The Cost of a Physics Prior Is Bounded by the Ablation Gap

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The Cost of a Physics Prior Is Bounded by the Ablation Gap

arXiv:2608.21059v1 [cs.AI] 21 Aug 2026

Boris Kriuk

Abstract—Physics-informed and shape-constrained machine learning routinely reports an accuracy cost of enforcing a prior and treats that number as a property of the prior. We show it is primarily a property of the free features and the validation split. Let P be the excess risk of restricting a hypothesis class to functions obeying a shape constraint on a feature subset S, and let ∆ be the excess risk of the ablated model that ignores S. Because a function constant in xj is simultaneously non-decreasing and nonincreasing in xj , the ablated class is contained in the constrained class, so 0 ≤ P ≤ ∆ for every risk functional, with no convexity, smoothness or realizability assumption. In the language of an experiment, the bound collapses to a single sign test: a constrained model must never be beaten by its own ablation. We instantiate this on an ordinal wildfire-severity task (N = 26 681, K = 3) with hard monotone constraints on four meteorological drivers, geographic coordinates left free, and a three-rung validation ladder from i.i.d. resampling to 2◦ spatial blocking, across a nineconfiguration grid. Four consequences are non-obvious. First, coordinates act as a shield: alone they recover 92.9% of the full model’s macro-F1 under spatial blocking, which collapses ∆ from 0.1288 to 0.0427 and forces every prior on the drivers to look cheap; the same screened prior costs 0.0473 shielded and 0.3470 unshielded, a ratio of 7.3 with identical physics. Second, because ∆ is protocol-dependent it is not transferable: coarsening blocks from 1◦ to 10◦ drives ∆ from 0.0942 to 0.0050, so two of our spatial configurations were unidentifiable before any constrained model was trained. Third, empirical inversions of the certified nesting bound the pipeline’s additive resolution in the reported metric, and maximizing over 318 certified comparisons yields a self-calibrating floor ŝ = 0.0220 macro-F1 , below which no price cell can be interpreted — including four of twenty-one in our own headline configuration. Fourth, constraint cost and constraint compliance are logically independent: the unconstrained model violates the textbook prior at rate 0.48–0.49 while enforcing it costs 0.0473. We close with a two-fit screen that rejects unidentifiable constraint-cost experiments before any constrained model is trained. Index Terms—Shape constraints, monotone classification, ordinal regression, physics-informed machine learning, inductive bias, spatial cross-validation, ablation analysis, measurement resolution, wildfire severity.

I. I NTRODUCTION recurring experiment in scientific machine learning proceeds as follows. A practitioner holds a model and a piece of domain knowledge: fire severity should not decrease with temperature, nor increase with relative humidity; permafrost fraction should fall as air temperature rises; aftershock intensity should decay with elapsed time. The knowledge is imposed as a hard shape constraint or a soft penalty, the model

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B. Kriuk is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowloon, Hong Kong SAR, China (e-mail: bkriuk@connect.ust.hk). Code, run configurations and raw artefacts: https://github.com/BorisKriuk/ The-Cost-of-a-Physics-Prior-Is-Bounded-by-the-Ablation-Gap.

is retrained, and the change in a held-out metric is reported as the cost of the prior. When small, the prior is declared essentially free, and the conclusion is generalized: physics can be respected at negligible expense [10], [11], [9]. Such claims appear throughout the physics-informed literature and its application-facing branches, including our own earlier hybrid physics–ML risk model for pan-Arctic permafrost infrastructure, which blends learned climate–permafrost relationships with a prescribed physical sensitivity in a fixed mixture to preserve extrapolative behaviour [13]; and POSEIDON, which embeds Gutenberg–Richter and Omori–Utsu scaling as learnable constraints and reports that the physical parameters land in scientifically plausible ranges while enhancing rather than compromising predictive accuracy [14]. Parallel claims are standard in the monotonicity-constrained tabular literature [4], [5], [6], [7]. The reported quantity is typically of the form “∆F1 = −0.01” or “R2 unchanged to three decimals.” This paper makes a structural observation that changes how such numbers must be read, and then — more importantly — shows that the observation is already computed by any study that runs an ablation baseline. Consider a constraint requiring f to be non-decreasing in xj . A function that does not depend on xj is non-decreasing in xj — weakly, but the constraint is stated weakly — and is therefore admissible. The same function is also nonincreasing in xj , hence admissible under the opposite prior, and admissible under convexity, concavity, Lipschitz bounds and unimodality in xj . So the worst a shape constraint on S can do is force the learner to discard S (Fig. 1): P(σ) | {z }

price of the prior on S

≤

∆(S) | {z }

.

(1)

value of S given the rest

We do not claim (1) is deep; its proof is three lines. We claim that its right-hand side is a quantity constrainedlearning papers already measure, under the name “ablation” or “feature-importance baseline,” and that reading the two numbers together dissolves several confusions. Concretely, in our own experimental harness the bound reduces to the statement that the column ∆F1 versus the ablated baseline must be non-negative for every constrained model — a sign check on a column the pipeline already writes. A. Contributions 1) A certified ceiling (Theorem 9, Corollary 13). For any risk functional and any class closed under feature ablation, the excess risk of a degenerate-admissible constraint family on S is at most the ablation gap of S. This covers hard constraints, penalized objectives

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H

MC (0.6029)

Hσ

MONO sc (0.5850)

H⊥S

GEO ONLY (0.5602)

∆−P

B. What this paper is not It is not a new constrained-learning algorithm, and it does not argue that shape constraints are useless. They buy extrapolation, auditability and regulatory defensibility [8], [4], none of which in-distribution accuracy measures. It argues that the accuracy cost of such constraints, as currently reported, is largely a statement about the free features and the split.

P

Fig. 1. Geometry behind Theorem 9, with the model names of our experimental zoo attached and their measured out-of-fold macro-F1 under 2◦ spatial blocking. The class H⊥S of hypotheses that ignore the constrained features S lies inside the shape-constrained class Hσ , because a function constant in xj satisfies every sign pattern on xj vacuously. Hence the constrained optimum can never be worse than the ablated optimum, and the price of the constraint is bounded by the ablation gap: here P = 0.0179 ≤ ∆ = 0.0427. The bound holds for arbitrary risk functionals.

(Corollary 14), monotonicity under arbitrary partial orders, convexity, Lipschitz and unimodality constraints. It is attained (Proposition 15) and so cannot be improved. 2) An operational reformulation (Corollary 11): ablation dominance. The bound is equivalent to requiring each constrained model to be no worse than its own ablation. This yields a falsifier computable from an existing results table with no additional training. 3) Shielding (Proposition 17), measured. If any S-blind surrogate is near-optimal, ∆(S) ≈ 0 and the constraint is automatically almost free, regardless of whether the prior is correct. In our design the shield is {lat, lon}; it alone attains 92.9% of the full model’s macro-F1 under spatial blocking, and it makes the identical monotone prior appear 7.3× cheaper than when it is removed. 4) Bounded sign controversy (Corollary 18). The entire spread of achievable risk over all 2|S| sign patterns, and over the choice of sign for an auxiliary channel, is bounded by the same ∆. We report the one regime in which our measured spread exceeds the bound, and show that the excess decomposes exactly into two elementary nesting inversions of magnitude ≤ ŝ. 5) A self-calibrating noise floor (Corollary 20). Since (1) holds at the population level, an empirical inversion of any certified nesting measures the pipeline’s own resolution. Maximizing over 318 certified comparisons in our grid gives ŝ = 0.0220 macro-F1 . Crucially ˆ < ŝ is checkable before any constrained model is ∆ trained (Algorithm 1); it fires on two of our twenty-one configuration-protocol cells. 6) Cost and compliance are independent (Proposition 21). We construct families with P = 0 and violation rate approaching 1, and we measure the same dissociation: the unconstrained model violates the textbook prior at rate 0.4795–0.4825 while the price of enforcing that prior is 0.0143–0.0549. 7) Empirical calibration on an ordinal wildfire-severity task with hard monotone constraints, four drivers, a three-rung validation ladder, an exogeneity gate on auxiliary channels, and paired cluster bootstrap inference across a nine-configuration grid (Sections VI–VIII).

II. R ELATED W ORK A. Monotone and shape-constrained learning Monotonicity as an inductive bias goes back to BenDavid [1] and Sill’s monotonic networks [2], and has developed into partially monotone neural networks [3], calibrated interpolated lookup tables [4], deep lattice networks [5], and certification for unconstrained architectures [6]. Monotone splits are standard in production gradient boosting [30], [28], [29], which is what makes hard constraints cheap to impose in our study. Cano et al. [7] survey algorithms and evaluation measures for monotonic classification. Across this literature the accuracy penalty is reported per dataset; to our knowledge it is never reported alongside the ablation gap of the constrained features, which Theorem 9 shows is the only quantity that makes it interpretable. B. Physics-informed and hybrid modeling Physics-informed neural networks [9] and the broader programme of embedding scientific knowledge into learning [10], [11] motivate constraints from conservation laws and empirical scaling relations. Our own prior work makes the “inexpensive physics” claim explicit in two settings: the pan-Arctic permafrost risk framework, where learned relationships are mixed with a prescribed physical temperature sensitivity specifically to survive extrapolation [13]; and POSEIDON, where seismological scaling laws enter as learnable constraints [14]. The present paper is complementary rather than adversarial to both: it identifies the additional number (∆, computed over the covariates that the prescribed component touches) that turns such a statement from unfalsifiable into informative, and the regime (∆ < ŝ) in which it cannot be tested at all. The same accounting applies to self-configuring learners whose inductive biases are set by the algorithm rather than the user, such as adaptive tree morphing [17] and epigenetically adapted evolutionary search [16], where the free-feature set itself moves during training; and to the broader question of which stage of adaptation an inductive bias belongs to [20]. C. Validation protocol and spatial structure That random k-fold cross-validation inflates apparent skill on spatially autocorrelated data is well established [21], [22], [23], [24], [25]. The usual conclusion is that reported accuracies are optimistic — and we reproduce it: our unconstrained model falls from 0.8316 to 0.6029 macro-F1 between i.i.d. and 2◦ -blocked splits. We add a second-order consequence: because protocol choice changes ∆, it changes the admissible range of every constraint-cost figure, so protocol sensitivity propagates into claims about inductive bias and interpretability,

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not merely into headline metrics. Related cautions about correlation- and attention-derived structure under non-i.i.d. resampling appear in [18], [19]. The wildfire corpus and its meteorological joins follow [15].

Definition 7 (Violation rate). For f ∈ H, coordinate j, step hj > 0, and a distribution µ on X , let Dj (x) = f (x + hj ej ) − f (x − hj ej ) and let τ = ε range(f ) be an activity threshold. Then VR(f ; j, σj ) = Pr [sign Dj (x) ̸= σj | |Dj (x)| > τ ] . x∼µ

III. S ETUP AND N OTATION Let X = j=1 Xj with each Xj ⊆ R totally ordered, let Y = {0, . . . , K − 1} carry the natural order, and let D be a distribution on X × Y. Write V = {1, . . . , d} for the available features, S ⊆ V for the constrained subset, and x−S for the coordinates outside S.

We distinguish VRpartial , evaluated on the model’s internal score, from VRe2e , evaluated end-to-end through any preprocessing the deployed pipeline applies; the latter can be positive when the former is zero if a preprocessing stage reintroduces dependence on S. Throughout we use hj = 0.05 × IQR(xj ) and ε = 10−3 .

Definition 1 (Risk functional). A risk is any functional R : H → R ∪ {+∞} with lower values preferred. For G ⊆ H put R⋆ (G) = inf f ∈G R(f ).

IV. M AIN R ESULT

Qd

No structure is assumed on R. It may be an expected loss, a negated macro-F1 , an AUC deficit, a quantile of a perregion error distribution, or a cross-validation estimate of any of these. This generality is what makes the result usable for the metrics applied papers actually report; our own metric, macro-F1 , is not an expected loss and is not decomposable over examples. Definition 2 (Ordinal scoring function). A hypothesis is a pair (f, dec) where f : X → R is a score and dec a fixed decoding of f into Y. Concretely PK−2 we use the cumulative parameterization f (x) = k=0 Ŝk (x) with Ŝk (x) ≈ Pr[y > k | x] and Ŝ0 ≥ · · · ≥ ŜK−2 enforced by running isotonization in k; the label is the argmax of the induced class probabilities. Shape constraints are imposed on each Ŝk and hence on f , and isotonization in k preserves monotonicity in x. Definition 3 (Sign pattern and shape class). For σ ∈ {−1, 0, +1}d with support S(σ) = {j : σj ̸= 0},  Hσ = f ∈ H : ∀j ∈ S, ∀x−j , t 7→ σj f (x−j , t) non-decr. . Definition 4 (S-blind class). H⊥S = {f ∈ H : f (x) = f (x′ ) whenever x−S = x′−S }.

Lemma 8 (Degenerate admissibility). For every σ {−1, 0, +1}d with support S, H⊥S ⊆ Hσ .

∈

Proof. Let f ∈ H⊥S , j ∈ S, and fix x−j . For t ≤ t′ , f (x−j , t) = f (x−j , t′ ) since f does not depend
on coordinate j ∈ S. Hence σj f (x−j , t′ ) − f (x−j , t) = 0 ≥ 0, so t 7→ σj f (x−j , t) is weakly non-decreasing. As j ∈ S was arbitrary, f ∈ Hσ . Theorem 9 (The price of a shape constraint is bounded by its headroom). Under Assumption 5, for every risk functional R and every σ with support S, 0 ≤ P(σ) ≤ ∆(S).

(4)

Equivalently R⋆ (Hσ ) ≤ R⋆ (H⊥S ). Proof. By Lemma 8, H⊥S ⊆ Hσ ⊆ H. The infimum over a larger set is no larger, so R⋆ (H) ≤ R⋆ (Hσ ) ≤ R⋆ (H⊥S ). Subtract R⋆ (H). Remark 10 (On triviality). The proof uses nothing beyond monotonicity of the infimum. We state it as a theorem not because it is hard but because its right-hand side is routinely computed and never connected to its left-hand side. The content of this paper is in Corollaries 11–20 and Proposition 21, and in the calibration of ŝ in Section VIII-H.

Assumption 5 (Ablation closure). H⊥S ̸= ∅. Assumption 5 is satisfied by every class used in practice. It is worth verifying it constructively for the class we use, since the verification is what makes the theorem non-vacuous here: a histogram gradient-boosted tree ensemble that never splits on any coordinate in S is constant in S, and therefore satisfies any per-feature monotonicity specification on S, including contradictory ones. The constrained solver is thus free to reach the ablated model, and does so whenever the constraint is sufficiently costly. Definition 6 (Price and headroom). P(σ) = R⋆ (Hσ ) − R⋆ (H), ⋆

∆(S) = R (H

⊥S

⋆

) − R (H).

(2) (3)

Both are non-negative by nesting [35]. ∆(S) is exactly the number an applied paper reports when it writes “removing temperature costs 0.13 macro-F1 .”

A. The operational form: ablation dominance Write the metric in higher-is-better form, F1 , so that P = F1 (unconstrained) − F1 (constrained) and ∆ = F1 (unconstrained) − F1 (ablated). Corollary 11 (Ablation dominance). Inequality (4) is equivalent to F1 (constrained) ≥ F1 (ablated), (5) i.e. the constrained model must not be beaten by the model that discards the constrained features. In particular the test requires no knowledge of F1 (unconstrained) and no subtraction of correlated estimates. Proof. Subtract the two displayed definitions: P ≤ ∆ iff −F1 (constrained) ≤ −F1 (ablated). Corollary 11 is the form we recommend for practice, because it is a single sign check on a difference that experiment

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harnesses already tabulate, and because it is numerically better behaved: it compares two quantities directly rather than differencing each against a common reference.

B. How far the bound generalizes Definition 12. A family C ⊆ H is S-degenerate-admissible if H⊥S ⊆ C. Corollary 13 (General shape constraints). If C is Sdegenerate-admissible then R⋆ (C) − R⋆ (H) ≤ ∆(S). This covers, for each j ∈ S: monotonicity of either sign and isotonicity with respect to any partial order on XS ; convexity or concavity; L-Lipschitz dependence for any L ≥ 0; unimodality, quasi-convexity and bounded total variation; bounded sensitivity ∥∂f /∂xj ∥∞ ≤ κ; and pairwise dominance or individual-fairness constraints x ⪯ x′ ⇒ f (x) ≤ f (x′ ). Proof. Each listed property holds for every function constant in the coordinates of S: a constant is non-decreasing and nonincreasing, convex and concave, 0-Lipschitz, unimodal, of zero total variation, has vanishing partial derivative, and satisfies every dominance requirement with equality. Hence H⊥S ⊆ C and Theorem 9 applies verbatim. Corollary 14 (Penalized and soft constraints). Let Ω : H → [0, ∞) be a penalty with Ω ≡ 0 on H⊥S , and Rλ = R + λΩ, λ ≥ 0. Then Rλ⋆ (H) − R⋆ (H) ≤ ∆(S) for every λ ≥ 0. Proof. Rλ = R on H⊥S , so Rλ⋆ (H) ≤ inf H⊥S Rλ = R⋆ (H⊥S ). This matters because most physics-informed pipelines use soft penalties rather than hard feasibility [12]: the ceiling is independent of the penalty weight, so no amount of λ-tuning can make the physics term cost more than the constrained features are worth. Proposition 15 (Sharpness). The bound is attained. Let d = 1, X1 = {0, 1}, Y = {0, 1}, D uniform on {(0, 1), (1, 0)}, H all real functions on X , R(f ) = Pr[1{f (x) > 0} ̸= y], σ = (+1). Then R⋆ (H) = 0 and P(σ) = ∆({1}) = 12 . Proof. f (x) = 12 − x attains risk 0. Any non-decreasing f induces a non-decreasing decision rule, which must assign a label to x = 1 no smaller than to x = 0; the Bayes labels are y(0) = 1, y(1) = 0, so every such rule errs on at least one of two equiprobable atoms, giving R⋆ (Hσ ) = 21 . Every S-blind f is constant and errs on exactly one atom, so R⋆ (H⊥S ) = 1 2. Proposition 15 identifies the worst case: tightness occurs precisely when the prior is maximally anti-aligned with the ˆ is evidence mechanism. Hence a measured P̂ close to ∆ ˆ against the physics; P̂ ≪ ∆ is weak evidence for it; and ˆ ≈ 0 is evidence about the feature set, not the physics. ∆ Section VIII-C exhibits all three regimes in one corpus: ˆ = 0.82 unshielded, 0.37 shielded, and ∆ ˆ = 0.0050 P̂/∆ under coarse blocking.

V. C ONSEQUENCES A. Constraint costs do not transfer across protocols Write RΠ for the risk induced by an evaluation protocol Π, and PΠ , ∆Π accordingly. Corollary 16 (Protocol non-transferability). For every Π, PΠ (σ) ≤ ∆Π (S). Consequently: (i) if ∆Π′ (S) ≤ ε then PΠ′ (σ) ≤ ε for every prior supported on S, including priors contradicting the data; (ii) PΠ carries no information about PΠ′ beyond the joint behaviour of ∆Π , ∆Π′ , and the ratio PΠ′ /PΠ is unbounded above and below. Proof. Apply Theorem 9 with R = RΠ and R = RΠ′ separately; cross-validation estimators are risk functionals. (i) is immediate. For (ii), Proposition 15 realizes P = ∆ and Proposition 17 realizes P = 0 for the same σ; embedding one behaviour in Π and the other in Π′ gives arbitrary ratios. Two papers reporting “monotonicity costs 0.01” under random splits and “monotonicity costs 0.09” under blocked splits are therefore not in disagreement, and are not measuring the same thing. In our grid the same prior on the same corpus costs 0.0456 at 1◦ blocks and −0.0139 at 10◦ blocks, a sign reversal driven entirely by the split. B. Shielding: how to make a prior look free Proposition 17 (Shielding). If there exists f † ∈ H⊥S with R(f † ) ≤ R⋆ (H) + ε, then ∆(S) ≤ ε and hence P(σ) ≤ ε for every σ supported on S. Moreover ∆(S) is non-increasing under enlargement of the free feature set V \ S, and nondecreasing under enlargement of S. Proof. The first claim is Definition 6 with Theorem 9. For the second, R⋆ (H) is fixed once V is fixed, while R⋆ (H⊥S ) is non-increasing as the free set grows and non-decreasing as S grows. The uncomfortable corollary for practice: if the feature set contains a proxy that reconstructs the signal carried by S — coordinates standing in for climatology, a station identifier for local calibration, a timestamp for seasonality [25], [26], [27] — then ∆(S) is small and the prior is guaranteed cheap, unconditionally on whether it is correct. Reporting a small P̂ in such a configuration is not evidence that the model respects physics; it is evidence that the model did not need the constrained features. We call the free proxy set the shield; in our design the shield is {lat, lon} and its strength is measured directly by the ablated baseline. The monotonicity in |S| is also observed: promoting insolation from auxiliary channel to fifth ˆ from 0.1288 to 0.1399 under i.i.d. splitting. driver raises ∆ C. The spread over sign patterns is bounded too Corollary 18 (Bounded sign controversy). Let Σ be any set of sign patterns supported on S. Then sup R⋆ (Hσ ) − inf R⋆ (Hσ ) ≤ ∆(S). σ∈Σ

σ∈Σ

(6)

The same holds for a single auxiliary channel c ∈ / S: with C0 ⊇ C+ ∪ C− the classes obtained by leaving the sign of c

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free, fixing it to +1, and to −1, all three contain the c-blind class, so max F1 − min F1 ≤ F1 (C0 ) − F1 (c-blind). sign

sign

(7)

Proof. All Hσ with σ supported on S satisfy H⊥S ⊆ Hσ ⊆ H, so their optima lie in the interval [R⋆ (H), R⋆ (H⊥S )] of length ∆(S). For the second claim, C± and C0 all contain the c-blind class by Lemma 8, and C± ⊆ C0 ; hence the maximum over signs is F1 (C0 ) and the minimum is at least F1 (c-blind). Corollary 18 has a deflationary reading we consider useful. A substantial literature is devoted to which sign a physical prior should take when the textbook sign and the data-driven sign disagree. Whatever the resolution, the total accuracy at stake is capped by the ablation gap. If the ablation gap is below the noise floor, the disagreement cannot be adjudicated by accuracy at all — only by extrapolation behaviour or external theory. Section VIII-G reports the empirical spread, the seven cells in which it exceeds the bound, and the exact decomposition of each excess into two elementary inversions. D. Slack as an instrument Theorem 9 is an identity about infima. Empirically we fit one model per class by regularized empirical risk minimization on each training fold and evaluate a pooled out-of-fold statistic; the fitted model is not the minimizer of that statistic over its class. Every certified nesting therefore yields a testable ordering whose empirical inversion measures the combined finite-sample estimation and optimization gap. Definition 19 (Certified comparison). A pair (A, B) of arms is a certified comparison if the hypothesis class of A is contained in that of B, so that F1 (B) ≥ F
1 (A) at the population level. Its inversion is F1 (A) − F1 (B) + . Corollary 20 (Slack lower-bounds pipeline error). Define
ŝ = max F1 (A) − F1 (B) + (8) (A,B)∈K

over all certified comparisons in a grid K of configurations. Then ŝ lower-bounds the additive error of the pipeline in the reported metric, and: 1) (reporting floor) any |P̂c | < ŝ is unresolvable and must not be interpreted; ˆ c < ŝ then configuration c cannot 2) (a priori futility) if ∆ resolve the price of any prior on S, since the whole admissible interval [0, ∆c ] lies inside the noise; 3) (pre-training screen) the test in (2) needs only the unconstrained fit and its S-ablation, neither of which requires a constrained solver. Proof. At the population level F1 (A) ≤ F1 (B). Writing Fˆ1 (A) = F1 (A) + ηA and Fˆ1 (B) = F1 (B) + ηB , a positive observed difference gives Fˆ1 (A) − Fˆ1 (B) ≤ ηA − ηB ≤ |ηA | + |ηB |; maximize over K. Claims (1)–(3) follow by comˆ c ] with ŝ. Restricting K to ablation pairs recovers paring [0, ∆ Corollary 11 and gives the narrower floor ŝabl ≤ ŝ.

Algorithm 1 Admissibility screen for a constraint-cost experiment Require: features V , constrained subset S, protocol Π, metric F1 , slack ŝ (or a grid K to estimate it) 1: fit unconstrained model on V ; record F1unc 2: fit ablated model on V \ S; record F1abl ˆ ← F unc − F abl 3: ∆ ▷ certified ceiling, Thm. 9 1 1 ˆ < ŝ then 4: if ∆ 5: report “price of any prior on S is unidentifiable under Π” 6: stop ▷ no constrained training required 7: end if 8: fit constrained model; record F1con 9: if F1con < F1abl − ŝ then 10: flag solver or estimation failure ▷ violates Cor. 11 11: end if ˆ ŝ) and the free feature set V \ S 12: report (P̂, ∆,

Unlike a bootstrap confidence interval, ŝ derives from an ordering the experiment is obliged to satisfy, so it captures finitesample estimation error and constrained-solver suboptimality simultaneously — error sources that resampling the data leaves invisible [32]. It does not capture seed variance [33]: all runs use seed 0, and b2 confirms only bytewise determinism at that seed. A seed sweep would plausibly raise ŝ and disqualify further cells, so we regard ŝ = 0.0220 as a lower bound on the pipeline’s resolution, and we do not correct it for multiplicity across the grid [34]. We regard this as the most practically useful consequence of Theorem 9. A corollary of the corollary: a grid K containing no heavily shielded configuration will report ŝ = 0 vacuously, so probing configurations should be included deliberately. Our grid (Table II) includes a block-size sweep spanning 1◦ to 10◦ and a gate-forced-open configuration that adds a free channel; the maximizing comparison turns out to lie in the headline configuration itself. E. Compliance is not price Proposition 21 (Independence of violation rate and price). For every δ > 0 there exist (D, H, σ, µ) with P(σ) = 0 and VR(fˆ; j, σj ) ≥ 1−δ for some risk minimizer fˆ ∈ arg minH R. Symmetrically there exist instances with VR(fˆ; j, σj ) = 0 and P(σ ′ ) > 0 for a different prior on the same coordinate. Proof sketch. Take d = 2, y a deterministic function of x2 alone, x1 independent of (x2 , y). Every f (x) = g(x2 )+ϵh(x1 ) attains the same risk for a decision-based metric provided ϵ is small enough not to change any argmax. The minimizer is therefore not unique, and one may choose h strictly decreasing on a set of µ-measure ≥ 1 − δ, giving VR ≥ 1 − δ under σ1 = +1. Since H⊥{1} contains a risk-optimal hypothesis, ∆({1}) = 0 and Theorem 9 forces P(σ) = 0. The symmetric statement follows by making y depend monotonically on x1 and choosing the opposite sign. Proposition 21 dissolves a common rhetorical move. Some papers argue a model “has learned the physics” because enforcing the physics is cheap; others argue the opposite

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TABLE I C ORRESPONDENCE BETWEEN THE THEORY OF S ECTION III AND THE FITTED MODELS . D = {temp, rh, wind, precip} ARE THE METEOROLOGICAL DRIVERS , G = {lat, lon} THE GEOGRAPHIC SHIELD , c THE RESIDUALIZED AUXILIARY CHANNEL ( INSOLATION ). σ nv = (+1, −1, +1, −1) IS THE TEXTBOOK ( NAIVE ) SIGN PATTERN , σ sc THE FOLD - WISE SCREENED ONE . Model

Features

Constraint

Role

MAJORITY GEO ONLY WX ONLY MC MONO naive MONO sc PHYS PHYS sc MIB MIB fp MIB fm

— G D D∪G D∪G D∪G D D D ∪ G ∪ {c} D ∪ G ∪ {c} D ∪ G ∪ {c}

— none none none σ nv on D σ sc on D σ nv on D σ sc on D σ sc on D σ sc , c: + 1 σ sc , c: − 1

H⊥D for PHYS* H⊥D for MONO*, MIB* H for PHYS* H for MONO* nv Hσ sc σ H ; c-blind arm unshielded prior unshielded prior C0 in (7) C+ C−

because the unconstrained model violates the physics often. Both inferences are invalid: VR does not appear in (4), and the two quantities can be set independently. A high VR with a small P is the generic situation whenever the constrained features are partially redundant, and it is exactly what we measure (Section VIII-F). VI. I NSTANTIATION : M APPING THE T HEORY ONTO A M ODEL Z OO The bound is only useful if the abstract classes correspond to models one actually fits. Table I gives the correspondence used throughout our experiments. All arms share one architecture — a cumulative ordinal ensemble of K −1 histogram gradientboosted binary classifiers [30] with identical hyperparameters (300 iterations, learning rate 0.06, 31 leaves, minimum 40 samples per leaf, L2 regularization 1.0, no early stopping, seed 0) — and differ only in feature set and in the per-feature monotonicity specification, which is a hard constraint of the split finder rather than a penalty. Consequently the nesting of Fig. 1 holds exactly at the level of representable functions, and no penalty weight is tuned anywhere. Four families of certified ceilings follow, and they are the falsifiable content of the empirical section. Proposition 22 (Instantiated ceilings). At the population level, for every protocol: F1 (MONO*), F1 (MIB*) ≥ F1 (GEO ONLY),

(R1)

F1 (PHYS*) ≥ F1 (best constant),

(R2)

F1 (MIB*) ≥ F1 (MONO sc),

(R3)

F1 (MIB) ≥ F1 (MIB fp), F1 (MIB fm), (R4) F1 (MC) ≥ F1 (MONO*), F1 (WX ONLY) ≥ F1 (PHYS*). Moreover maxctx sign F1 − minctx sign F1 F1 (MONO sc).

(R5) ≤

F1 (MIB) −

Proof. (R1)–(R3) are Corollary 11 applied to the corresponding rows of Table I: within the D ∪ G architecture H⊥D is exactly the class of ensembles that never split on D, i.e. the GEO ONLY class, and within the D-only architecture it is the constants; the c-blind subclass of any MIB arm is the MONO sc class, since an ensemble that never splits on c satisfies any sign specification on c. (R4) is C± ⊆ C0 and (R5) is Hσ ⊆ H. The final claim is (7). Remark 23 ((R2) is deliberately conservative). We test a loosened ceiling, so that a violation is correspondingly stronger evidence of pipeline error. Under macro-F1 the majority-class predictor is not optimal among constants — predicting a rare class can raise the macro average — so F1 (MAJORITY) ≤ F1 (best constant). (R1) is conservative for the MIB arms for the same reason: their true D-ablation is “coordinates plus a free channel,” which contains the GEO ONLY class. Remark 24 (MIB versus MC is not a constraint cost). MIB carries a feature MC does not, so neither class contains the other and no certified ordering holds between them. Differences we report in that column are the joint effect of the D-prior and the added channel, not a price; an unconstrained model on D ∪ G ∪ {c} was not fitted, so a feature-matched reference for the MIB arms does not exist in this zoo. This is why MIB exceeding MC in Table III is not an inversion, and why we exclude those cells from (8). Remark 25 (A degeneracy check built into the harness). When the exogeneity gate of Section VII rejects all auxiliary candidates, the channel c is absent and Lemma 8 predicts that MIB, MIB fp and MIB fm must be identical to MONO sc, not merely close: constraining the sign of a feature that does not exist is vacuous. Our implementation asserts bytewise equality of out-of-fold predictions in this case and aborts otherwise. The assertion passes in all three protocols of both gate-empty configurations (closed and drv), and signature deduplication reports the arms as aliases rather than as small insignificant differences. This converts a corollary of the theorem into a runtime test of the constraint plumbing, and it is the cheapest such test we know. VII. E XPERIMENTAL D ESIGN The experiments are not intended to demonstrate a new method; they calibrate the quantities of Theorem 9 and measure ŝ on a real environmental prediction task where monotone physical priors are uncontroversial. A. Data and task We use the Eurasian wildfire corpus of [15]: georeferenced fire records joined to reanalysis meteorology at the time and place of detection. The target is an ordered severity/type variable with K = 3 levels, declared in increasing severity as Controlled burn ≺ Uncontrolled burn ≺ Wildfire, so the ordinal cumulative parameterization of Section III applies. After class filtering and completeness filtering on drivers and coordinates, N = 26 681 records remain, with class counts [333, 2711, 23 637] (shares 0.0125, 0.1016, 0.8859)

7

and a majority-class macro-F1 baseline of 0.3132. Drivers are D = {temp, rh, wind, precip}; the shield is G = {lat, lon}. Two properties of the corpus constrain what this design can test: the absence of a parsable date column, so there is no temporal rung on the ladder and calendar harmonics are unavailable to the exogeneity gate; and the near-surjectivity of coordinates on records, so grouping by exact coordinate removes little information relative to i.i.d. resampling. Effectively the ladder has two informative rungs: unblocked (iid, point) and spatially blocked.

B. Priors The textbook pattern is σ nv = (+1, −1, +1, −1) on (temp, rh, wind, precip). The screened pattern σ sc is obtained inside each training fold as the sign of the mean central finitedifference partial effect of the unconstrained MC model, with a 400-replicate clustered bootstrap interval reported per fold so that non-identified signs are visible rather than silently adopted. Screening on the training fold only is essential: screening on the pooled data would leak the test fold into the constraint specification. In the headline configuration σ sc coincided bytewise with σ nv in 9 of 15 training folds (2/5 under iid, 4/5 under point, 3/5 under spatial), and MONO sc was consequently fitted as MONO naive in those folds, detected by signature deduplication.

C. Auxiliary channel and its exogeneity gate Candidate auxiliary channels (dewpoint, apparent temperature, pressure, cloud cover, insolation, and calendar harmonics when a parsable date exists) are admitted only if they are not reconstructible from the drivers. Each candidate c is regressed on a quadratic design in D — intercept, linear, squares and all pairwise products — and admitted iff R2 ≤ 0.90 and the residual relative standard deviation is ≥ 0.10. Only one candidate is present in this corpus, insolation, and it is admitted comfortably: R2 = 0.2646, residual relative standard deviation 0.8576. Admitted channels enter the MIB arms in residualized form, c − Ab̂, with b̂ estimated on training data only. The motivation is that a partially monotone certificate on D is meaningless if a free channel is a smooth function of D: the deployed pipeline can then reintroduce arbitrary dependence on D through the free channel, which is exactly what VRe2e detects. The gate opened in every one of the 15 training folds of every gate-auto configuration, a fact that turns out to neutralize our screening-leak probe (Section VIII-I).

D. Validation ladder Three protocols in increasing severity: iid stratified 5-fold; point, grouping by exact coordinate so no location is split across folds; and spatial, grouping by blocks of a chosen angular size (429 blocks at 2◦ , 1080 at 1◦ , 115 at 5◦ , 40 at 10◦ ). Grouped folds use stratified group k-fold. This ladder is the standard remedy for autocorrelation-inflated skill [21], [22], [23].

TABLE II C ONFIGURATION GRID K. E ACH ROW IS A COMPLETE RUN OF THE THREE - PROTOCOL LADDER OVER ALL ELEVEN ARMS OF TABLE I, WITH 5 FOLDS , SEED 0 AND B = 2000 CLUSTER BOOTSTRAP REPLICATES . CTX , B 2 AND GLOB COINCIDE BYTEWISE ; SEE S ECTION VIII-I. Tag

Setting

Purpose

ctx b1 b2 b5 b10 closed drv glob k5

headline: 2◦ blocks, gate auto/fold 1◦ blocks (1080) 2◦ blocks (429) 5◦ blocks (115) 10◦ blocks (40) gate forced closed insolation as 5th driver gate decided on pooled data K=4 label granularity

headline numbers shield strongest determinism check (≡ctx) block sweep headroom collapse; ŝ probe MIB*≡MONO sc test driver-set perturbation, |S|=5 screening-leak probe label-resolution perturbation

E. Configuration grid Because ŝ in (8) is a maximum over configurations, we run the full ladder under each of the settings in Table II. Nine tags were executed; three of them (ctx, b2, glob) produce bytewise identical point estimates, leaving seven numerically distinct configurations. We designate ctx the headline configuration; all figures are generated from it. The k5 configuration re-splits the majority class, yielding K = 4 ordered levels Controlled burn ≺ Uncontrolled burn ≺ Forest fire ≺ Natural fire with counts [333, 2711, 18 415, 5195] on N = 26 654 records and a majority baseline of 0.2043. It is included because sign identifiability turns out to depend far more on label resolution than on protocol. F. Inference All paired comparisons use one shared set of B = 2000 cluster bootstrap replicates [31], resampling spatial blocks, so that every pairwise difference is computed on identical replicates and is therefore directly comparable; replicates missing a class are discarded. All 2000 replicates survived in every configuration and protocol, so no interval in this paper rests on a degenerate resample. Predictions are pooled out-of-fold and coverage is asserted. Models whose predictions coincide bytewise are detected by signature deduplication, fitted once, and reported as identical rather than as a small insignificant difference — a distinction that matters because Lemma 8 predicts exact coincidence in specific configurations. Note that the bootstrap clusters are the same spatial blocks that define the spatial folds, so the block-size sweep changes interval widths on the iid and point rungs even though their point estimates are invariant; the half-width of the MONO sc price under iid grows from 0.0228 at 2◦ to 0.0246 at 10◦ . VIII. R ESULTS Table III is the headline configuration in full; Table IV is the whole grid. Six runtime falsifiers passed in every configuration: WX ONLY present in all cost tables (C1); intervals computed against all three tabular references (C2); VRown = 0 for every constrained arm (C3a) and undefined

8

Fig. 2. Out-of-fold macro-F1 across the validation ladder, headline configuraˆ tion. The gap between MC and GEO ONLY is the certified ceiling ∆(D) on the cost of any monotone prior on the drivers (0.1288 → 0.1165 → 0.0427); the gap between MC and MONO sc is the measured price (0.0473 → 0.0555 → 0.0179). Theorem 9 requires the MONO sc curve never to fall below the GEO ONLY curve, which it does not. Note that WX ONLY crosses below GEO ONLY at the spatial rung: under blocking, coordinates alone beat all four drivers together by +0.0572 [+0.0252, +0.0814].

for every unconstrained arm (C3b); no bootstrap degeneration (C4); MIB* ≡ MONO sc bytewise whenever the gate is empty (C5); and complete out-of-fold coverage (C6). A. Protocol ladder and the collapse of headroom Fig. 2 shows macro-F1 for the key arms across the ladder. The unconstrained model falls from 0.8316 (iid) through 0.8311 (point) to 0.6029 (spatial), a split-leakage term of +0.2287; under 10◦ blocks it reaches 0.5218, a term of +0.3098. The quantity that governs everything downstream is not that drop but the vertical gap between MC and GEO ONLY, ˆ i.e. ∆(D): it falls from 0.1288 under iid to 0.1165 under point to 0.0427 under 2◦ spatial blocking, a factor of 3.0. By Corollary 16, the price of every prior on D is forced to shrink with it. The apparent disappearance of constraint cost under strict validation is thus a theorem, not a finding. B. Price against certified ceiling Fig. 3 reports paired differences with 95% cluster bootstrap intervals against the three references of Table I. The panel against GEO ONLY is the direct test of Corollary 11: every constrained arm on D ∪ G must sit to the right of zero, and in the headline configuration all fifteen such cells do, with margins from +0.0029 (MIB fm, spatial) to +0.0941 (MIB fp, iid). ˆ is the Reading Table III as prescribed, the ratio P̂/∆ fraction of the drivers’ value that the prior destroys: 0.367 for MONO sc under iid, 0.476 under point, 0.419 under spatial. The corresponding unshielded ratios are 0.821, 0.822 and 0.694. That contrast is the subject of Section VIII-C. C. Shielding, measured The shield’s strength is F1 (GEO ONLY) = 0.7028 under iid against a majority baseline of 0.3132, and 0.5602 under spatial blocking. Expressed as a fraction of the full model

Fig. 3. Paired macro-F1 differences with 95% cluster bootstrap intervals, by protocol, headline configuration. PHYS* versus WX ONLY and MONO* versus MC are the only feature-matched comparisons and hence the only ones that measure a constraint cost (cf. Remark 24 for the MIB arms); the panel against GEO ONLY is the ablation dominance test of Corollary 11, where every constrained arm must lie to the right of zero. Note how the PHYS* points sit an order of magnitude further left than the MONO* points against their respective unconstrained references — identical physics, different free feature set.

it rises with protocol severity: coordinates alone recover 84.5% of MC under iid and 92.9% under blocking. Under blocking they also beat the four drivers together by +0.0572 [+0.0252, +0.0814]. The consequence for constraint accounting is stark. The same screened monotone prior costs 0.0473

shielded: MONO sc versus MC,

0.3470

unshielded: PHYS sc versus WX ONLY,

a ratio of 7.34 under iid; the ratio is 6.43 under point and 7.36 under spatial, and 7.02 for the textbook prior under iid. Nothing about the physics differs between the two arms; only the free feature set does. Nor is the effect a pure rescaling by ˆ the normalized price P̂/∆ ˆ also more than doubles, from ∆: 0.367 shielded to 0.821 unshielded. Under Proposition 15 the unshielded ratio of 0.82 is close to the anti-alignment regime, i.e. on the drivers alone the monotone prior destroys most of what the drivers are worth — a conclusion the shielded arm hides completely. Had we reported the shielded configuration alone — the natural choice, since coordinates improve absolute performance — we would have concluded that the prior is nearly free. This is Proposition 17 in numbers, and it is the single most important reason we recommend reporting the free feature set alongside every constraint-cost figure. The block-size sweep isolates the mechanism, and inverts the naive expectation. Coarsening blocks might be expected ˆ to weaken the coordinate shield and so raise ∆(D); instead F1 (GEO ONLY) moves only from 0.5592 to 0.5168 while F1 (MC) collapses from 0.6534 to 0.5218 (Table IV, spatial rows), so ˆ ∆(D) :

0.0942 → 0.0427 → 0.0184 → 0.0050.

Coordinates transfer across coarse block boundaries far better than the drivers do: F1 (WX ONLY) falls from 0.5686 to 0.4410 over the same sweep. What Proposition 17 guarantees is monotonicity of ∆ in the feature sets, not in the protocol; Corollary 16 is what governs the sweep, and it bites hard. The measured price tracks the ceiling down and then inverts: 0.0456, 0.0179, 0.0229, −0.0139 for MONO sc. At 10◦ the ceiling 0.0050 is a fifth of ŝ, so the last two entries are not measurements of anything.

9

TABLE III ˆ IS MC − GEO ONLY FOR ARMS H EADLINE CONFIGURATION ( CTX : 2◦ BLOCKS , GATE AUTO / FOLD ), ALL ELEVEN ARMS , ALL THREE PROTOCOLS . ∆ ON D ∪ G AND WX ONLY − MAJORITY FOR ARMS ON D ALONE (R EMARK 23). P̂ IS THE NEGATED DIFFERENCE AGAINST THE FEATURE - MATCHED UNCONSTRAINED REFERENCE . “ DOMINANCE ” IS THE SIGN TEST OF C OROLLARY 11 AGAINST THE ARM ’ S OWN ABLATION AND MUST BE ≥ 0. † MARKS |P̂| < ŝ = 0.0220, I . E . CELLS THAT CARRY NO INTERPRETATION . ‡ MARKS THE MIB ARMS , WHOSE REFERENCE IS NOT FEATURE - MATCHED (R EMARK 24). VR COLUMNS ARE END - TO - END VIOLATION RATES AGAINST THE ARM ’ S OWN DECLARATION , THE TEXTBOOK PRIOR , AND THE SCREENED PRIOR . F1 (OOF)

F1 fold ± sd

ˆ ∆

P̂

ˆ P̂/∆

dominance

VRown

VRnv

VRsc

F1 per class

iid

MAJORITY GEO ONLY WX ONLY MC MONO naive MONO sc PHYS PHYS sc MIB MIB fp MIB fm

0.3132 0.7028 0.7358 0.8316 0.7815 0.7843 0.3842 0.3888 0.7888 0.7968 0.7809

0.3132 ± 0.0000 0.7026 ± 0.0147 0.7356 ± 0.0116 0.8312 ± 0.0097 0.7813 ± 0.0160 0.7843 ± 0.0095 0.3842 ± 0.0100 0.3885 ± 0.0138 0.7881 ± 0.0219 0.7969 ± 0.0095 0.7808 ± 0.0161

— — — 0.1288 0.1288 0.1288 0.4226 0.4226 0.1288 0.1288 0.1288

— — — — 0.0501 0.0473 0.3516 0.3470 0.0428‡ 0.0347‡ 0.0507‡

— — — — 0.389 0.367 0.832 0.821 0.332 0.269 0.394

— — — — +0.0787 +0.0815 +0.0710 +0.0756 +0.0860 +0.0941 +0.0781

n/a n/a n/a n/a 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

n/a n/a 0.4849 0.4795 0.0000 0.2068 0.0000 0.3855 0.2280 0.2245 0.2035

n/a n/a 0.4953 0.4957 0.1572 0.0000 0.0852 0.0000 0.0000 0.0000 0.0000

0.000 / 0.000 / 0.940 0.406 / 0.733 / 0.969 0.597 / 0.646 / 0.964 0.700 / 0.816 / 0.979 0.605 / 0.766 / 0.973 0.602 / 0.776 / 0.974 0.000 / 0.211 / 0.941 0.000 / 0.225 / 0.942 0.586 / 0.803 / 0.978 0.616 / 0.797 / 0.977 0.593 / 0.776 / 0.974

point

MAJORITY GEO ONLY WX ONLY MC MONO naive MONO sc PHYS PHYS sc MIB MIB fp MIB fm

0.3132 0.7146 0.7476 0.8311 0.7761 0.7755 0.3888 0.3907 0.8070 0.8023 0.7790

0.3132 ± 0.0004 0.7143 ± 0.0203 0.7471 ± 0.0132 0.8309 ± 0.0196 0.7755 ± 0.0175 0.7750 ± 0.0184 0.3888 ± 0.0086 0.3907 ± 0.0087 0.8066 ± 0.0098 0.8021 ± 0.0056 0.7786 ± 0.0115

— — — 0.1165 0.1165 0.1165 0.4344 0.4344 0.1165 0.1165 0.1165

— — — — 0.0549 0.0555 0.3588 0.3569 0.0241‡ 0.0288‡ 0.0521‡

— — — — 0.471 0.476 0.826 0.822 0.207 0.247 0.447

— — — — +0.0616 +0.0609 +0.0756 +0.0775 +0.0924 +0.0877 +0.0644

n/a n/a n/a n/a 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

n/a n/a 0.4922 0.4799 0.0000 0.0747 0.0000 0.1661 0.0880 0.0912 0.0826

n/a n/a 0.4928 0.4868 0.0494 0.0000 0.0326 0.0000 0.0000 0.0000 0.0000

0.000 / 0.000 / 0.940 0.433 / 0.741 / 0.970 0.610 / 0.667 / 0.966 0.691 / 0.823 / 0.980 0.586 / 0.769 / 0.973 0.581 / 0.771 / 0.974 0.000 / 0.225 / 0.942 0.000 / 0.230 / 0.942 0.637 / 0.806 / 0.978 0.634 / 0.795 / 0.977 0.592 / 0.772 / 0.974

spatial

MAJORITY GEO ONLY WX ONLY MC MONO naive MONO sc PHYS PHYS sc MIB MIB fp MIB fm

0.3132 0.5602 0.5030 0.6029 0.5886 0.5850 0.3679 0.3713 0.6125 0.6159 0.5630

0.3109 ± 0.0069 0.5544 ± 0.0617 0.5045 ± 0.0628 0.6014 ± 0.0721 0.5883 ± 0.0635 0.5856 ± 0.0639 0.3643 ± 0.0271 0.3692 ± 0.0326 0.6105 ± 0.0765 0.6102 ± 0.0886 0.5626 ± 0.0516

— — — 0.0427 0.0427 0.0427 0.1898 0.1898 0.0427 0.0427 0.0427

— — — — 0.0143† 0.0179† 0.1351 0.1317 −0.0096†‡ −0.0130†‡ 0.0399‡

— — — — 0.335 0.419 0.712 0.694 −0.225 −0.305 0.934

— — — — +0.0284 +0.0248 +0.0547 +0.0581 +0.0523 +0.0557 +0.0029

n/a n/a n/a n/a 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

n/a n/a 0.4904 0.4825 0.0000 0.2650 0.0000 0.3462 0.2885 0.2863 0.2494

n/a n/a 0.4888 0.4968 0.1904 0.0000 0.0638 0.0000 0.0000 0.0000 0.0000

0.000 / 0.000 / 0.940 0.082 / 0.640 / 0.958 0.130 / 0.433 / 0.946 0.208 / 0.640 / 0.961 0.194 / 0.613 / 0.959 0.183 / 0.613 / 0.959 0.006 / 0.161 / 0.937 0.000 / 0.174 / 0.939 0.192 / 0.681 / 0.964 0.204 / 0.679 / 0.964 0.127 / 0.604 / 0.958

Protocol

Model

D. Feature-set and label-set perturbations Two configurations perturb S and Y rather than the split, and both behave as Proposition 17 requires. Promoting insolation from auxiliary channel to fifth driver (drv) enlarges S from four to five features. The ceiling rises accordingly, from 0.1288 to 0.1399 under iid and from 0.4226 to 0.4675 for the unshielded arms, while the price rises less (0.0473 → 0.0431 shielded, 0.3470 → 0.3578 unshielded), so the normalized price falls to 0.308 shielded and 0.765 unshielded. The channel that the exogeneity gate classified as auxiliary is in fact the single best-identified driver in the whole study: its sign is positive in 15/15 folds, agrees with the textbook prior in 15/15, and its bootstrap interval excludes zero in 12/15 — against 0–6/15 for the four meteorological drivers. Exogeneity (R2 = 0.2646 on the driver design) is a statement about redundancy, not about predictive value, and the gate is doing the job it was designed for rather than the job of feature selection. Refining the label set to K = 4 (k5) leaves the ceiling essentially unchanged (0.1203, 0.1303, 0.0655 across the ˆ for MONO sc ladder) but raises the price sharply: P̂/∆

becomes 0.451, 0.537 and 0.534 versus 0.367, 0.476 and 0.419 at K = 3. Under Proposition 15 this is the expected reading: at finer label resolution the monotone prior is a worse approximation to the mechanism. Consistently, this is also the configuration in which the drivers’ signs become identifiable (Table V). E. Sign identification, and why its stakes are bounded Fig. 4 shows fold-wise mean partial effects with clustered intervals; Table V aggregates. In the headline configuration the screened sign agrees with the textbook sign in 53 of 60 driverfolds (88.3%) and the interval excludes zero in only 13 of 60 (21.7%). Temperature and relative humidity agree in 15/15 folds each; wind in 14/15; precipitation in only 9/15, with the screened sign coming out positive — against the textbook −1 — in 6 of 15 folds while being identified in just 2. Precipitation is the one driver on which a practitioner would face a genuine textbook-versus-data dispute, and it is precisely the driver the data cannot resolve. By Corollary 18 the accuracy at stake in that dispute is ˆ = 0.0427 under spatial blocking and 0.1288 bounded by ∆

10

TABLE IV C OMPLETE OUT- OF - FOLD MACRO -F1 ACROSS THE GRID . CTX IS THE HEADLINE CONFIGURATION ; B 2 AND GLOB REPRODUCE IT BYTEWISE AND ARE NOT REPEATED . ROWS MARKED ∗ HAVE AN EMPTY EXOGENEITY GATE , SO MIB* ≡ MONO SC BY L EMMA 8 ( VERIFIED BYTEWISE ). ˆ G = MC − GEO ONLY IS THE CERTIFIED CEILING FOR THE SHIELDED ARMS AND ∆ ˆ M = WX ONLY − MAJORITY FOR THE UNSHIELDED ARMS . ∆ B ECAUSE BLOCK SIZE ALSO SETS THE BOOTSTRAP CLUSTERING , THE IID AND POINT ROWS OF B 1/ B 5/ B 10 REPRODUCE CTX EXACTLY IN POINT ESTIMATE AND DIFFER ONLY IN INTERVAL WIDTH . Config

Protocol

MAJORITY

GEO ONLY

WX ONLY

MC

MONO naive

MONO sc

PHYS

PHYS sc

MIB

MIB fp

MIB fm

ˆG / ∆ ˆM ∆

ctx

iid point spatial

0.3132 0.3132 0.3132

0.7028 0.7146 0.5602

0.7358 0.7476 0.5030

0.8316 0.8311 0.6029

0.7815 0.7761 0.5886

0.7843 0.7755 0.5850

0.3842 0.3888 0.3679

0.3888 0.3907 0.3713

0.7888 0.8070 0.6125

0.7968 0.8023 0.6159

0.7809 0.7790 0.5630

0.1288 / 0.42 0.1165 / 0.43 0.0427 / 0.18

b1

iid point spatial

0.3132 0.3132 0.3132

0.7028 0.7146 0.5592

0.7358 0.7476 0.5686

0.8316 0.8311 0.6534

0.7815 0.7761 0.6061

0.7843 0.7755 0.6078

0.3842 0.3888 0.3711

0.3888 0.3907 0.3996

0.7888 0.8070 0.6182

0.7968 0.8023 0.6381

0.7809 0.7790 0.5982

0.1288 / 0.42 0.1165 / 0.43 0.0942 / 0.25

b5

iid point spatial

0.3132 0.3132 0.3132

0.7028 0.7146 0.5670

0.7358 0.7476 0.4975

0.8316 0.8311 0.5854

0.7815 0.7761 0.5846

0.7843 0.7755 0.5625

0.3842 0.3888 0.3443

0.3888 0.3907 0.3865

0.7888 0.8070 0.5696

0.7968 0.8023 0.5847

0.7809 0.7790 0.5516

0.1288 / 0.42 0.1165 / 0.43 0.0184 / 0.18

b10

iid point spatial

0.3132 0.3132 0.3132

0.7028 0.7146 0.5168

0.7358 0.7476 0.4410

0.8316 0.8311 0.5218

0.7815 0.7761 0.5419

0.7843 0.7755 0.5357

0.3842 0.3888 0.3439

0.3888 0.3907 0.3569

0.7888 0.8070 0.5488

0.7968 0.8023 0.5411

0.7809 0.7790 0.5460

0.1288 / 0.42 0.1165 / 0.43 0.0050 / 0.12

closed∗

iid point spatial

0.3132 0.3132 0.3132

0.7028 0.7146 0.5602

0.7358 0.7476 0.5030

0.8316 0.8311 0.6029

0.7815 0.7761 0.5886

0.7843 0.7755 0.5850

0.3842 0.3888 0.3679

0.3888 0.3907 0.3713

0.7843 0.7755 0.5850

0.7843 0.7755 0.5850

0.7843 0.7755 0.5850

0.1288 / 0.42 0.1165 / 0.43 0.0427 / 0.18

drv∗

iid point spatial

0.3132 0.3132 0.3132

0.7028 0.7146 0.5602

0.7807 0.7850 0.5371

0.8427 0.8443 0.6207

0.7956 0.7965 0.6018

0.7996 0.7991 0.6136

0.4208 0.4282 0.4133

0.4229 0.4293 0.4263

0.7996 0.7991 0.6136

0.7996 0.7991 0.6136

0.7996 0.7991 0.6136

0.1399 / 0.46 0.1297 / 0.47 0.0606 / 0.22

k5 (K=4)

iid point spatial

0.2043 0.2043 0.2043

0.6701 0.6684 0.4608

0.6763 0.6887 0.4112

0.7904 0.7987 0.5262

0.7361 0.7288 0.5058

0.7361 0.7288 0.4912

0.2979 0.2956 0.2935

0.2979 0.2956 0.2912

0.7549 0.7616 0.5148

0.7448 0.7428 0.5071

0.7385 0.7355 0.5074

0.1203 / 0.47 0.1303 / 0.48 0.0655 / 0.20

under iid, regardless of how it is resolved — and by 0.0050 at 10◦ blocks, where it is unresolvable in principle. Note that this is a bound on accuracy consequences only; a misspecified sign can still be arbitrarily damaging out of distribution, which no in-distribution metric detects. Table V also exposes an artefact that deserves its own warning. “Identified” counts folds whose 95% clustered interval excludes zero, and the clusters are the spatial blocks. Refining blocks therefore manufactures identification: temperature is identified in 9/15 folds at 1◦ (1080 clusters), 5/15 at 2◦ (429), 4/15 at 5◦ (115) and 6/15 at 10◦ (40). Sign identifiability is thus partly a property of the analyst’s blocking choice, not of the physics. What is not an artefact is the effect of label resolution: at K = 4 every driver’s mean effect grows by a factor of 2–40 and precipitation becomes identified in 15/15 folds with mean effect −0.01144 against −0.00028 at K = 3.

Fig. 4. Fold-wise identification of driver signs, headline configuration: mean central-difference partial effect of the unconstrained model with 95% clustered intervals, three protocols × five folds per driver. Circles mark folds where the interval excludes zero (13 of 60); crosses mark non-identified signs. Temperature and wind sit above zero, relative humidity below; precipitation straddles zero, which is why its screened sign contradicts the textbook prior in 6 of 15 folds. A non-identified sign implies little headroom and hence, by Corollary 18, little accuracy at stake in choosing it.

F. Compliance is not price, empirically Constrained arms satisfy their own declaration exactly, VRown = 0.0000 in every arm, protocol and configuration; the harness asserts this and aborts otherwise, so it is a plumbing test rather than a result (Table III, ninth column). The informative comparison is between the unconstrained arms’ violation rates and the measured prices. The unconstrained MC model violates the textbook prior on the drivers at rate 0.4795 (iid), 0.4799 (point) and 0.4825 (spatial), computed over 40 873–45 071 active partial-effect evaluations; against the screened prior the rates are 0.4957, 0.4868 and 0.4968. Per driver under blocking they are 0.432 (temp), 0.492 (rh), 0.548 (wind) and 0.463 (precip), i.e.

indistinguishable from the 0.5 of a coin flip (Fig. 5). The measured price of enforcing that same prior is 0.0143–0.0555. A rate of 0.48 would ordinarily be reported as “the model has not learned the physics”; a price of 0.02 would ordinarily be reported as “the physics is free.” Both statements hold simultaneously here, exactly as Proposition 21 predicts, because the drivers are largely redundant given coordinates: the sign of a partial effect is unconstrained wherever the effect does not matter. The one place where the rates are informative is the comparison across declarations. MONO sc violates the textbook prior at 0.2068 (iid) and 0.2650 (spatial) — it must, since

11

TABLE V S IGN SCREENING ACROSS 15 TRAINING FOLDS (3 PROTOCOLS × 5 FOLDS ) PER CONFIGURATION . “AGREE ” COUNTS FOLDS WHERE THE SCREENED SIGN MATCHES THE TEXTBOOK PRIOR ; “ IDENT.” COUNTS FOLDS WHOSE 95% CLUSTERED INTERVAL EXCLUDES ZERO ; “n+ ” COUNTS FOLDS WITH A POSITIVE SCREENED SIGN ; “ EFF .” IS THE MEAN PARTIAL EFFECT ON E[y]. T EXTBOOK SIGNS ARE temp+, rh−, wind+, precip−, solar+. Config

Driver

agree

ident.

n+

eff.

ctx (headline)

temp rh wind precip

15/15 15/15 14/15 9/15

5/15 6/15 0/15 2/15

15/15 0/15 14/15 6/15

+0.00126 −0.00129 +0.00146 −0.00028

b1 (1◦ )

temp rh wind precip

15/15 15/15 14/15 9/15

9/15 8/15 3/15 4/15

15/15 0/15 14/15 6/15

+0.00129 −0.00130 +0.00153 −0.00048

b5 (5◦ )

temp rh wind precip

15/15 14/15 14/15 8/15

4/15 2/15 0/15 2/15

15/15 1/15 14/15 7/15

+0.00124 −0.00128 +0.00135 −0.00022

b10 (10◦ )

temp rh wind precip

15/15 15/15 15/15 8/15

6/15 2/15 0/15 1/15

15/15 0/15 15/15 7/15

+0.00121 −0.00133 +0.00123 −0.00032

drv (|S|=5)

temp rh wind precip solar

15/15 15/15 14/15 8/15 15/15

2/15 1/15 1/15 1/15 12/15

15/15 0/15 14/15 7/15 15/15

+0.00049 −0.00085 +0.00105 +0.00013 +0.00166

k5 (K=4)

temp rh wind precip

14/15 15/15 15/15 15/15

6/15 12/15 13/15 15/15

14/15 0/15 15/15 0/15

+0.00336 −0.00274 +0.00501 −0.01144

its own declaration differs from the textbook one in 6 of 15 folds — while MONO naive violates the screened prior at 0.1572 and 0.1904. PHYS sc, which has no shield to hide behind, disagrees with the textbook prior most strongly of all, at 0.3855 and 0.3462. Those numbers quantify how far apart the two priors are in deployment, and they are the only VR figures in this paper that are comparable across arms. Fig. 6 additionally distinguishes the certificate from the deployed pipeline. When the auxiliary channel is admitted the residualizer sits upstream of the constrained learner, so a nonzero VRe2e against the declared prior would reveal that the certificate does not survive preprocessing. No such leakage was observed: VRown = 0.0000 end-to-end for all three MIB arms in all three protocols of all five gate-open configurations. Residualizing the channel against a quadratic design in D does not reintroduce a sign violation on D. G. The auxiliary channel: gate, and bounded sign sensitivity Of the five candidate auxiliary channels the harness looks for, only insolation is present in this corpus; dewpoint, apparent temperature, pressure and cloud cover are absent from the join, and calendar harmonics are unavailable because no parsable date column exists. Insolation passes the exogeneity

Fig. 5. Violation rate of the unconstrained MC model against the textbook and the screened prior, per driver, under 2◦ spatial blocking (overall 0.4825 naive, 0.4968 screened, over 45 071 active evaluations). Every bar sits near the dashed 0.5 line, indicating a partial effect whose sign is essentially arbitrary in the unconstrained fit. The only visible gap is precip, whose screened rate (0.527) exceeds its naive rate (0.463) because the screened sign flips in some folds. By Proposition 21 these rates carry no implication for the price of enforcing the prior, which is 0.0179 in the same cell.

Fig. 6. Certificate versus deployed pipeline, headline configuration (c = insolation admitted). Left: violations of each model’s own declared sign pattern, evaluated end-to-end through the residualizer. Every bar in this panel is exactly 0.0000 — the constrained arms are feasible by construction of the hard split constraints and remain feasible after preprocessing — and the four unconstrained arms declare nothing, hence n/a. The panel is therefore flat by design; it is a passed plumbing test, not a missing plot. Right: violations against the common textbook prior, which is comparable across all arms. MONO naive and PHYS read 0.0000 there because the textbook prior is their own declaration.

gate decisively (R2 = 0.2646 against the threshold 0.90; residual relative standard deviation 0.8576 against the threshold 0.10) and was admitted in all 15 training folds of every gateauto configuration. Two regimes must then be distinguished, and only one of them is informative. In the gate-closed regime (closed, and drv where insolation has been promoted to a driver so no candidate remains), MIB, MIB fp and MIB fm are identical to MONO sc, and the sign-sensitivity spread is exactly 0.0000 in all three protocols. This is the predicted degeneracy of Lemma 8, verified bytewise on out-of-fold predictions by check C5; it is a successful test of the implementation and simultaneously a null result about the auxiliary channel. In the gate-open regime, Corollary 18 predicts that the spread over the three sign choices cannot exceed F1 (MIB) − F1 (MONO sc). Table VI reports both quantities. The bound holds in 8 of the 15 gate-open cells and is violated in 7, by up to 0.0295. The violations are not mysterious: each one decomposes exactly into the two elementary nesting inversions

12

TABLE VI AUXILIARY- CHANNEL ARMS IN THE FIVE GATE - OPEN CONFIGURATIONS . “ SPREAD ” IS max − min OVER {MIB, MIB FP, MIB FM}; “ BOUND ” IS F1 (MIB) − F1 (MONO SC) AS REQUIRED BY (7). C ELLS WHERE THE SPREAD EXCEEDS THE BOUND ARE MARKED ×; THE EXCESS EQUALS THE SUM OF THE R4 AND R3 INVERSIONS LISTED IN TABLE VII. G ATE - CLOSED CONFIGURATIONS ( CLOSED , DRV ) HAVE SPREAD = BOUND = 0.0000 EXACTLY AND ARE OMITTED . Config

Protocol

spread

bound

excess

ctx ctx ctx b1 b1 b1 b5 b5 b5 b10 b10 b10 k5 k5 k5

iid point spatial iid point spatial iid point spatial iid point spatial iid point spatial

0.0160 0.0280 0.0529 0.0160 0.0280 0.0399 0.0160 0.0280 0.0331 0.0160 0.0280 0.0077 0.0164 0.0261 0.0077

0.0045 0.0315 0.0275 0.0045 0.0315 0.0104 0.0045 0.0315 0.0071 0.0045 0.0315 0.0131 0.0188 0.0328 0.0236

× 0.0114 — × 0.0254 × 0.0114 — × 0.0295 × 0.0114 — × 0.0260 × 0.0114 — — — — —

of Proposition 22 that produce it,
excess = F1 (MIB fp) − F1 (MIB) + | {z }

R4
+ F1 (MONO sc) − F1 (MIB fm) + , | {z } R3

to four decimal places in all seven cells — for instance 0.0295 = 0.0199 + 0.0096 at b1/spatial and 0.0254 = 0.0034 + 0.0220 at ctx/spatial. Each summand is an inversion of a certified ordering and is therefore already accounted for in ŝ; the composite excess must not be added to ŝ separately, on pain of double counting. Interpreted through Corollary 20, the finding is that the sign imposed on an auxiliary channel changes the outcome by more than the channel is worth, which is only possible because the pipeline’s resolution is coarser than the channel’s value: F1 (MIB)−F1 (MONO sc) = 0.0045 under iid and 0.0275 under blocking, against ŝ = 0.0220. The honest conclusion about the auxiliary channel in this corpus is that its contribution is at or below the noise floor, and that its sign therefore cannot be chosen on accuracy grounds. H. Calibrating the slack We enumerate every certified comparison of Proposition 22 over the seven numerically distinct configurations and three protocols: 87 distinct R1 tests (after collapsing the 18 alias cells of the gate-closed runs), 42 R2, 63 R3, 42 R4 and 84 R5, for 318 orderings that the pipeline is obliged to satisfy. Eighteen are inverted; Table VII lists all of them. Hence ŝ = 0.0220 macro-F1 ,

(9)

attained at ctx/spatial/MIB fm versus MONO sc, i.e. in the headline configuration itself. Restricting to ablationdominance pairs alone gives the narrower floor ŝabl = 0.0154,

at b5/spatial/MIB fm versus GEO ONLY. No R2 inversion occurred: the loosened majority-constant ceiling of Remark 23 was never approached, with margins of +0.0307 to +0.1150. For comparison, the fold standard deviation at the maximizing cell is 0.0516 and the largest anywhere in the grid is 0.0889 (drv/spatial/MONO naive); the widest cluster bootstrap halfwidth among the constrained-arm dominance intervals is 0.0793 (b5/spatial/MONO sc versus GEO ONLY). So ŝ is 0.43× the local fold spread and 0.28× the widest interval — but 0.87× the half-width of the very quantity it invalidates, the spatial MONO sc price (0.0254). The point of ŝ is not that it is larger than a bootstrap interval; it is that it is derived from an ordering the experiment must satisfy, so it survives in cells where the bootstrap is silent. The most striking illustration is inversion #17–#18 in Table VII: at 10◦ blocks the constrained MONO naive beats its own unconstrained superclass MC by 0.0202 with interval [−0.0128, +0.0529] — a violation the ˆ = 0.0050 and nothing bootstrap cannot see, in a cell where ∆ was measurable to begin with. Consequences, stated as Corollary 20 prescribes: 1) Reporting floor. Four of the twenty-one price cells in Table III satisfy |P̂| < ŝ and carry no interpretation: MONO naive (0.0143), MONO sc (0.0179), MIB (−0.0096) and MIB fp (−0.0130), all under spatial blocking. All four would ordinarily be reported as “the constraint is nearly free,” and two of them as “the constraint helps.” Across the grid the same test disqualifies the entire spatial rung of b5 and b10. 2) A priori futility. Two of the twenty-one (configuration ˆ < ŝ: b5/spatial (∆ ˆ = 0.0184) × protocol) cells have ∆ ˆ and b10/spatial (∆ = 0.0050). Their ten shieldedarm price cells were uninformative before any constrained model was trained, so Algorithm 1 would have terminated them after two unconstrained fits. A third, ˆ = 0.0427, clears the floor by only a ctx/spatial with ∆ factor of 1.9. Two clarifications on interpretation. First, a negative measured price is not by itself an error: hard constraints act as regularizers and can improve generalization, and the fitted arm is not the minimizer of the out-of-fold statistic over its class, so the population ordering P ≥ 0 does not transfer to estimates. The largest such effect in our grid, b10/spatial/MIB beating MC by 0.0270 with interval [+0.0071, +0.0496] and p = 0.003, is excluded from (8) anyway because those two arms are not feature-matched (Remark 24). Second, and symmetrically, every inversion we do count admits the same regularization reading — the ablated class is smaller still, so it too can win by variance reduction. That is precisely why ŝ should be read as a resolution, i.e. the scale below which the pipeline cannot separate a genuine excess-risk gap from an estimation artefact, rather than as a bug count. I. The screening-leak probe returns a null The glob configuration differs from the headline run only in deciding the exogeneity gate on the full dataset rather than on each training fold. The resulting inflation is ∆F1 (MONO sc) = +0.0000,

∆F1 (MIB) = +0.0000,

13

TABLE VII C OMPLETE SLACK LEDGER : ALL 18 INVERTED ORDERINGS AMONG THE 318 CERTIFIED COMPARISONS OF THE GRID . “ RELATION ” REFERS TO P ROPOSITION 22. ŝ = 0.0220 IS THE MAXIMUM OF THE LAST COLUMN . I NTERVALS ARE THE PAIRED 95% CLUSTER BOOTSTRAP DIFFERENCE OF THE SUBCLASS AGAINST THE SUPERCLASS WHERE THE HARNESS INSTRUMENTS THAT REFERENCE ; R3 AND R4 ARE NOT INSTRUMENTED AGAINST MIB IN THE EMITTED ARTEFACTS AND ARE MARKED —. C ONFIGURATIONS B 2 AND GLOB REPRODUCE CTX BYTEWISE AND WOULD TRIPLE ROWS #1, #5, #9, #10; THEY ARE NOT LISTED . #

Config

Protocol

Comparison (subclass ⪯ superclass)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

ctx b1 b5 b10 ctx b1 b5 b10 ctx ctx b1 b1 b5 b5 b5 b5 b10 b10

iid iid iid iid iid iid iid iid spatial spatial spatial spatial spatial spatial spatial spatial spatial spatial

MIB fm ⪯ MONO sc MIB fm ⪯ MONO sc MIB fm ⪯ MONO sc MIB fm ⪯ MONO sc MIB fp ⪯ MIB MIB fp ⪯ MIB MIB fp ⪯ MIB MIB fp ⪯ MIB MIB fm ⪯ MONO sc MIB fp ⪯ MIB MIB fm ⪯ MONO sc MIB fp ⪯ MIB MONO sc ⪯ GEO ONLY MIB fm ⪯ GEO ONLY MIB fm ⪯ MONO sc MIB fp ⪯ MIB MONO naive ⪯ MC MONO sc ⪯ MC

exactly, in all three protocols; glob reproduces ctx bytewise on every arm. Two reasons combine. First, the gate is unanimous: insolation was admitted in 15/15 training folds, so pooling the gate decision changes nothing. Second, in this implementation the pooled-screening flag governs only the gate; the sign pattern σ sc is screened per training fold in both arms by construction. The probe therefore covers leakage in channel admission only, and not the more consequential error of screening the sign pattern itself on pooled data, which this design forecloses rather than measures. The b2 configuration, identical to ctx by construction, likewise reproduces it bytewise on all 33 model-protocol cells, confirming that the harness is deterministic under seed 0 and that no result in this paper depends on run-to-run variation in fitting. IX. D ISCUSSION A. What should be reported Any claim about the accuracy cost of a domain constraint ˆ ŝ) and an explicit should be accompanied by the triple (P̂, ∆, statement of the free feature set. The first number alone is uninterpretable: our headline price of 0.0473 becomes 0.3470 when the shield is removed and −0.0139 when the blocks are coarsened, with the physics held fixed. The second bounds it and is nearly free to compute, being the standard ablation. The third says whether either is above the noise; in our grid it disqualifies four of twenty-one headline price cells and ten cells before training. For hybrid architectures with a fixed mixing weight between learned and prescribed components [13], ∆ should be computed with respect to the covariates entering the prescribed component. For models with learnable physical parameters [14], the relevant S is the set of covariates whose functional form the physical law pins down. In both cases

relation

F1 sub

F1 super

inversion

R3 R3 R3 R3 R4 R4 R4 R4 R3 R4 R3 R4 R1 R1 R3 R4 R5 R5

0.7809 0.7809 0.7809 0.7809 0.7968 0.7968 0.7968 0.7968 0.5630 0.6159 0.5982 0.6381 0.5625 0.5516 0.5516 0.5847 0.5419 0.5357

0.7843 0.7843 0.7843 0.7843 0.7888 0.7888 0.7888 0.7888 0.5850 0.6125 0.6078 0.6182 0.5670 0.5670 0.5625 0.5696 0.5218 0.5218

0.0034 0.0034 0.0034 0.0034 0.0080 0.0080 0.0080 0.0080 0.0220 0.0034 0.0096 0.0199 0.0045 0.0154 0.0109 0.0151 0.0202 0.0139

the ablation is already implemented; only the comparison is missing. Two secondary recommendations follow from findings we did not anticipate. First, report the granularity of the bootstrap clustering alongside any claim that a driver’s effect sign is “identified,” since refining our blocks from 10◦ to 1◦ moves temperature from 6/15 to 9/15 identified folds without touching the data. Second, report the label resolution, since moving from K = 3 to K = 4 raised every driver’s mean partial effect and every normalized price while leaving the ceiling essentially fixed — the prior became measurably more costly purely because the target became finer. B. On reporting our own negative cells Four of our twenty-one headline price cells fall below ŝ and are reported as uninterpretable rather than as evidence that the prior is cheap. This is not a weakness of the instantiation but the intended use of Corollary 20: a screen that never fires on its author’s own grid provides no evidence that it fires at all. The same applies to the seven gate-open cells in which the spread exceeds its bound (Table VI): each excess decomposes exactly into two elementary inversions already counted in ŝ, which is what a resolution-limited measurement of a small quantity is expected to look like. C. Why constraints can still be worth imposing Nothing above argues against shape constraints. Theorem 9 caps the in-distribution price and says nothing about extrapolation — where the S-blind surrogate that made the constraint look cheap is precisely the component that fails, since a coordinate-based shield cannot transfer to unseen territory [24]. Our own numbers make this concrete: the

14

arm that recovers 92.9% of full-model accuracy under spatial blocking does so using latitude and longitude, which carry no mechanism whatsoever and cannot be evaluated outside the sampled domain. That is exactly the motivation for prescribing a physical sensitivity in the permafrost setting [13]: the prescribed term is not there to improve interpolation. Monotonicity additionally buys auditability and satisfies requirements no accuracy metric encodes [8]; our constrained arms carry a certificate that survives the deployed residualizer exactly (VRown = 0 end-to-end), which is a property of the artefact, not of its macro-F1 . Our claim is narrower: in-distribution accuracy cost is the wrong instrument for evaluating a physical prior, because its dynamic range is set by the ablation gap rather than by the prior. X. C ONCLUSION A shape constraint on a feature can never cost more than the feature is worth, because ignoring the feature is always an admissible way to satisfy the constraint. Operationally: a constrained model must never be beaten by its own ablation. This elementary bound turns the widely reported “cost of physics” into a quantity whose scale is fixed by the ablation gap, and hence by the free feature set and the validation protocol rather than by the physics. On an ordinal wildfireseverity task the same monotone prior costs 0.0473 macroF1 with coordinates left free and 0.3470 without them; the certified ceiling that bounds both falls from 0.1288 to 0.0050 as the spatial blocks coarsen, taking every price with it; the unconstrained model violates the prior at rate 0.48 while enforcing it costs 0.02; and the pipeline’s own inversions of orderings it is obliged to satisfy calibrate a resolution of ŝ = 0.0220 that renders four of twenty-one headline price cells uninterpretable and ten cells of the grid unidentifiable before a single constrained model is trained. We recommend replacing the single reported cost with the triple (price, ceiling, slack) plus the free feature set, and screening constraint-cost experiments with two unconstrained fits before training any constrained model. R EFERENCES [1] A. Ben-David, “Monotonicity maintenance in information-theoretic machine learning algorithms,” Machine Learning, vol. 19, no. 1, pp. 29–43, 1995. [2] J. Sill, “Monotonic networks,” in Advances in Neural Information Processing Systems, vol. 10, 1998, pp. 661–667. [3] H. Daniels and M. Velikova, “Monotone and partially monotone neural networks,” IEEE Trans. Neural Netw., vol. 21, no. 6, pp. 906–917, 2010. [4] M. Gupta, A. Cotter, J. Pfeifer, K. Voevodski, K. Canini, A. Mangylov, W. Moczydlowski, and A. van Esbroeck, “Monotonic calibrated interpolated look-up tables,” J. Mach. Learn. Res., vol. 17, no. 109, pp. 1–47, 2016. [5] S. You, D. Ding, K. Canini, J. Pfeifer, and M. Gupta, “Deep lattice networks and partial monotonic functions,” in Advances in Neural Information Processing Systems, 2017, pp. 2981–2989. [6] X. Liu, X. Han, N. Zhang, and Q. Liu, “Certified monotonic neural networks,” in Advances in Neural Information Processing Systems, 2020, pp. 15427–15438. [7] J.-R. Cano, P. A. Gutiérrez, B. Krawczyk, M. Woźniak, and S. Garcı́a, “Monotonic classification: An overview on algorithms, performance measures and data sets,” Neurocomputing, vol. 341, pp. 168–182, 2019. [8] S. Wang and M. Gupta, “Deontological ethics by monotonicity shape constraints,” in Proc. Int. Conf. Artificial Intelligence and Statistics, 2020, pp. 2043–2054.

[9] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” J. Comput. Phys., vol. 378, pp. 686–707, 2019. [10] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,” Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021. [11] J. Willard, X. Jia, S. Xu, M. Steinbach, and V. Kumar, “Integrating scientific knowledge with machine learning for engineering and environmental systems,” ACM Computing Surveys, vol. 55, no. 4, pp. 1–37, 2022. [12] A. S. Krishnapriyan, A. Gholami, S. Zhe, R. M. Kirby, and M. W. Mahoney, “Characterizing possible failure modes in physics-informed neural networks,” in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 26548–26560. [13] B. Kriuk, “Hybrid physics-ML framework for pan-Arctic permafrost infrastructure risk at record 2.9-million observation scale,” in Proc. 2026 IEEE Mediterranean and Middle-East Geoscience and Remote Sensing Symposium (M2GARSS), Marrakech, Morocco, 2026, pp. 214–218. [14] B. Kriuk and F. Kriuk, “POSEIDON: Physics-optimized seismic energy inference and detection operating network,” arXiv preprint arXiv:2601.02264. [15] B. Kriuk, “Advancing Eurasia fire understanding through machine learning techniques,” arXiv preprint arXiv:2502.17023, Feb. 2025. [16] B. Kriuk, K. Sulamanidze, and F. Kriuk, “ELENA: Epigenetic learning through evolved neural adaptation,” Evolutionary Intelligence, vol. 18, art. no. 50, 2025, doi:10.1007/s12065-025-01034-w. [17] B. Kriuk, “MorphBoost: Self-organizing universal gradient boosting with adaptive tree morphing,” arXiv preprint arXiv:2511.13234, Nov. 2025. [18] B. Kriuk, L. Ng, and Z. Al Hossain, “DeepSupp: Attention-driven correlation pattern analysis for dynamic time series support and resistance levels identification,” arXiv preprint arXiv:2507.01971, Jul. 2025. [19] B. Kriuk and F. Kriuk, “ORCA: Online regime correlation analyzer,” arXiv preprint arXiv:2604.17251. [20] B. Kriuk, “Artificial adaptive intelligence: The missing stage between narrow and general intelligence,” arXiv preprint arXiv:2605.16844. [21] D. R. Roberts, V. Bahn, S. Ciuti, M. S. Boyce, J. Elith, G. GuilleraArroita, S. Hauenstein, J. J. Lahoz-Monfort, B. Schröder, W. Thuiller, D. I. Warton, B. A. Wintle, F. Hartig, and C. F. Dormann, “Crossvalidation strategies for data with temporal, spatial, hierarchical, or phylogenetic structure,” Ecography, vol. 40, no. 8, pp. 913–929, 2017. [22] P. Ploton et al., “Spatial validation reveals poor predictive performance of large-scale ecological mapping models,” Nature Communications, vol. 11, art. no. 4540, 2020. [23] R. Valavi, J. Elith, J. J. Lahoz-Monfort, and G. Guillera-Arroita, “blockCV: An R package for generating spatially or environmentally separated folds for k-fold cross-validation of species distribution models,” Methods in Ecology and Evolution, vol. 10, no. 2, pp. 225–232, 2019. [24] H. Meyer and E. Pebesma, “Predicting into unknown space? Estimating the area of applicability of spatial prediction models,” Methods in Ecology and Evolution, vol. 12, no. 9, pp. 1620–1633, 2021. [25] H. Meyer, C. Reudenbach, S. Wöllauer, and T. Nauss, “Importance of spatial predictor variable selection in machine learning applications — moving from data reproduction to spatial prediction,” Ecological Modelling, vol. 411, art. no. 108815, 2019. [26] R. Geirhos, J.-H. Jacobsen, C. Michaelis, R. Zemel, W. Brendel, M. Bethge, and F. A. Wichmann, “Shortcut learning in deep neural networks,” Nature Machine Intelligence, vol. 2, no. 11, pp. 665–673, 2020. [27] S. Kaufman, S. Rosset, C. Perlich, and O. Stitelman, “Leakage in data mining: Formulation, detection, and avoidance,” ACM Trans. Knowl. Discov. Data, vol. 6, no. 4, art. no. 15, 2012. [28] T. Chen and C. Guestrin, “XGBoost: A scalable tree boosting system,” in Proc. 22nd ACM SIGKDD Int. Conf. Knowledge Discovery and Data Mining, 2016, pp. 785–794. [29] G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T.Y. Liu, “LightGBM: A highly efficient gradient boosting decision tree,” in Advances in Neural Information Processing Systems, 2017, pp. 3146– 3154. [30] J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” Annals of Statistics, vol. 29, no. 5, pp. 1189–1232, 2001. [31] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap. New York, NY, USA: Chapman & Hall, 1993. [32] C. Nadeau and Y. Bengio, “Inference for the generalization error,” Machine Learning, vol. 52, no. 3, pp. 239–281, 2003.

15

[33] X. Bouthillier et al., “Accounting for variance in machine learning benchmarks,” in Proc. Machine Learning and Systems (MLSys), vol. 3, 2021, pp. 747–769. [34] J. Demšar, “Statistical comparisons of classifiers over multiple data sets,” J. Mach. Learn. Res., vol. 7, pp. 1–30, 2006. [35] V. N. Vapnik, Statistical Learning Theory. New York, NY, USA: Wiley, 1998.

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