 24-hour mortality is not scored; the mortality secondaries use the hour-48 landmark only (3.11); E1 excluded. Effect: none. D7 From: item 7 and plan F4. This read: IC and SH not run (C1). Effect: the isolation statement and personalization stay eICU VAL evidence. D8 From: plan WP11 families F1 to F3; master plan wave 3. This read: superseded (the WP9 model was never built). Effect: none. D9 From: the eICU v2 inference (2,000 stay-level draws, p without add-one, Holm over 12). This read: whole-person 10,000 draws, add-one, Holm over 9; the v2 construction as sensitivity (C3). Effect: counts comparable, p finite. D10 From: P1 inference (2,000 draws, gate). This read: 10,000 draws, one-sided test with Holm over two (C8). Effect: a test the eICU numbers never had. D11 From: P1 recipe. This read: as run, except: E1 never labeled (the loss pools three endpoints; the E1 logit, index 0, gets no gradient); P1’s early-stopping evaluator refuses an endpoint without labelled inner cells, so a declared copy requires them for E2 to E4 only (2.1); fit set = all MIMIC-IV TRAIN less rescued stays (no TRAIN-tune split); five seeds. Effect: the pooled loss and early-stopping weights change; the copy equals P1’s function when all four endpoints are labelled. D12 From: v2 calibration on eICU VAL. This read: none (C2). Effect: no calibrated metric. D13 From: v2 built labels in the prediction pass. This read: separate label stage after the freeze. Effect: seal is by stage, not only by procedure. D14 From: eICU terminal contract (unit discharge offset and status). This read: MIMIC-IV contract: ceil of ICU LOS hours; hospital death flag, not an ICU death; exits merged. Effect: the family-A and family-B labels code both exits 0, so they are invariant to the cause; of the secondaries, LOS-X_H uses the clock only and MORT-W and MORT-H read the flag as hospital death (D28). D15 From: legacy clock after the terminal. This read: 75.2 % of MIMIC-IV TRAIN stays have a defining channel measured after the terminal clock (eICU TRAIN-fit 58.5 %); labels stop at the terminal; a record can reach hour 30 through rows after the terminal. Effect: disclosed; the hour-30 entry can include such stays. D16 From: input distribution of the frozen model. This read: MIMIC-IV observed-hour ratio to eICU: arterial saturation 0.02 (1.2 % of hours against 49.2 %), heart rate 0.79, non-invasive MAP 0.62, most laboratory values 1.4 to 4.4; no remapping (C9). Effect: a transfer failure is possible and is reported as such. D17 From: 12 eICU settings. This read: 9 (arterial saturation excluded). Effect: Holm over 9. D18 From: producer v2’s encoding-equality check. This read: replaced by explicit pins (3.1). Effect: declared exception. D19 From: zero-prior-read TEST authorities. This read: the disclosure schema of 7.2. Effect: prior reads bound, not denied. D20 From: the v2 bundle committed a stay-level lock. This read: nothing stay-level committed (7.8). Effect: none. D21 From: E4 channel. This read: hourly bedside SpO2 in MIMIC-IV against nurse-charted SpO2 in eICU. Effect: different ascertainment. D22 From: eICU TRAIN-tune persons were seen by the pretrained encoder. This read: MIMIC-IV persons were not. Effect: the tilt in favor of f and h is gone. D23 From: eICU TRAIN-fit 119,330 stays. This read: MIMIC-IV TRAIN 66,007. Effect: smaller fit set for every arm. D24 From: WP8a used per-cell rosters as primary. This read: hour-30 rosters primary, as the paper’s headline; per-cell as sensitivity. Effect: none. D25 From: the eICU last-value producer computed LOCF for roster stays only and skipped stays without a reading. This read: LOCF computed in the predict stage for every stay reaching L, with a missing flag; the roster is applied at scoring. Effect: the scored LOCF values are the same computation on the same roster stays. D26 From: P0 ran the physical-unit guard and invalid-value rule on TRAIN only. This read: also on TEST, in the label stage. Effect: an additional check; a TEST failure stops family B only (C12).

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D27 From: P0/P1 and the paper’s eICU contrast used one tree seed. This read: primary unchanged (C6); paired tree-seed sensitivity and the b range added. Effect: seed dependence is visible, not tested. D28 From: the eICU death forecast was scored against death in the unit within 48 hours of hour 48 (revision_v3_joint_ factual_export.py:228-229). This read: MORT-W: hospital death with unit exit within 48 hours; MORT-H: hospital death at any time (3.11). Effect: MIMIC-IV records the time of in-hospital death, but the legacy harmonized root used here keeps only a hospital death flag without a time, and this read adds no exporter (4.1); MORT-W contains the eICU label and adds stays that left the unit alive in the window and died later in the hospital; the death forecast targets death in the unit before the head’s crossing, so a mean below either observed rate arises in part by construction, as it did against the eICU label; the text states the mean forecast against the observed rate (below, above or within sampling variation), gives that reason only with a shortfall, keeps AUROC as the headline, and 9.3 rewords the paper’s eICU sentence to the same standard (3.11). D29 From: the earlier hospital-mortality reads at C = 48 h: corrected clock and LOS ≥ 48 h (TRB-M4-20, 7,016 stays of 5,446 persons), or the legacy clock (legacy reads), with models trained on MIMIC-IV. This read: legacy clock; terminal clock > 48 and a record reaching hour 48; the frozen eICU model. Effect: the same outcome in a population defined by a different clock and entry rule; its size and its overlap with the TRB-M4-20 stays are not known before the read (SC counts it after the marker, 3.11, and the result reports it); no model comparison is made or tested. D30 From: the earlier LOS reads regressed remaining ICU LOS (TRB-M4-21; legacy los_h cells). This read: leaving the unit within 6, 12, 24 or 48 hours of hour 48, from the merged-exit forecast (3.11). Effect: a threshold of remaining LOS; not comparable with their absolute errors; the exit forecast targets leaving the unit before the head’s crossing, a part of every exit, so a mean below the observed rate arises in part by construction (3.11). D31 From: the eICU export read the forecasts at position 48 of one pass over the whole record (revision_v3_joint_ factual_export.py:449-459, :175). This read: one pass over the 48-hour prefix, in the allowlist view of stage AS. Effect: the same computation under causal masking, up to floating-point order; test (ii) of 7.3 checks that nothing after hour 48 is read. D32 From: the eICU death interval: five-seed mean with a t interval across seeds, and 2,000 per-seed person draws (revision_v3_joint_factual_postprocess.py:42-45). This read: the same t interval, plus 10,000 whole-person draws of the five-seed mean. Effect: the paper prints the whole-person interval, which includes sampling of patients; the across-seed interval is in the Supplementary Information.

The following deviations and corrections are not in protocol section 8, whose text has not changed since commit P. Items 1 to 8 are listed in section 4 of the result note. Items 2, 3, 5 and 7 arose after commit P and before the freeze at C2, and items 1, 4 and 6 arose after the freeze. Item 8 concerns the design of the verification stage. Item 9 arose after the read. 1. Relaunch. The first chain of jobs (13138 to 13148) was cancelled before any stage started. The read ran under route 1 of protocol section 7.7 from relaunch commit D1 (0e8b093b), which changed only the dispatcher’s resource block. The memory requests of stages BX, R, BP, LB, SC and VF fell from 64 GB to 48 GB, against measured peaks of 0.9 to 12.3 GB in the analogous earlier jobs. 2. Job steps. Package commits A6 and A7 made the dispatcher run the clone and checkout (A6) and its remote checks and submissions (A7) inside Slurm job steps, because the login session’s disk writes were throttled. This changed only where the commands ran. 3. Fixed width of 48 hours. Family B built the encoder states at a fixed width of 48 hours instead of the variable width of probe P1, which moves predictions by about 10−7 . The scholar acknowledged this as a deviation from protocol section 2.1 (decision item 21). 4. Dispatcher tests after D1. On the tree after D1, scripts/test_revision_v3_mimic4_read_dispatch_ v1.py fails 11 tests (9 failures and 2 class-setup errors). All of them pin the literal 64 GB values of the registered stage table, which D1 changed by design. Route 1 allows no other change, so the tests were not edited. The read’s own stages were written to run their native tests first and to stop on a failure. Every stage ended with exit code 0:0. Their logs were not shipped, so the passing of these tests rests on the exit codes and the launcher (Provenance and execution, above). The release and common modules pass their 92 tests on the back-filled tree. 5. Reused code without a run of its tests. Stage BF reuses fit_fixed_catboost of revision_v3_ landmark_trajectory_core_v1, whose tests no stage runs because the cluster environment lacks pytest (validation record, item 38). 6. Shipping. The shipping script that copied the aggregates was written after the protocol (Provenance and execution, above). 7. Validation record. The validation record pinned at C2 (reports/revision_v3/MIMIC_IV_READ_ VALIDATION_RECORD_V1.md) contains two stale statements, which are corrected here because the pinned 113

record is not edited. Its line 13 says that no cluster job was run, whereas its items 54 and 55 cite the infrastructure-test jobs 13104, 13105 and 13118, which read no data. Its item 1 says that the package pin names A5, whereas the pin is A7 (80c86ec7). 8. Verification. Stage VF calls the same scoring function as stage SC (score_all). The verification therefore shows that the scoring is deterministic; it does not test an independent implementation. 9. Revision of the manuscript. Decision item 22 of the decision record (reports/revision_v3/ROUTE_B_ DECISION_20260926_V1.md) records the scholar’s decisions on the revision of the manuscript after the lab’s editorial review, taken on 2 and 7 October 2026 with the read’s outcomes known. It also records the departures from protocol section 9 that these decisions cause. The paper has no contribution list, so the registered family-B contribution clause has no host. In a later revision of the manuscript for clarity and length, decided by the scholar on 20 October 2026 (for the Abstract, on 6 November 2026), the registered sentences of this read were reworded for readability and some were moved, for example from Results to Methods; every number, outcome and caveat is unchanged, and the registered wording is given in the read’s outcomes.json (source_data/Supplementary_Tables_S20_S27/mimic4_read_6f9c4a6_13172_ v1/outcomes.json), which the read’s result note (MIMIC_IV_READ_RESULT_20261002_V1.md) names.

Limitations stated in advance (protocol section 10) Protocol section 10 is reproduced in full as written before the read. In it, plain section numbers refer to the protocol, C6 and C12 to the design choices of protocol section 0.3, and D16, D29 and D30 to items of protocol section 8 (Deviations, above). TRB-M4, REG-C01 and PREPROC identifiers refer to rows of Supplementary Table S27, and decision items to the scholar’s decision record (reports/revision_v3/ROUTE_B_DECISION_ 20260926_V1.md). • MIMIC-IV TEST was read before (section 2.3): zero-shot transfers of earlier frozen eICU twins (TRB-M407, -13 to -16, -22), fine-tunes of the same joint lineage (REG-C01), the hypotension rule that E2 repeats, legacy KDIGO AKI (the analogue of E3), legacy hypoxemia (overlapping E4 and the SpO2 target), the HR > 130 and SpO2 < 90 events, and raw future heart and respiratory rate; the analysts know those results. • Family A tests transfer of the frozen eICU model under one input mapping; it does not test synchronization in a model adapted to MIMIC-IV. • The frozen model meets a large input shift (D16) that it was never trained on; nothing is adapted. • The targets are threshold crossings of the MIMIC-IV TRAIN mean in stays selected by measurement (the observation-gap censor), not clinical events; the hour-30 entry conditions the shorter settings on a longer record, and on the legacy clock a record can reach hour 30 through rows after the terminal. • The label and model clocks are the legacy one-based clock, not discharge-bounded; the terminal is the ceiling of the ICU LOS hours and the cause is the hospital flag (merged, so the family-A and family-B labels are unaffected; the mortality secondaries read the flag as hospital death, 3.11). • The legacy root’s clip and fill quantiles use all splits (PREPROC). • The prefix view is procedural within one process: the governed loader holds the previously read TEST mortality and LOS columns in memory during prediction, and the predict processes hold labeldetermining hours (13 to 30 in AP, 13 to 48 in BP, and 13 to 48 in AS, which makes no family-A or family-B prediction), unused by construction and verified by the per-landmark invariance test (7.3), not physically absent. Test (iii) of 7.3 checks the predict modules’ own text, not the modules they import, which include label modules. • The secondaries (3.11) are descriptive, with no criterion, on outcomes read on this split many times before (section 2.3), with results the analysts know. The death summary targets death in the unit before the head’s crossing. MIMIC-IV records the time of in-hospital death (admissions.deathtime, ROUTE_B_PREDICTION_PLAN_20260927_V1.md:58), but the legacy harmonized root used here keeps only hospital_expire_flag, without a time, and this read adds no exporter for it (decision item 19), so endpoint E1 stays excluded (4.1) and the death summary is scored against hospital death: MORT-W adds later in-hospital deaths of stays that left the unit alive in the window, and MORT-H has a longer window than the forecast, so neither calibration nor discrimination is on the eICU estimand (death in the unit within 48 hours, every such death counted). The exit summary targets leaving the unit before the head’s crossing, not every unit exit, so it is not an estimand match for LOS-X_H either. A mean forecast below 114

the observed rate of MORT-W, MORT-H or LOS-X_H therefore arises in part by construction, as the reworded eICU sentence states for eICU (9.3). The heads were trained only at origins where their target had not yet crossed; at hour 48 they are read outside those origins in every stay whose target has already been measured past the threshold, as the eICU death summary was. The death and exit summaries are the largest of target-specific CIFs, one of whose heads reads a channel that is nearly empty in MIMIC-IV (the three-head sensitivity shows its weight). The earlier MIMIC-IV results for these outcomes come from different models, clocks and, for LOS, estimands (D29, D30), and are context, not a comparator. In a final attempt the secondaries can stop while family A is scored (C12); they are then reported as stopped, not scored later. • Family B’s comparator is an untuned fixed recipe; the P1 recipe keeps its documented learning-rate confound between f and g and its epoch cap; h uses in-sample TRAIN states. • E2 overlaps a rule already scored on this split; E4 is ascertained on hourly SpO2 in MIMIC-IV and on nurse charting in eICU; E4 is underpowered. • Family B’s intervals condition on the fitted models; the primary test conditions on one tree seed (C6), and the tree-seed spread and the paired tree-seed sensitivity are reported but enter no test. • The power analysis rests on normal approximations, on eICU standard errors rescaled by event counts, on an approximate design effect for repeat stays, and on assumed correlations between settings and endpoints; it was not checked by resampling any real panel. • Family A’s pass criterion is a count rule: a failed outcome can coexist with an eICU-sized gain in most settings (5.3), and a shown outcome with gains well below the eICU size. • The isolation arms and all causal analyses are not repeated on MIMIC-IV.

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Supplementary Note 8: Earlier transfer analyses on MIMIC-III Scope and status. No result of the main text uses MIMIC-III, but its test split was read in 21 earlier campaigns between May and September 2026. This note puts those reads on record and reports how earlier versions of the model transferred to this second database. Each read is disclosed in the test-read registry (rows TRB-M401, TRB-M4-05 to TRB-M4-16 and TRB-M4-18 to TRB-M4-25; Supplementary Table S28). None had a registry row before it took place, so all 21 are reported as disclosed and not registered. Three of them (TRB-M4-19 to TRB-M4-21) followed rules fixed in their own records before the read. Because those rules were not entered in the registry, we treat the three reads like the others. Twelve of the reads concerned models or encoders trained on eICU-CRD and applied to MIMIC-III with or without fine-tuning. The other nine concerned models trained within MIMIC-III, a semi-synthetic dataset built on it, or early runs recorded only as output directories. No listed read applied the model of the main text to MIMIC-III. We report numbers only for reads whose aggregate tables are committed to the project repository, which will be released with the code (Code availability of the main text), and list the other reads without results. Cohort and split. The MIMIC-III cohort comprises 28,394 CareVue-era stays of 23,692 patients who do not appear in MIMIC-IV (Supplementary Table S18). They were split by person into a training split of 20,007 stays of 16,661 patients, a validation split, and a test split of 4,244 stays of 3,531 patients. The earlier campaigns selected the test split through a legacy split column. For MIMIC-III this column equals the person-level column by construction, although no direct recount was made. Models and readouts. Apart from the withdrawn comparison of TRB-M4-05 (below), the reported transfer cells used A0_dyn, an earlier version of the model with time-to-event, competing-risk and dynamics heads. It was trained on eICU-CRD with three seeds (13, 17 and 23) under a legacy marginal-status objective. None of these cells used the corrected competing-risks model of the main text. The cells were scored with the legacy readout on the legacy endpoints. These included the harmonized mortality label (hospital death) and length of stay at landmarks of 6, 12, 24 and 48 hours, and three threshold events (systolic blood pressure below 90 mmHg, SpO2 below 90% and heart rate above 130 beats per minute). The endpoints also included incident onsets of acute kidney injury, hypotension, hypoxemia, high lactate and a sepsis proxy within 24 hours of the same landmarks. Since 30 August 2026 the project’s evaluator has refused to score cumulative risk from models trained with this objective, so the cells run in September used the June 2026 evaluator (commit fc1e613). Their numbers are therefore comparable with each other but not with any result of the main text. Zero-shot transfer and fine-tuning. The models trained on eICU-CRD discriminated the mortality label on MIMIC-III without any fine-tuning, and their discrimination improved as they were fine-tuned on growing shares of the MIMIC-III training split. Applied to the test split without fine-tuning (zero-shot transfer; TRB-M4-22), the three A0_dyn models had mortality AUROCs of 0.762 to 0.808 over seeds and landmarks. Fine-tuned on 1% of the MIMIC-III training split (TRB-M4-08), they reached a mean AUROC of 0.794 over seeds and landmarks, and fine-tuned on all of it, 0.864. Mean discrimination rose at every step of this fine-tuning ladder (1, 5, 10, 25, 50 and 100% of the training split) for mortality, for the threshold events (AUROC 0.809 to 0.872) and for the incident onsets (0.653 to 0.702). Over the same steps, however, the expected calibration error of the mortality forecast rose from 0.059 to 0.078. Per-patient weight adaptation. Each A0_dyn transfer cell also scored the same model with its weights updated online for each patient. The main text kept the twin’s weights frozen after a separate eICU-CRD evaluation, in which online adaptation of the forecasting heads of an earlier model version did not improve forecasts (Supplementary Fig. S1). In the reported transfer cells on MIMIC-III, no improvement from the updates survived Holm correction either. Mechanism records show that the updates ran at every step, with no rollback and no non-finite update, in the zero-shot cells and in the crossed cells described below. On the fine-tuning ladder, none of the 216 differences between the adapted and the unadapted model on MIMIC-III and MIMIC-IV survived Holm correction within the campaign, in either direction. Without correction, 32 of the 108 differences on MIMIC-III had intervals across seeds that excluded zero. Eleven of them favored the adapted model, including threshold-event discrimination at 1% of the training data (AUROC +0.0028; 95% interval across seeds 0.0017 to 0.0040) and the onset log-loss, Brier score or calibration error at fractions from 1% to 25%. Twenty-one favored the unadapted model, among them the mean absolute error of length of stay from 10% of the training data on (+0.15 to +0.22 hours), the Brier score of the mortality forecast from 10% on (+0.0004 to +0.0006) and threshold-event discrimination at 25% (AUROC −0.0011). In the zero-shot cells, none of the 36 differences on the two databases survived Holm correction. On MIMIC-III, three differences had intervals across seeds that excluded zero without correction, and all three favored the adapted model (threshold-event AUROC +0.0043

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and AUPRC +0.0117, onset log-loss −0.1497). In the crossed cells (TRB-M4-23), a KL-anchored objective replaced the dynamics objective of these updates on models fine-tuned on 1% and 100% of the training split. The onset log-loss difference then changed sign, from −0.1499 and −0.0889 under the dynamics objective at the same fractions of the fine-tuning ladder to +0.0477 and +0.0803, with intervals including zero. The expected calibration error of the threshold events rose by 0.0090 at both fractions, and both rises survived Holm correction over the 36 differences of the MIMIC-III crossed cells. Pretraining against random initialization. TRB-M4-05 compared a model pretrained on eICU-CRD, with its head fine-tuned on MIMIC-III, against the same model trained from random initialization. The pretrained model discriminated the mortality label with AUROC 0.874 to 0.888 at the four landmarks. We withdraw the comparison because the model trained from random initialization failed to train: its AUROC was 0.572 at 24 hours and 0.197 at 48 hours on MIMIC-III, and at or below chance in five of the 12 mortality cells across the three databases. Tabular predictors trained within MIMIC-III. Two corrected-clock reads (TRB-M4-20 and TRB-M4-21) scored gradient-boosted and other tabular models fitted on the training data of each database. Neither read involved an ICare-DT model. At 48 hours, the MIMIC-III test split held 2,289 stays of 1,930 patients with an ICU length of stay of at least 48 hours, and 353 of these stays had a positive mortality label (rate 0.154). Nine tabular models discriminated the mortality label with AUROC 0.878 to 0.882. For remaining length of stay, the read’s candidate (frozen_oof_simplex_cell_median_calibrated_v1) was a weighted blend of gradientboosted models whose weights were fixed on the training data before the test read. Its mean absolute error was 129.1 hours, against 177.2 hours for the training median. Five of the other 12 models of the read had lower errors, from 128.4 to 128.8 hours. The 48-hour MIMIC-IV cells of both reads are given in Supplementary Note 7 (Supplementary Table S25e,f). Interpretation. The transfer cells describe how earlier versions of the model, trained on eICU-CRD, behaved on a second database. Their discrimination of mortality, threshold events and incident onsets rose with the amount of MIMIC-III training data used for fine-tuning, while the calibration of the mortality forecast worsened. Per-patient weight adaptation gave no difference that survived Holm correction within its campaign, apart from a calibration harm under one objective. The reads do not test the model of the main text, its synchronization with frozen weights or any registered claim, and their endpoints and readout differ from those of the main text. The test split was read by many models, and four reads informed later choices of scope or comparators (TRBM4-06, TRB-M4-10, TRB-M4-12 and TRB-M4-19). For these reasons, and because each transfer cell rests on three seeds, we draw no conclusion about MIMIC-III from these reads.

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Supplementary Table S28: Earlier reads of the MIMIC-III test split. Rows of the test-read registry (reports/revision_v3/TEST_READ_REGISTRY.md) whose read included MIMIC-III test data, with dates and contents as recorded there. All are disclosed, and none had a row in this registry before its read. A0_dyn and A0_dyn_thr are earlier model versions, trained on eICU-CRD except in TRB-M4-11, where A0_dyn was trained on MIMIC-III. In the Results column, “Note 8” marks results reported in this note from committed aggregate tables, “Internal notes” marks results recorded only in the project’s working notes (icare_dt/RESULTS.md, icare_dt/CAMPAIGN.md) and not reported, and “Outputs only” marks reads recorded only as output directories. Registry row

Date (2026)

Analysis

Results

TRB-M4-01

23 to 27 May

Outputs only

TRB-M4-05

7 to 14 June

TRB-M4-06

15 June

TRB-M4-07

June

TRB-M4-08

June

TRB-M4-09

3 July

TRB-M4-10

July

TRB-M4-11

July

TRB-M4-12

July

TRB-M4-13

July

TRB-M4-14

July

TRB-M4-15

10 July

Baselines and early model variants: mortality, length of stay and legacy onsets eICU-pretrained model with its head fine-tuned on each database, against the same model trained from random initialization; conventional baselines; mortality and length of stay at 6 to 48 hours Outcome-regression head of an earlier model version on each database; future values of the regression targets Frozen eICU-trained A0_dyn applied without fine-tuning, with and without online weight adaptation A0_dyn fine-tuned on 1% to 100% of the MIMIC-III training split, with and without online weight adaptation XGBoost, LSTM, GRU-D and Transformer trained, calibrated and tested within each database; 28 tasks The same baselines and LightGBM and CatBoost trained on eICU-CRD and tested on MIMIC A0_dyn trained on MIMIC-III, with and without online weight adaptation LightGBM and CatBoost trained within each database; six-model aggregate Frozen eICU-trained A0_dyn_thr (threshold head) applied without fine-tuning, with and without online weight adaptation Re-read of transfer predictions by risk, error, stay length and pre-onset strata Re-read of the TRB-M4-13 predictions by stratum

TRB-M4-16

10 July

TRB-M4-18

From 11 July

TRB-M4-19

21 to 23 July

TRB-M4-20

26 August

TRB-M4-21

26 August

TRB-M4-22

19 September

TRB-M4-23 TRB-M4-24

19 to 20 September 23 to 25 May

TRB-M4-25

22 to 23 May

The TRB-M4-13 models with a threshold-calibration adaptation; closed negative LightGBM on the frozen states of three eICU-trained encoders; mortality, length of stay, lactate and sepsis onset at 24 hours Shared low-rank readout on the same frozen encoders and tasks; selected on validation, and its test result closed a follow-up Tabular predictors of the mortality label among stays with a length of stay of at least 6, 12, 24 or 48 hours (corrected clock); no ICare-DT model Tabular predictors of remaining ICU length of stay (corrected clock); no ICare-DT model Frozen eICU-trained A0_dyn applied without fine-tuning (three seeds), with and without online weight adaptation; June 2026 evaluator A0_dyn fine-tuned on 1% and 100% of the MIMIC-III training split, adapted online with a KL-anchored objective; June 2026 evaluator Online personalization configurations D1 and D3 of an early model version, run per database Counterfactual evaluations on a semi-synthetic dataset built on MIMIC-III

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Note 8 (withdrawn) Internal notes Internal notes Note 8 Internal notes Internal notes Internal notes Internal notes Internal notes Internal notes Internal notes Internal notes Internal notes Internal notes Note 8

Note 8 Note 8

Note 8 Outputs only Outputs only

Supplementary Note 9: Registered analyses of the washout panels added before submission A. Realism audit of the washout panels Realism audit of the washout panels. Placebo copies, in which the memory-route response is switched off but the simulated dosing and its direct and mediated routes remain, gave audit AUC 0.712 to 0.723 (largest difference from the panels 0.004, in the replication panel; 95% interval of that paired difference −0.005 to 0.009), and the audit separated the panels from their own placebo copies with AUC 0.508 to 0.510, above its label-swap null band in the discovery, replication and fixed-schedule panels. Paired comparisons of the same stays detected systematic differences in the written channels in every panel (Hotelling’s test, P < 0.05). In the panels, the lag-1 autocorrelation of 1 to 2 of the 12 outcome channels fell more than 0.02 below that of the real records. Near the threshold the verdict depends on sampling and on the assignment of stays to folds: at an AUC of 0.75 with 400 stays per group the Hanley-McNeil standard error is about 0.017, and Supplementary Table S29 gives each panel’s bootstrap interval and the range over 100 random fold assignments. Wording. The protocol fixed the sentences that report this audit. In the revision of the manuscript for clarity and length (20 October 2026; for the Abstract, 6 November 2026), some of these sentences were reworded for readability, shortened in figure captions or moved, for example from Results to Methods; every number, outcome and caveat is kept in the paper, and the registered wording is given in section 7 of the audit’s result note (WASHOUT_PANEL_REALISM_AUDIT_RESULT_20261014_V1.md). Compute. The registered run (06b23b75ed33_20261014t202905z_washout_realism_audit) executed its six stages as six chained Slurm jobs, 13464 to 13469, one per stage, on 14 October 2026; they were submitted by a dispatch job that cloned the pinned commit, which the shipped accounting summary does not cover. The scheduler’s accounting, as summarized in the shipped file job_log_summary.json, records elapsed times of 1 min 45 s for S0 (job 13464; authentication, pairing, channel contract and census), 6 min 4 s for S1 (job 13465; construction of the 26 record sets and the plumbing check G1), 1 min 24 s for S2 (job 13466; the three primary audits), 5 min 51 s for S3 (job 13467; the other 21 audits and the plateau diagnostics), 2 min 33 s for S4 (job 13468; the descriptive refits, with 16 parallel workers) and 1 min 24 s for S5 (job 13469; rendering, identifier scan and shipping). Each stage also ran its native test suite before its entry point (result note, section 2.1). Jobs 13464 to 13468 ended in state COMPLETED with exit code 0:0. The summary was written by S5 while S5 was still running, so S5’s row records the state RUNNING and the elapsed time at that moment; S5’s stage receipt records its completion. The stage receipts record completion times from 20:32:43 (S0) to 20:50:01 UTC (S5). Of the scheduler’s accounting, the shipped summary carries elapsed time, state and exit code only; it records no CPU time, so none is reported here. No stage stopped the run. Supplementary Table S29: Realism audits of the washout panels. Every audit of the registered protocol WASHOUT_PANEL_REALISM_AUDIT_PROTOCOL_20261008_V1.md as amended by WASHOUT_PANEL_REALISM_AUDIT_PROTOCOL_ 20261011_V2.md (tag washout-realism-audit-protocol-v2), with its descriptive outputs. The panels were audited after their other analyses had been run, including the exploratory construction campaign from which the registered cell was chosen; the registered runs of the in-support control and the g-computation comparator followed the audit. Each audit applies the locked realism audit unchanged to two sets of 400 records cut to the same window, hours 24 to 163 of the stay (hours 24 to 113 for A1r): the 644 summaries of 161 channels (per-stay mean, standard deviation, missing fraction and lag-1 autocorrelation), a five-fold ridge logistic discriminator grouped by stay, and five checks, audit AUC at most 0.75, median absolute standardized mean difference (|SMD|) at most 0.25, its 90th percentile at most 0.50, median absolute missingness gap at most 0.05 and label match at least 0.999. The label tables of the constructed record sets carried no outcome label: they held the stay identifier, source database, age, sex and split, so the label check matches exactly by construction. Precision stated before the run (protocol section 7.3): at an audit AUC of 0.75 with 400 real and 400 generated records the Hanley-McNeil standard error is about 0.017; it treats the two groups as independent, and the paired design and the identical coordinates change the true value. On the probe of the generated dataset the pair-bootstrap standard deviation was 0.0129, 30 random pair-to-fold assignments moved the audit from 0.630 to 0.646, and the shipped 0.641 was reproduced only as 0.630 in another environment (Supplementary Note 5, Proposition 3). The audit assigns stays to folds deterministically from the sorted stay identifiers, without shuffling. A panel whose audit AUC lies within about 0.03 of 0.75 can therefore pass or fail by sampling and by the fold assignment; the locked rule still decides; panel b measures both sources, and its NEAR_THRESHOLD flag reports the case. Panel a lists every audit; panels b to e give the registered descriptive outputs and the plateau diagnostic, none of which has a decision role; panel f gives values that are not audit values; two earlier realism reads of an earlier washout design are disclosed below the panels. The protocol, the result note (WASHOUT_PANEL_REALISM_AUDIT_RESULT_20261014_V1.md) and its release review (WASHOUT_PANEL_REALISM_AUDIT_ RELEASE_REVIEW_20261014_V1.md) are in reports/revision_v3/. Files named without a path are in source_tables/ revision_v3/washout_realism_audit_d68e077_13469_v1/.

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a Every audit. The row above each group states its construction. G1 compares the discovery roster’s real windows with an exact copy of themselves and is exact by construction. G2 is the discovery panel generated without the measurement hold, so that every hour receives a fresh value, under A1; it and CN15 are the sensitivity gates (detected: audit AUC above 0.75). CN05 to CN15 add independent noise to the real windows at each real measurement of each of the 12 outcome channels, held until the next one; 0.15 reference SD equals the panels’ own noise term. CS025 to CS10 raise heart rate by 0.25, 0.5 and 1 reference SD on every finite cell. A1, the primary audit, combines the panel’s generated outcome values, set to missing wherever the real record is missing, with the real record for every other channel, the treatment channels included. A1r is A1 on window hours 0 to 89, which the estimators read. A2 is the panel as stored, with filled cells and the simulated dosing as its treatment channels, completed with the real record’s other channels. A3 is A1 built from the placebo copy, in which the memory-route response is switched off and the simulated dosing and its other routes remain; AP takes the placebo copy’s A3 record as real against the panel’s A1 record. B1 is the generated dataset that passed the audit, on the panel stays and windows, one audit per roster. The fixed-schedule panel uses the discovery roster. Rows: hours per stay in the window. Pass: all five checks. Every report was written and valid (binding and construction checks passed), and no audit was left not run. The median and 90th-percentile |SMD| and the median missingness gap are taken over the 161 channels and cannot register a difference confined to the 12 written channels, so the discriminator is the operative check. Category Q4_PASS_INSENSITIVE: the three A1 audits passed, G2 was detected and CN15 was not. Sources: outcomes.json (table_s29), audits/*/realism_report.json and calibration.csv. Audit

Panel or roster

Stays

Rows

Median Audit AUC Validity |SMD|

90th pct. Median Label |SMD| missing gap match

Pass

G1, plumbing check: the real windows against an exact copy G1 discovery roster 400 140 0.500 valid

0

0

0

1

yes

G2, gate: the discovery panel written at every hour, under A1 G2 discovery roster 400 140 0.959 valid

0

0

0

1

no (detected)

CN05, CN10, CN15: measurement-time noise of 0.05, 0.10 and 0.15 reference SD (CN15 a gate) CN05 discovery roster 400 140 0.513 valid 0 0 0 CN10 discovery roster 400 140 0.555 valid 0 0 0 CN15 discovery roster 400 140 0.609 valid 0 0 0

1 1 1

yes yes yes (not detected)

CS025, CS05, CS10: heart rate raised by 3.75, 7.5 and 15 beats per minute CS025 discovery roster 400 140 0.550 valid 0 CS05 discovery roster 400 140 0.599 valid 0 CS10 discovery roster 400 140 0.699 valid 0

0 0 0

0 0 0

1 1 1

yes yes yes

A1, primary: generated outcome values, real missing cells, real other channels A1 discovery 400 140 0.714 valid 0 A1 replication 400 140 0.727 valid 0 A1 fixed schedule 400 140 0.720 valid 0

0 0 0

0 0 0

1 yes 1 yes 1 yes

A1r: A1 on window hours 0 to 89 A1r discovery 400 A1r replication 400 A1r fixed schedule 400

90 0.737 valid 90 0.713 valid 90 0.738 valid

0 0 0

0 0 0

0 0 0

1 yes 1 yes 1 yes

A2: the panel as stored A2 discovery A2 replication A2 fixed schedule

400 400 400

140 0.970 valid 140 0.978 valid 140 0.968 valid

0 0 0

0.014 0.021 0.019

0 0 0

1 no 1 no 1 no

A3: the placebo copy under A1 A3 discovery 400 A3 replication 400 A3 fixed schedule 400

140 0.712 valid 140 0.723 valid 140 0.722 valid

0 0 0

0 0 0

0 0 0

1 yes 1 yes 1 yes

AP: the placebo copy as real against the panel AP discovery 400 140 0.509 valid AP replication 400 140 0.508 valid AP fixed schedule 400 140 0.510 valid

0 0 0

0 0 0

0 0 0

1 yes 1 yes 1 yes

B1: the generated dataset on the panel stays B1 discovery roster 400 140 0.581 valid B1 replication roster 400 140 0.591 valid

0 1.9 × 10−11 0 1.5 × 10−11

0 0

1 1

yes yes

b Precision of the A1 audits (descriptive). Fixed-score pair bootstrap: draws of the 400 pairs with both records of a pair together and the out-of-fold scores held fixed. Refit pair bootstrap: draws of the 400 pairs with the copies of a stay kept in one fold, refitted per draw. Fold band: random assignments of the 400 stays to five folds of 80, refitted per assignment; its minimum and maximum are given to four decimals so that no value is rounded onto 0.75. Percentile intervals. NEAR_THRESHOLD is set when the refit interval or the fold band’s central range contains 0.75. In the discovery and fixed-schedule panels the registered fold assignment gave a lower AUC than every random assignment, and in the fixed-schedule panel at least one random assignment gave an AUC above 0.75. Sources: intervals.csv and outcomes.json. Discovery

Replication

Fixed schedule

A1 audit AUC (registered folds) 0.714 0.727 0.720 Fixed-score pair bootstrap, 95% interval (2,000 draws) 0.692 to 0.739 0.705 to 0.749 0.699 to 0.744 Refit pair bootstrap, 95% interval (200 draws) 0.678 to 0.737 0.675 to 0.732 0.680 to 0.739 Fold band, central 95% range (100 assignments) 0.718 to 0.742 0.721 to 0.742 0.726 to 0.745 Fold band, minimum to maximum 0.7148 to 0.7477 0.7195 to 0.7496 0.7247 to 0.7541 NEAR_THRESHOLD flag not set not set not set

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c Label-swap nulls, placebo copies and paired tests (descriptive). Label-swap null: 200 swaps of the two labels within pairs, refitted on the registered folds, for each A1 and each AP audit. A1 − A3: the point difference with the percentile 95% interval of a refit pair bootstrap of 200 draws, the same draws for both audits. Hotelling’s T 2 on the paired differences (panel minus real) of the 36 means, standard deviations and lag-1 autocorrelations of the 12 written channels in the A1 records of the 400 stays, with q coordinates kept and F on q and 400 − q degrees of freedom. Sources: nulls.csv, intervals.csv and hotelling.csv. Discovery A1 audit AUC 0.714 A1 label-swap null, mean (SD) 0.500 (0.012) A1 label-swap null, 95th percentile 0.518 A3 audit AUC 0.712 A1 − A3 [paired 95% interval] 0.002 [−0.006, 0.009] AP audit AUC 0.509 AP label-swap null, mean (SD) 0.500 (0.004) AP label-swap null, 95th percentile 0.506 Hotelling: coordinates kept, q of 36 36 Hotelling: F (36, 364) 44.6 Hotelling: P 2.0 × 10−111

Replication

Fixed schedule

0.727 0.720 0.499 (0.011) 0.500 (0.013) 0.517 0.520 0.723 0.722 0.004 [−0.005, 0.009] −0.001 [−0.004, 0.009] 0.508 0.510 0.500 (0.003) 0.500 (0.004) 0.505 0.506 36 36 49.9 56.3 1.3 × 10−118 1.7 × 10−126

d Plateau diagnostic on the A1 records. Outcome channels whose lag-1 autocorrelation in the panel falls more than 0.02 below that of the real records. Sources: plateau/PL_*/receipt.json and plateau/PL_*/channel_persistence_summary.csv. Discovery Outcome channels with a lag-1 deficit, of 12 2 Channel: lag-1 gap, panel minus real invasive mean arterial pressure: −0.038; heart rate: −0.034

Replication

Fixed schedule

1 heart rate: −0.038

2 invasive mean arterial pressure: −0.032; heart rate: −0.028

e Decomposition (descriptive). The locked pipeline refitted on subsets of the 644 summaries of A1 and A2: single summary families (161 summaries each) and channel groups (the 12 written outcome channels, 48 summaries; the 5 treatment channels, 20; the 144 other channels, 576). All 644 is the audit AUC; the refit on all 644 summaries reproduced it with a gap of 0 for each of the 12 A1, A2, A3 and AP audits. Refitted on the 48 summaries of the written channels alone, the pipeline gives an AUC above 0.75 for A1 in every panel; in the full A1 audit, 36 of these 48 summaries can differ within a pair (the 12 restored missing fractions are equal), and they stand beside the 596 summaries of the 149 copied channels, which are identical in each pair, so 608 of the 644 summaries carry no difference. The same file gives, per written channel and summary, the paired difference, its paired t and the audit’s SMD (not shown). † Given to four decimals because it rounds to 1.000. Sources: decomposition.csv and stage_receipts/S4.json. Summary family Audit

Panel

All 644

Means

Missing SDs fractions

A1

discovery replication fixed schedule discovery replication fixed schedule

0.714 0.727 0.720 0.970 0.978 0.968

0.612 0.627 0.614 0.889 0.915 0.889

0.690 0.680 0.694 0.825 0.878 0.828

A2

0.500 0.500 0.500 0.998 0.996 0.998

Channel group Lag-1 0.602 0.610 0.602 0.650 0.683 0.646

5 12 written outcome treatment 0.781 0.781 0.789 0.990 0.993 0.990

0.500 0.500 0.500 0.994 0.998 0.9995†

144 other 0.500 0.500 0.500 0.500 0.500 0.500

f Not audit values. A flexible classifier, scikit-learn’s histogram gradient boosting (200 iterations, random state 13, other defaults), on the same 644 summaries and folds as each A1 audit, with its native handling of missing values. These AUCs are not audit values and enter no decision (the A1 audit AUCs are in panel a). Source: flexible.csv. Panel

Flexible classifier AUC (not an audit value)

Discovery Replication Fixed schedule

0.922 0.909 0.925

Earlier realism reads (disclosure). Two realism reads of an earlier washout design exist; they are not audits of these panels and are not listed above. Both used synthetic source tapes (smooth curves, not eICU-CRD records) and an earlier version of the construction, NaCl scored on creatinine. In the first, the locked discriminator recipe separated the synthetic source tape from the generated record with natural dosing, far above the threshold of 0.75, and did not separate the generated record without the injected response from the record with it. In the second, a discriminator trained to tell the record with the injected response from its copy without it stayed below 0.75. Source: protocol section 2.2.

B. The in-support control and the g-computation comparator Registration and earlier reads. Both analyses were registered in one protocol (WASHOUT_Q5Q6_PROTOCOL_ 20261008_V1.md, amended by WASHOUT_Q5Q6_PROTOCOL_20261011_V2.md, tag washout-q5q6-protocol-v2) after the results at hour 90 were known, at the request of an internal review; the scored hours and every pin were frozen (tag washout-q5q6-freeze-v1) before any clinical outcome was computed. The true effects at hour 90 had been read by earlier analyses of the three panels: the exploratory construction campaign from

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which the registered cell was chosen (disclosure TRB-EI-12 of the test-read registry), the identifiability map, the target restoration, the oracle-structure learner and the response controls, and the static and sequence benchmarks; the ICare-RH placebo gate stopped before any target read. The production run of the map reproduces one of the exploration checks byte for byte (jobs 10684 and 10690), so the two are one measurement. The panels’ source stays include 50, 67 and 50 stays of the ICare-DT test split. Before the protocol was written, the analysts knew every learner’s result at hour 90, and they knew that the vendored G-Net is numerically fragile on the intensive care cohort, from an earlier operational run on training and validation data and the repairs it required. They also knew a label-free diagnostic at hour 42: the Causal Transformer’s predicted effects varied as little across patients at hour 42 as at hour 90 (per fold, standard deviation of the seed-mean prediction 0.033 to 0.086 at hour 42 and 0.031 to 0.086 at hour 90; Spearman correlation between its predictions at the two hours 0.31 to 0.83). A second label-free diagnostic, the Spearman correlation between the static learners’ predictions at hours 42 and 90 (per fold 0.08 to 0.91; pooled over folds 0.27 to 0.84; undefined in 8 folds of the S-learner), was recorded on the cluster and available to them; whether anyone read it is not recorded. Both candidate hours, 30 and 42, were therefore registered and censused without an order. Neither analysis is confirmatory. Wording. The protocol fixed the sentences that report these analyses, and they stand in the paper as registered, except the heading of the Results section on standard learners, which was shortened for readability without a change of meaning; the registered heading is “None of 16 standard treatment-effect learners outperforms predicting zero effect at a query outside the treatment support” (section 7 of the result note, WASHOUT_ Q5Q6_RESULT_20261022_V1.md). G-Transformer. A second g-computation sequence model, G-Transformer (Xiong et al., 2024), was not run. On 8 October 2026 no released code was found in the sources searched: the proceedings page, the first and latest arXiv versions, the PubMed Central manuscript, the corresponding author’s laboratory page, the proceedings text, three GitHub repository searches and five web searches. The first arXiv version had promised code upon publication. A public repository named G-transformer implements a different method and shows no license on its landing page. The dated record is WASHOUT_Q5Q6_GTRANSFORMER_CODE_SEARCH_20261008_ V1.md; code released later is not used. G-Net as run. G-Net (Li et al., 2021) was run from the Causal Transformer repository’s reimplementation (third_party/CausalTransformer/src/models/gnet.py), with the changes of earlier project work, each listed as a deviation with a test that it is inert or declared, among them a recurrent cell that clamps only finite states, a residual bank restricted to holdout trajectories with at least one transition, and a masked loss algebraically identical to the original. Three corrections were registered for these panels: losses on measured hours only, as for the other sequence models; residuals set to zero where a channel was not measured, so that the simulation injects no artefact of the measurement hold; and one seeded person-level holdout split of 10%, shared by every input, so that G-Net was fitted on 90% of the persons the other sequence models used. Three candidates were compared in each fold on a factual criterion, the error of 12-step simulated heart-rate trajectories from hours 6 to 72 on validation persons: the published configuration with early stopping (P0-ES), the same with the learning rate one step lower on the authors’ grid (P1-ES), and the authors’ fixed schedule (P0-F, 200 optimizer steps on this subset). The selected candidate was refitted with three seeds, and its predicted effects were averaged over seeds. The effect at a query was obtained by simulating every recorded variable forward under each arm, averaging 100 Monte Carlo trajectories whose random draws were the same in both arms, a choice that favours the model. Synthetic gates. Before any clinical read (campaign S, jobs 13479 to 13486), G-Net passed both registered gates on synthetic tapes without escalation. In the positive control, with an effect standard deviation near 1, its predictions correlated 0.60 with the true effects (gate 0.5). In a g-formula test under time-varying confounding, it estimated the average effect as 1.022 against a true 1.078 (relative error −5.2%, tolerance 15%), where a naive outcome regression on the same data was biased by 102% of the true effect. On the registered cell’s synthetic scenarios at hours 90 and 30 it was neither informative nor failed. The native test suites passed in a sandbox that hid the data roots. The census and the in-support control. The control was to score the same fitted learners at sustained dosing for 12 hours from hour 30 or from hour 42, inside the dosing period, with true effects computed by the generator. An hour was to be scored only if, before any of its effects was computed, it met two criteria in both the discovery and the replication panel: at least 20 query patients received each regime (POS-1), and at least half had an estimated propensity between 0.05 and 0.95 (POS-2). At hour 90, where no patient is dosed, the 122

propensity criterion passed in the replication panel (median fraction 0.725), so it could not separate supported from unsupported queries and was replaced, as registered, by a criterion on the recorded treatments alone (POS-2′ ): the median over the five folds of the fraction of query patients whose bin of recent doses (0, 1 to 3, 4 to 8 or 9 to 12 in the preceding 12 hours) held at least 10 treated and 10 untreated training windows starting within three hours of the hour. POS-2′ was 0 at hour 90 in both panels. POS-1 held at both candidate hours, but POS-2′ failed at hour 30 in the discovery panel and at hour 42 in both panels (Supplementary Table S30), so no in-support hour was scored: no in-support effect was computed and no learner was scored inside the support. The synthetic checks of the in-support code passed. Before the freeze, the static learners, the Causal Transformer, RMSN, the oracle-structure learner and the reference arms were refitted or replayed at hour 90 in the three panels, and all 19 arms reproduced their registered predictions exactly, without reading an evaluation target. G-Net at hour 90. In this run the true effects at hour 90 were opened once, after G-Net’s predictions were sealed. The oracle-structure learner was better than G-Net in all six registered comparisons (Holm correction over 6; adjusted p = 0.0012 each), each over a comparator that did not itself beat zero; the registered headline is in Supplementary Note 1. Against predicting zero (Holm correction over 2), G-Net’s centered PEHE exceeded that of predicting zero by 0.0091 in the discovery panel (95% interval 0.0016 to 0.0165; worse than predicting zero after correction, adjusted p = 0.033) and by 0.0058 in the replication panel (−0.0020 to 0.0140; unresolved), and, descriptively, by 0.0161 in the fixed-schedule panel (0.0067 to 0.0256). Its correlations with the true effects were 0.05, 0.10 and 0.04 (Supplementary Table S4; only the replication interval, 0.009 to 0.197, excluded zero), and its predictions spread 0.34, 0.35 and 0.42 times as widely as the true effects (metrics.csv of the aggregates). Mechanism record. All 15 fits (three panels, five folds) completed without a declared failure. In every fold P1-ES had the lowest selection criterion, and no selection fit was stopped by the epoch cap; the refits ran 172 to 231 epochs, one optimizer step each, against 200 steps in the authors’ schedule, and each fit took 850 to 926 seconds. Every fit decoded 48,000 of 48,000 Monte Carlo trajectories as finite, and the median per-patient Monte Carlo standard error of the predicted effect, 0.0011 to 0.0025, was small against the spread of the predictions (standard deviation 0.079 to 0.097). Each refit seed alone was further above predicting zero than the average over seeds. Precision and what the null means. Computed before any data, with 400 patients per panel: a null of this kind is evidence against a calibrated correlation of about 0.27 or more in a panel (80% power) and says little about correlations below about 0.2; a gain under the registered rule (Holm correction over 2, ranking in both panels and at least 2% of the standard deviation of the true effects) needed a calibrated correlation of about 0.22. G-Net’s correlation intervals ended at 0.148, 0.197 and 0.135. By Proposition 6, with correlations of at most 0.104 no rescaling of its predictions could have lowered its centered PEHE below that of predicting zero by more than 0.54% of the true effects’ standard deviation, so its spread alone does not account for the null. Because no query inside the support met the overlap criteria, the benchmark cannot distinguish a failure of the learners from the lack of treatment variation at the decision query. The result concerns this cell, a query that no patient received, and this implementation; the three panels are dependent (their true effects correlate 0.999 row by row), so agreement between the discovery and replication panels is not an independent replication. Forecast. The protocol recorded a forecast before any data, as its author’s judgment: that G-Net would most likely not beat predicting zero at hour 90 and might need the registered escalation to pass its positive control (it did not need it); and, inside the support, that most likely no learner would beat predicting zero, an outcome that presupposed a scored hour, whereas the census ended the control earlier. Compute. From the scheduler’s accounting (elapsed time, and elapsed time times allocated processor cores, each run with its dispatch job): the synthetic campaign (jobs 13470 and 13479 to 13486) 4.3 task-hours and 31.8 allocated CPU-hours; the census and the replays (13534 and 13541 to 13546) 18.8 and 150.9; the clinical run (13761 and 13772 to 13781) 3.7 and 26.1, almost all in the 15 G-Net fits of about 15 minutes each; 26.8 task-hours and 208.8 allocated CPU-hours in all. Disclosures. A first dispatch of the synthetic campaign was cancelled before it started, after a check before start found defects that were fixed by an amendment of the protocol (V2) before any registered run. A nonregistered test job of 11 October 2026 ran the packages’ tests outside the sandbox, and one test of the realism-audit package executed that package’s first stage on the panels’ archives (disclosure TRB-EI-19); no

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code of this protocol read panel content in it. The cluster’s account association lapsed more than once during the runs, which delayed jobs but changed no input. The result note and its release review (WASHOUT_Q5Q6_ RESULT_20261022_V1.md, WASHOUT_Q5Q6_RELEASE_REVIEW_20261022_V1.md) are in reports/revision_v3/, and the aggregates in source_tables/revision_v3/washout_q5q6_56dc33f_13776_v1/. Supplementary Table S30: The in-support control. Label-free census of the registered candidate hours 30 and 42, inside the dosing period, and of hour 90, the decision query (negative control), on the factual records of the three panels, pinned before any in-support effect could be computed (registered protocol WASHOUT_Q5Q6_PROTOCOL_20261008_V1.md as amended by WASHOUT_Q5Q6_PROTOCOL_20261011_V2.md). W = 1 and W = 0: query patients, of 400 per panel, whose factual 12-hour window from the hour was fully dosed or undosed (partial: the rest). POS-1: at least 20 patients with each of W = 1 and W = 0. POS-2: median over the five folds of the fraction of query patients with an estimated propensity between 0.05 and 0.95. POS-2′ : median over the five folds of the fraction of query patients whose bin of recent doses held at least 10 treated and 10 untreated training windows starting within three hours of the hour; it replaced POS-2 because POS-2 passed at hour 90 in the replication panel, where no patient is treated. An hour was to be scored if POS-1 and POS2′ held in both the discovery and the replication panel (threshold 0.50); none did, so no in-support effect was computed and the table has no learner row. Domain classifiers (descriptive): person-grouped classifiers separating the query patients’ covariates from training units pooled over hours 6 to 42, from treated or untreated units only, and from units starting within three hours of the hour (undefined at hour 90, where no unit starts within 87 to 93); ranges over the five folds. The fixedschedule panel is reported but does not enter the gate. Treated training units in the query’s recency cell at hour 90 (dosed earlier, no dose in the preceding 12 hours), per fold: 1, 1, 0, 1, 1 (discovery), 3, 3, 2, 1, 3 (replication) and 0, 0, 0, 0, 0 (fixed schedule). Sources: census.json and selection.json in source_tables/revision_v3/washout_insupport_prep_v1/. Hour W = 1 W = 0 Partial POS-1 POS-2 POS-2′

Panel Discovery Discovery Discovery

30 42 90

84 108 0

35 50 400

281 yes 242 yes 0 no

0.4375 0.4000 0.1875

0.2750 0.2875 0.0000

Replication Replication Replication

30 42 90

80 108 0

39 49 400

281 yes 243 yes 0 no

0.5875 0.5250 0.7250

0.6375 0.3000 0.0000

Fixed schedule Fixed schedule Fixed schedule

30 42 90

0 0 0

400 400 400

0 no 0 no 0 no

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

Panel

Hour

Pooled AUC

Against treated units, AUC

Against untreated Same hour units, AUC AUC

Discovery Discovery Discovery

30 0.755 to 0.821 42 0.807 to 0.845 90 0.9996 to 0.9999

0.824 to 0.905 0.788 to 0.902 0.835 to 0.915 0.843 to 0.899 0.9966 to 0.9999 0.995 to 0.998

Replication Replication Replication

30 0.727 to 0.814 0.822 to 0.888 42 0.799 to 0.882 0.791 to 0.887 90 0.99974 to 0.99997 0.999 to 1.000

0.787 to 0.854 0.854 to 0.916 0.980 to 0.999

0.626 to 0.751 0.581 to 0.698 –

Fixed schedule Fixed schedule Fixed schedule

30 0.986 to 0.988 42 0.985 to 0.987 90 1.000

0.970 to 0.976 0.969 to 0.975 1.000

0.917 to 0.941 0.864 to 0.886 –

1.000 1.000 1.000

124

0.634 to 0.715 0.536 to 0.729 –

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