Residual Algebra for Representation-Preserving Learning

Residual Algebra for Representation-Preserving Learning

Yao Wu wuyao@westlake.edu.cn

arXiv:2608.07349v1 [cs.LG] 7 Aug 2026

Abstract Learning from heterogeneous representations is usually reduced to feature concatenation, which erases which representation produced an error. We instead algebraize the residual: a representation is a typed object that owns both a coordinate system and the residual it leaves unresolved, and learning is an ordered composition of operators that preserve or deliberately erase that type. Fold realizes the objects as point-in-time conditional-mean fields on 10×10 rank grids. FPRC-PQ realizes the algebra as relax–aggregate–close: each field is relaxed by a correction fitted to its own residual in its own coordinates; corrected fields meet at a fixed mean that is the sole identity-erasure boundary; and a shared learner closes only the aggregate’s fresh residual. The composition telescopes exactly into representation, local residual estimate, and residual-of-residual estimate. Its aggregate is a learned control-variate interface with population variance reduction, while refitting the closer along perturbations of the backbone yields firstorder coupled-path mean orthogonality. As an analytical extension, a reflective rumination operator reads the displacement of a global reconstruction from the aggregate anchor, reflects it, and fixes its gain by a unique orthogonal projection rather than return-tuned grid search. On 3.67M Chinese A-share stock-day observations (2023–2026) under a frozen point-in-time protocol, the evaluated base algebra raises net-of-cost return from 13.52% to 19.10% and Sharpe from 1.42 to 2.09. Matched-capacity, unified-residual, identity-free two-stage, and pairwiseonly controls all trail it. The gain is therefore not explained by more features or more trees, but by making residual ownership and composition explicit while representation identity is still available.

1

Introduction

Cross-sectional return prediction is a canonical hard tabular problem: several thousand entities are re-ranked daily from a few dozen noisy covariates whose joint distribution drifts continuously [15, 9, 11]. On such data, gradient-boosted decision trees remain a reference model class [7, 14, 26, 31], while deep tabular alternatives often only match them [12, 17, 4]. Yet both are commonly asked to learn from one flat feature vector. Concatenation preserves values but destroys provenance: after heterogeneous representations are pooled, the learner no longer knows which representation generated which error. The central question of this paper is therefore: Core question (residual algebra). Can representation-specific errors be made composable without first erasing the identity of the representation that owns them? Simply growing one learner conflates heterogeneous error populations; simply averaging experts discards their identities [18, 32]. Answering it requires two ingredients: explicit representation objects and operators with controlled type boundaries. Objects. An interaction must become a stable object with its own coordinates and residual, rather than an implicit collection of tree paths. 1

Operators. Local correction must act before identity erasure; aggregation must expose an explicit boundary; and any shared correction must target only the fresh residual left by the preceding operators. We call this construction residual algebra: residuals are typed by their generating representations, and valid learning stages are defined by their domains, targets, and composition order. Representation layer—Fold (§2). Fold maps a factor pair to a 10×10 grid of daily rank deciles. Each cell is a discrete interaction state, and its point-in-time conditional mean defines a factor field Fm . In the population, Fm is a projection onto the information carried by that grid; r − Fm is therefore exactly what the representation leaves unresolved. Three heterogeneous fields provide distinct local coordinate systems and three typed residual populations. Residual algebra—FPRC-PQ (§3). Field-Preserving Residual Closure instantiates three ordered operators. Relaxation maps every (Fm , r − Fm ) to a corrected but still typed field Hm ; aggregation maps the product of typed fields to one shared backbone B and is the only identity-erasure step; closure adds a model of r − B without rereading local state. The algebra exposes exact telescoping, a control-variate correction, and a coupled-path orthogonality of the final mean. Its value is structural rather than terminological: changing the residual owner, operator order, information boundary, or capacity placement produces a different model that can be tested directly. Analytical extension—reflective rumination (§3.1). A global reconstruction F2 = T (F ) defines a displacement q = F − F2 . Reflecting this displacement about the aggregate anchor and imposing orthogonality of the remaining fresh residual uniquely determines an analytical rumination gain λ⋆ . The construction assigns the q direction to second-stage rumination and its orthogonal complement to terminal closure. It is a mechanism-level extension, not an additional result row in the present evaluation. Evidence discipline. All architectural choices, factor selections, and hyperparameters were frozen on data through 2022; evaluation covers 2023–2026 under annual point-in-time refits, realistic costs, and paired 21-day block-bootstrap inference [22, 24]. Five operator audits and four matched structural controls isolate residual ownership from total tree capacity, pairwise interaction class, and generic staging. Eight extra meta-information channels also fail, several significantly, showing that side information cannot repair an incorrectly placed correction stage. Our contribution is a representation-learning principle with an executable form: algebraize the residual before optimizing it. Fold supplies explicit representation objects; FPRC supplies typed operators and a controlled identity-erasure boundary; reflective rumination supplies an analytically gauge-fixed allocation of a reconstruction direction; and the frozen audits test neighboring base algebras rather than presenting only a winning pipeline. The resulting account separates why the system works from which market variables instantiate it.

2

Fold: from continuous factor pairs to interaction states

Construction. Let Ut be the tradable universe on day t and let xA,i,t , xB,i,t be two continuous factors. Within each day, both factors are rank-transformed and binned into deciles,     ai,t = 10 rankpctt (xA,i,t ) − 1, bi,t = 10 rankpctt (xB,i,t ) − 1, (1) yielding a discrete interaction state si,t = (ai,t , bi,t ) ∈ {0, . . . , 9}2 . A factor field is the point-intime conditional-mean map   b PIT ri,t si,t = s , Fm (s) = E (2) where ri,t is the 5-day forward return and the estimate uses only history observable at deployment time (annual refresh, sample weights with a 252-day half-life). Equivalently, Fold is a double sort read as a one-hot encoding (Fig. 1): with es ∈ {0, 1}100 the indicator of the active cell, the field ⊤ is the linear readout Fm (s) = wm es whose weight table holds the PIT cell means. The interaction is therefore a first-class object rather than a path carved implicitly through a tree ensemble. With ∼4,000 stocks per day, every cell receives ∼40 fresh observations daily, so equation 2 is dense and statistically stable while remaining a 100-entry table that can be plotted, audited, and diffed across years. 2

a

daily cross-section

b double sort → state (a, b)

c one-hot e s

d

s = 10a + b rankt, bin

factor xB

decile b = 3

activate

read out

field: linear readout 9 20 10

6

0

⋮

3

−10 −20

0

factor xA (continuous)

decile a = 6

{0, 1}100

0

3

6

9

⊤ Fm(s) = wm es, wm = PIT cell means

Figure 1: Fold is a double sort read as a one-hot encoding. (a) A daily cross-section in continuous factor space; one stock highlighted. (b) Both factors are ranked within the day and cut into deciles; the rank pair selects exactly one of 10×10 interaction states (bands: the stock’s column decile a=6 and row decile b=3). (c) The induced indicator es ∈ {0, 1}100 has a single active entry at ⊤ s = 10a + b. (d) The field is the linear readout Fm (s) = wm es whose weight table holds PIT cell means of forward returns (surface illustrative). Three design properties matter downstream. (P1) Regime robustness: states are within-day order statistics, invariant to level and scale shifts of the raw factors [9]. (P2) Interpretability: each cell is a human-readable conjunction (“momentum decile 6, reversal decile 3”), and Fm is an explicit interaction surface rather than a path through a tree ensemble [25, 5]. (P3) Portability: the construction consumes any ordered feature pair; nothing in the residual algebra of §3 references financial semantics. The grid is therefore a concrete carrier of representation identity, not an assumption about a particular factor family. Heterogeneous fields. We instantiate M =3 fields with distinct economic mechanisms: momentum×short-term-reversal (F1 ), value×momentum (F2 ), and breakout×volume-confirmation (F3 ) (App. C). Their signals correlate only moderately (mean pairwise 0.54), which is precisely why identity preservation—not blending—becomes the central architectural question. Purified auxiliary axes. Two kinds of axes assist local correction: one shared apex P (turnover rank) and a selected collection of slow fundamental axes Q. Let KQ denote the number of purified Q axes; the evaluated instance uses KQ = 4 (volatility, profit growth, cash-flow quality, and profit stability). Their purification is deliberately asymmetric. Let Xpar = {a1 , b1 , . . . , aM , bM } denote the union of parent coordinates (duplicate factor identities are represented once). The single apex coordinate used by every expert is h i b t (P | Xpar ) . P⋆⊥ = RankPctt P − E (3) In parallel, three temporary field-local views of the same raw apex are formed, h i local b t (P | am , bm ) , Pm = RankPctt P − E m = 1, . . . , M,

(4)

but these coordinates are filters, not expert inputs. They appear only while purifying each candidate Qk against all parent and apex semantics, h
i local local b Q⊥ , . . . , PM , k = 1, . . . , KQ . (5) k = RankPctt Qk − Et Qk | Xpar , P1 b t is a same-date cross-sectional OLS projection; the implementation daily-ranks the raw Here E coordinate before the projection and ranks the residual again afterward. Thus it uses no forward local label and is point-in-time by construction. Once Q⊥ coordinates exit the 1:KQ is emitted, all Pm dataflow. Every base FPRC-PQ expert receives exactly h i local local Xm = am , bm , P⋆⊥ , Q⊥ dim Xm = KQ + 3, (6) 1:KQ , so the three experts share one definition of the apex while retaining different response functions. The temporary local views let Q be deconfounded against all three apex semantics without inflating 3

each learner from KQ + 3 to KQ + 5 coordinates. The Q candidates were scored by pre-2023 residual predictive ability with sign-consistency screening, frozen at 2022-12-31 (App. C); no selection touched the evaluation window. Projection view: what a field leaves unresolved. Let Gm = σ(sm ) be the information carried by field m’s grid and let Km = L2 (Gm ), equivalently the span of its 100 cell indicators inside H = L2 (Ω, F, P). The population field is the orthogonal projection Πm r = E[r | Gm ]; the PIT table Fm is its finite, shrinkage-regularized estimate. Hence the population residual (I − Πm )r is orthogonal to every function of the grid state alone. A useful
correction must therefore enlarge the local information set, here to Lm = L2 σ(sm , P⋆⊥ , Q⊥ ) ⊃ Km , while retaining which Km produced the residual. Because the field subspaces overlap and are not mutually orthogonal, their mean is not a joint projection; it is an explicit interface across which identity is deliberately erased. This projection-to-residual transition motivates the architecture that follows.

3

Residual algebra: field-preserving relaxation and closure

The evaluated instance, FPRC-PQ, seats the purified P and Q axes inside each field’s P local correc1 tion. Two naive extremes bracket its design space. Pooling the fields, F̄ = M m Fm , discards which field produced which opinion and earns 13.5% net over the evaluation window (Table 1). A single shared corrector on top of F̄ treats three heterogeneous residual populations as one and reaches 18.0%. FPRC-PQ instead makes the residual an algebraic object and composes three learned nonlinear operators—a field-preserving relaxation RPQ , an identity-erasing aggregation interface A, and a fresh-residual closure C (Fig. 2). Their type signatures are RPQ :

M Y

Km −→

m=1

M Y

Lm ,

A:

M Y

Lm −→ H,

m=1

m=1

(7)

C : H −→ H. The product type before A retains the field index; the scalar backbone after A does not. This is the controlled identity-erasure boundary. Composition is therefore part of the model definition, not notation that can be reordered freely:
Sb = C ◦ A ◦ RPQ (F), F = (F1 , . . . , FM ). (8) Expanding the operators makes their residual targets, information domains, and routing boundaries explicit:
(relax) RPQ,m (Fm ) = Fm + gm am , bm , P⋆⊥ , Q⊥ =: Hm , (9) 1:KQ ; r − Fm P M 1 (10) (aggregate) A(H1 , . . . , HM ) = M m=1 Hm =: BPQ ,
b (close) C(BPQ ) = BPQ + G Xpar ; r − BPQ =: S, (11) where gm , G are gradient-boosted regressors with hyperparameters frozen before evaluation, and Xpar = {a1 , b1 , . . . , aM , bM }

(12)

collects the parent factors—the raw axes from which each field is constructed. Notation g( · ; e) means “fit to target e”. Residual telescoping: what each stage actually fits. The routing P has an exact algebra. Per1 field fresh residuals average to the aggregate fresh residual, M m (r − Fm ) = r − F̄ with P 1 c F̄ := M m Fm , so—writing (·) for the fitted estimate and defining the aggregate local correction P 1 \ (r\ − F̄ ) := M m (r − Fm )—aggregation collapses the corrected fields into representation-plusresidual, BPQ = F̄ + (r\ − F̄ ). Feeding this to the closure, F̄ and the local estimate cancel inside the shared target, leaving a residual of the residual:
\ Sb = BPQ + G Xpar ; r − BPQ = F̄ + ρb + (ρ − ρb), ρ := r − F̄ , ρb := (r\ − F̄ ). (13) The shared learner never re-sees the fields or their corrections: its target r − BPQ = ρ − ρb is exactly what the local stage failed to remove. The fitted predictor and its terminal error can therefore be 4

written as stage-owned additive terms, r=

F̄ |{z}

representation

+

ρb |{z}

local closure

\ + (ρ − ρb) + | {z } shared closure

e2 , |{z}

\ e2 = (ρ − ρb) − (ρ − ρb),

(14)

final residual

a twice-applied “leftover” operator on ρ: capacity is partitioned across stages, not shared. This is stagewise boosting [10] with typed intermediate targets; post-hoc probes find no recoverable out-ofsample structure in e2 within the tested local and shared hypothesis classes (App. F). Control-variate aggregation and bias correction. Equation (10) is not merely an ensemble average. Because Hm = Fm + (r\ − Fm ), it has the exact interface M

BPQ =

1 X Hm = F̄ + (r\ − F̄ ) = F̄ + ρb, M m=1

(15)

where F̄ is a stable low-dimensional anchor and ρb is a regression adjustment. This is the learned analogue of a control variate [27]: the anchor carries the predictable level and the local residual fit removes variation left around it. In the population idealization ρb = ΠL ρ, the L2 projection of ρ onto the aggregate local correction class, the Pythagorean identity gives       (16) E (r − BPQ )2 = E (ρ − ΠL ρ)2 = E[ρ2 ] − E (ΠL ρ)2 ≤ E[ρ2 ]. Thus the population operator weakly reduces residual second moment relative to F̄ ; finite regularized GBDTs approximate rather than guarantee this projection identity. Neyman-style orthogonalization. For a backbone B, define cB (x) := E[r − B | Xpar = x] and SB := B + cB (Xpar ), so E[SB ] = E[r]. Along a perturbation Bt = B + th with the closer refit to the corresponding fresh residual, d = E[h] − E[E(h | Xpar )] = 0. (17) E[Bt + cBt (Xpar )] dt t=0 The backbone perturbation is cancelled by the induced change in the residual target. This is Neyman orthogonality only along the coupled refit path [8]; it does not cover independent perturbations of both learned functions or imply semiparametric efficiency. 3.1

Reflective rumination via orthogonal gauge fixing

The residual algebra admits a second, analytical allocation layer. Let F := F̄ be the lowdimensional aggregate anchor and retain the local aggregate correction ρb from Eq. (13). We call R1 := ρb ≈ r − F (18) the first-stage ruminant: it is the locally digested component that can enter an additive closure directly. Now let a shared-coordinate learner produce a smooth global reconstruction F2 := T (F ) = Gall,1 (Xsh ; F ), q := F − F2 , (19) where Xsh = Xpar in the present parents-only interface. The vector q is the rumination vector: it records how the reconstruction displaced the original anchor. The unit reflective update is F2 + R2 = F. (20) R2 := F + q = 2F − F2 = (2I − T )F, 2 Thus R2 is the mirror image of F2 about F . Importantly, q = F −F2 is only the reflective increment; the complete second-stage ruminant retains the anchor F . In the ideal unit-reflection case R1 = r − F , the additive ownership chain closes exactly: (r − F ) + (2F − F2 ) + (F2 − F ) = r. (21) | {z } | {z } | {z } R1

R2

terminal closure

More generally, anchor conservation gives a one-parameter affine family. For an operator aF − bF2 to return F whenever F2 = F , its coefficients must satisfy a − b = 1. Writing a = 1 + λ and b = λ yields   (λ) (λ) R(λ) Bλ := R1 + R2 = BPQ + λq. rum (F ) = R2 := F + λ(F − F2 ) = (1 + λ)I − λT F, (22) With e0 := r − ρb − F = r − BPQ , the fresh residual after rumination is eλ := r − Bλ = e0 − λq. The remaining scalar freedom can therefore be fixed without a grid search. 5

Proposition 3.1 (Optimal reflective rumination via orthogonal gauge fixing) Assume q ̸= 0 and take inner products over the point-in-time fitting sample. Requiring the fresh residual left for the downstream learner to have zero linear alignment with the rumination vector, ⟨eλ , q⟩ = 0, uniquely fixes the analytical rumination gain as λ⋆ =

⟨e0 , q⟩ . ⟨q, q⟩

(23)

For centered, non-degenerate variables this is λ⋆ = Cov(e0 , q)/ Var(q). Moreover, with Pq := (q ⊗ q)/⟨q, q⟩, λ⋆ q = Pq e0 , eλ⋆ = (I − Pq )e0 . (24) ⋆ If q = 0, define λ = 0 and recover the base backbone. Proof. Substituting eλ = e0 − λq into the orthogonality condition gives ⟨e0 , q⟩ − λ⟨q, q⟩ = 0; uniqueness and the projection identities follow immediately. □ The optimized backbone and its terminal target are therefore B ⋆ = BPQ + Projq (r − BPQ ),

r − B ⋆ = (I − Pq )(r − BPQ ),

(25)

which makes the residual ownership explicit: second-stage rumination owns span(q), while terminal closure owns its orthogonal complement. A parents-only closer preserves the original information boundary,
rbrum = B ⋆ + Gall,2 Xsh ; r − B ⋆ , Xsh = Xpar . (26) ⋆ The signed value of λ diagnoses both direction and scale: values above one amplify the unit reflection, values in (0, 1) attenuate it, one recovers Eq. (20), and a negative value reverses the proposed direction. Its mechanism strength is measured without another parameter by the rumination efficiency ⟨e0 , q⟩2 ∥e0 ∥2 − ∥eλ⋆ ∥2 = , (27) ηrum := 2 ∥e0 ∥ ∥e0 ∥2 ∥q∥2 equal to Corr2 (e0 , q) after centering. In short, anchor conservation supplies the affine family and orthogonal gauge fixing selects its unique member. This construction is point-in-time only if T , the inner products in Eq. (23), and the downstream learner are fitted on past or out-of-fold observations available at deployment, then frozen before prediction. Using evaluation labels to estimate λ⋆ would be leakage. Accordingly, reflective rumination is a theoretical extension in this paper: all reported returns evaluate base FPRC-PQ, equivalently the λ = 0 member of Eq. (22). 3.2

Operator principles and scope

Four algebraic principles are encoded: (D1) Typed local correction. Each gm sees only its own field’s residual, expressed in that field’s coordinates (am , bm ) plus purified axes. Corrections therefore cannot leak across mechanisms: an error of the value field is repaired as a value-field error. (D2) Controlled identity erasure. Fields enter (9)–(10) as unit-coefficient offsets; no gate or learned blend re-weights them. Their identities remain intact through relaxation and are erased exactly once, by the explicit mean. Aggregate drift therefore cannot silently reassign responsibility between mechanisms. (D3) Fresh-residual closure. The shared stage fits r − BPQ —the error remaining after local semantics are exhausted—from parent factors only. It neither re-reads local states nor the aggregate’s own rank (both were tested and rejected; §5.2). (D4) Staged information compression. Semantically local information (grid coordinates, purified axes) is consumed at the local stage; only generic, field-agnostic coordinates remain for closure. Capacity is placed where the corresponding error population lives, not where it is easiest to add. Scope of the projection view. The Hilbert-space argument belongs to Fold’s population interface, not to the fitted ensemble as an exact orthogonal decomposition. Empirical fields and both correctors 6

Residual algebra: one lens drained per operator each operator fits only what its predecessor left, in coordinates its predecessor never saw



parents only

∙



∙



Fold

typed ×3

grid state

unit weights

e2

residual seen in:

ρ − ρ̂

ρ

r

grid cells sm = (am, bm)

typed local axes P ⟂, Q ⟂ × state

each field corrected in its own coordinates — no cross-mechanism leakage

shared parents Xpar (5 factors)

after 

after →

ρ = r − F;̄ ρ:̂ typed local corrections, merged by  at unit weight

recoverable structure in this lens

after Fold

̂

the label

e2 is empty in every lens

2 post-hoc probes on r − S: Roos = −2.5×10−4 (typed-local), −1.3×10−3 (shared)

drained: nothing left to fit

schematic tiles; common colour scale within each row

Figure 2: Residual structure is routed through typed operators. The residual is sequentially consumed by three coordinate systems: field grids sm (top), typed local axes (middle), and shared parents Xpar (bottom), following Eq. 14. Fold removes grid-level structure; RPQ corrects each field in its own coordinates; A aggregates the corrected fields; and C operates only on the remaining aggregate residual. The staircase is schematic rather than an orthogonal decomposition. The terminal residual e2 contains no detectable out-of-sample signal within tested local and shared probe classes 2 (Roos ≤ 0; App. F). are finite, regularized learners; the field subspaces overlap; and A is not a projection onto their sum. Equations (9)–(11) should therefore be read as an ordered, finite-capacity routing hypothesis whose targets, boundaries, and order are tested empirically in §5.2. FPRC-PQ is deliberately not defended by a larger capacity budget. In the matched control, one direct GBDT receives exactly the sum of the three local learners’ and the shared closer’s trees in each deployment year; it still trails the typed composition (App. E). The architecture was not designed by intuition alone: five preregistered audits, each freezing a question and a decision rule before fitting, selected (9)–(11) over its neighbors (§5.2, Table 2).

4

Experimental setup

Data and protocol. We evaluate on the Chinese A-share market [23]: 844 trading days from 202301 to 2026-07, median 4,380 stocks per day, for a strict common book of 3,666,920 stock-day rows (a wider “available” book of 3,690,036 rows serves as a coverage-sensitivity check; conclusions must agree on both). The label is the 5-day forward return. Deployment is annual point-in-time: models for year Y are fit on data observable before Y with a purge gap covering label observability [24]. Every selection—fields, apex P , axes Q, learner settings, and architecture—was frozen on data through 2022-12-31; no 2023–2026 label influenced any choice. Learners and execution. gm and G are XGBoost regressors with frozen settings (depth 5, learning rate 0.035, ≤600 trees with early stopping on the preceding calendar year, min-child-weight 2000, λ=10). Signals are executed long-only (dropping the bottom quintile), with 21-day phased holdings 7

Table 1: Architecture ladder, strict book (3.67M rows, 2023–2026). Net is cumulative net-of-cost return; ∆ is versus the pooled-field baseline (Mean of three Fold fields). All rows share the same data, 10/15 bp cost model, and frozen learner hyperparameters. The top block gives two samefeatures baselines: a single gradient-boosted learner fit to the full pooled feature set Xfull (all factors and their derived inputs), either directly on the label (G(Xfull ; r)) or on the pooled-field residual (F̄ + G(Xfull ; r − F̄ ))—the same inputs, without Fold state or typed composition. The Operator column of the remaining rows composes Eqs. (9)–(11): A is equal-mean aggregation, R• a typed local relaxation seated on axes • (P or PQ), C a parents-only fresh-residual closure, and CQ a closure that incorrectly rereads Q. ICIR Sharpe Gross% Cost% Net% ∆ (pp)

Model

Operator

Direct GBDT, same features Residual GBDT, same features

G(Xfull ; r) 4.74 F̄ + G(Xfull ; r − F̄ ) 4.94

Mean of three Fold fields A Shared correction only (on F̄ ) C ◦A Typed local-P only A◦RP Typed local-PQ only A◦RPQ Local-P → shared-Q (predecessor) CQ ◦A◦RP FPRC-PQ C ◦A◦RPQ + shared-Q reread CQ ◦A◦RPQ

4.45 4.90 5.09 5.08 5.03 5.01 5.04

1.67 1.79

18.64 19.68

2.72 2.70

15.93 +2.41 16.98 +3.46

1.42 1.94 1.94 2.00 2.03 2.09 2.07

16.09 20.63 20.45 20.73 21.30 21.67 21.37

2.57 2.62 2.43 2.44 2.59 2.57 2.57

13.52 18.01 18.02 18.29 18.71 19.10 18.80

— +4.49 +4.50 +4.78 +5.19 +5.58 +5.28

and blended 10/15 bp transaction costs; we report cumulative net-of-cost return over the 42-month window, annualized Sharpe on daily net returns, and validation ICIR. Status of reflective rumination. The headline experiments use the base backbone BPQ (λ = 0); no reported return is attributed to the analytical family in §3.1. A future deployment must estimate T and λ⋆ strictly on past or out-of-fold data and freeze both before evaluation. Inference. All comparisons are paired on identical books and identical dates, with 21-day block bootstrap (10,000 draws) [22, 28]; we report observed deltas in percentage points (pp), Pr[∆ > 0], and 95% intervals. Because multiple preregistered variants exist, no method is promoted merely for ranking first [16, 3]; verdicts follow the registered contrasts. Reproduction audits require every frozen signal to be re-derivable bit-exactly (max difference 0; App. I).

5

Results

5.1

The architecture ladder: placement beats pooling and beats size

Table 1 and Fig. 3 trace the ladder from raw representation to full architecture on the strict book. Three comparisons isolate the design principles. Capacity control. On the same 3,666,920-row strict book, FPRC-PQ earns 19.098% net with Sharpe 2.089. A direct learner given exactly the combined annual tree budget of the three local learners and the shared closer reaches only 16.970% and Sharpe 1.963, trailing by 2.128 pp and losing in all four years. Extra direct capacity raises Sharpe relative to the ordinary direct control (1.882) but does not close the structural gap (App. E). Identity control. Collapsing the three fields into one concatenated representation and fitting a unified residual gives 16.824% net, 2.273 pp below FPRC-PQ in 4/4 years (95% CI [+0.123, +4.453] pp for FPRC minus control). A fresh but identity-free two-stage booster also fails (17.408%, −1.690 pp, 0/4 years). Thus two-stage boosting alone is insufficient: the intermediate target must retain which field produced it, and each correction must be fitted to that field’s own residual. Interaction control. An explicit pairwise-only histogram GBDT—a GA2 M-class control, not an EBM package benchmark—reaches 16.681% net and Sharpe 1.939, trailing FPRC-PQ by 2.417 pp in 4/4 years. The gain is not explained by total tree capacity, pairwise interaction modeling, or generic two-stage boosting. It appears only when each field retains its identity and receives a correction fitted to its own residual before aggregation. 8

a

b

cumulative net return (%)

20

FPRC-PQ Typed local-PQ only Shared correction only Mean of Fold fields

15

13.5

Mean of Fold fields

Sharpe 1.42

18.0

Shared correction only Sharpe 1.94

10

18.3

Typed local-PQ only Sharpe 2.00

Local-P → shared-Q

5

18.7 Sharpe 2.03

19.1

FPRC-PQ

0

Sharpe 2.09

2023

2024

2025

2026

14

16

18

20

2023–2026 net return (%)

Figure 3: (a) Cumulative net-of-cost return, strict book. Curves separate where the residual algebra predicts: typed local correction over pooling, then fresh closure over either stage alone. (b) The same architecture ladder with annualized Sharpe. Table 2: The operator decision chain. Five frozen audits; contrasts are paired block-bootstrap deltas in pp on the strict book (95% CI). Positive favors the selected structure. #

Frozen question

Selected structure

Key contrast

A1

Does typed local correction beat pooling and global-only correction? Does the equal-mean interface discard state the closer needs? Should cross-level edges (shared-P , dualQ) be reopened? Does Q belong locally, shared, or both?

local identity retained before aggregation plain mean; no disagreement coordinates asymmetric routing; no full closure local Q; shared slot closed

May the closer read the aggregate’s own rank uB ?

empty shared state slot

typed local-PQ vs. pooled fields: +4.78; composed vs. best single stage: +0.81 best disagreement variant vs. mean: +0.001 [−0.17, 0.17], Pr = 0.47 full reopening vs. FPRC routing: −0.12 [−0.89, 0.58], Pr = 0.31 local-only vs. dual-layer Q: +0.30 [−0.15, 0.73], Pr = 0.91; interaction −0.31 [−0.35, −0.01] self-state vs. empty: −0.45 [−0.80, −0.05], Pr = 0.02

A2 A3 A4

A5

5.2

Five frozen audits: why the closure looks the way it does

Each structural choice in (9)–(11) was selected by a preregistered audit with a frozen question, decision rule, and paired inference (Table 2; full tables in App. D). Three findings deserve emphasis. First, the negative interaction in A4 (Γ = −0.31 pp, CI [−0.35, −0.01]): once Q is consumed locally, re-exposing it at the shared stage is redundant and mildly destructive—capacity placement is a genuine either/or, not an additive menu (App. Fig. 5). Second, A5 is a significant rejection: letting the closer see the aggregate signal’s own cross-sectional rank over-calibrates (−0.45 pp, and −0.38 pp versus its permutation control), justifying D3’s strict fresh-residual discipline. Third, A2 shows the mean interface loses nothing measurable: orthogonal disagreement contrasts between corrected experts carry no incremental trading information, so the interpretable aggregate is also the sufficient one—and relaxing its unit coefficients, whether by an annually-refit optimal common scaling (−0.01 pp) or a two-coefficient analytic reweighting (+0.05 pp), moves net return within noise while doubling the anchor harms it, so the equal mean is not merely sufficient but sits at a flat coefficient optimum (App. F). 5.3

Boundary controls: side information cannot repair the wrong algebra

The positive result is not a license to append arbitrary state to a strong backbone. Eight preregistered channels—a local outlier alert C alert , a market-level map-drift scalar Dmap (short- vs. long-window field discrepancy), committee dispersion (std) and shape (skew), shared order entropy, transition surprise, and their combinations—produce deltas from −0.06 to −1.34 pp; five are significantly negative (Fig. 4). The drift scalar is instructive: it correlates with committee-level absolute error (ρ = 0.24) yet destroys 1.13 pp when the closer conditions on it, collapsing the closer’s fitted 9

Pr[Δ > 0]

+ DTS directed surprise (local)

0.31

+ field alert C alert (local)

0.07

+ expert-order entropy Hord (shared)

0.05

+ field std (shared)

0.01

+ field skew (shared)

0.01

+ map drift Dmap (shared)

0.01

+ drift+std+skew (shared)

0.003

+ all metadata (local+shared)

0.002

−2.0

−1.5

−1.0

−0.5

0.0

Δ 2023–2026 net return vs. FPRC-PQ (pp), 21-day block bootstrap 95% CI

Figure 4: Eight failed shortcuts. Paired deltas versus frozen FPRC-PQ with 95% block-bootstrap intervals, strict book. Amber denotes local channels and red shared channels; the right column gives Pr[∆ > 0]. Error-correlated metadata does not improve the model merely by being appended to a correction stage. complexity (mean trees 51 → 17). A covariate can diagnose error and still be harmful when exposed after identity erasure. Extra information is not a substitute for a correctly typed residual target. 5.4

Robustness and audits

The architecture conclusions replicate on the wider available book and across deployment years (App. H), including the purely forward 2026 stub. Frozen-signal reproduction audits pass bitexactly; native-missing gates, purge gaps, and point-in-time label availability are checked per feature (max reproduction difference 0.0; App. I). Wall-clock cost is modest: the full algebra fits in minutes per year on four GPU workers.

6

Related work

Tabular learning. GBDTs [10, 7, 20, 29] remain the reference on medium-to-large tabular tasks [14, 26, 31]; deep alternatives [2, 12, 17] narrow the gap partly by learning feature discretizations [13]. Fold makes the discretization explicit, supervised only by rank structure, and shared across all downstream stages. Interaction-explicit models (GA2 M/EBM, NAM) [25, 5, 1] pursue intelligibility for pairwise effects; Fold differs in making the interaction a representation that owns an intermediate residual, not merely a final additive term. Experts, stacking, and residual staging. FPRC relates to mixtures of experts [18, 19, 30] and stacking [32], but inverts their premise: experts are defined by fixed interpretable representations rather than learned gates; aggregation is an explicit identity-erasure boundary; and the audit trail (§5.2) treats where residual capacity lives as the experimental object. The relax–aggregate–close composition is closest to stagewise boosting [10], but differs by making intermediate targets representationtyped and by forbidding later stages from rereading consumed local state. Financial ML. Large-scale return prediction with ML is well established [15, 21, 23], as are warnings on multiple testing and leakage [16, 3, 24]. Our contribution to this literature is methodological: a frozen-protocol architecture study with preregistered controls, rather than a new predictor hunt. The name “factor fields” is also used for neural-field basis decompositions in vision [6]; the two lines are unrelated—our fields are discrete conditional-mean tables over tabular rank grids.

7

Conclusion

The central result is that representation learning benefits from an explicit residual algebra. Fold turns each interaction into a typed conditional-mean object; FPRC applies a local residual operator while that type is intact, erases identity only at a fixed aggregation boundary, and closes the aggregate with a genuinely fresh target. This ordering gives an exact telescoping decomposition and 10

a control-variate correction, and it moves the frozen system from 13.52% to 19.10% net (Sharpe 1.42→2.09). Capacity-matched, unified-residual, identity-free, pairwise-only, learned-aggregation, and side-information controls delimit the explanation: the gain does not follow from more trees, generic staging, or feature abundance. Reflective rumination extends the same ownership principle analytically: a reconstruction displacement is reflected about the anchor, a closed-form gain assigns its span to the second stage, and the terminal closer receives only the orthogonal complement. That extension is not part of the reported empirical headline and remains to be tested out of sample. Evidence still comes from one market and one asset class; the fields are two-dimensional; and the orthogonality statement concerns coupled refitting rather than a general efficiency theorem. The transferable object is the algebra itself: typed residual ownership, ordered composition, and one auditable identity-erasure boundary. Reproducibility statement All architectural decisions, factor selections, and hyperparameters were frozen before the evaluation window; the paper reports the registered contrasts rather than post-hoc winners. Appendices B–I specify the full protocol (bins, costs, learner settings, bootstrap), field and axis inventories, and the audit ledger, including bit-exact reproduction checks of every frozen signal. Construction code, derived signals, and field/state definitions will be released; raw vendor data cannot be redistributed, but the release includes the field list and derivation statistics needed to rebuild the panel from any licensed source. Ethics statement This work studies representation and architecture questions using historical market data. Reported returns are historical research measurements under a frozen protocol, not investment advice; realistic costs are included but capacity, impact, and implementation constraints of live trading are out of scope. The study uses no personal data.

11

References [1] Rishabh Agarwal, Levi Melnick, Nicholas Frosst, Xuezhou Zhang, Ben Lengerich, Rich Caruana, and Geoffrey E. Hinton. Neural additive models: Interpretable machine learning with neural nets. In Advances in Neural Information Processing Systems, volume 34, 2021. [2] Sercan Ö. Arık and Tomas Pfister. TabNet: Attentive interpretable tabular learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 6679–6687, 2021. [3] David H. Bailey, Jonathan Borwein, Marcos López de Prado, and Qiji Jim Zhu. The probability of backtest overfitting. Journal of Computational Finance, 20(4):39–69, 2017. [4] Vadim Borisov, Tobias Leemann, Kathrin Seßler, Johannes Haug, Martin Pawelczyk, and Gjergji Kasneci. Deep neural networks and tabular data: A survey. IEEE Transactions on Neural Networks and Learning Systems, 35(6):7499–7519, 2024. [5] Rich Caruana, Yin Lou, Johannes Gehrke, Paul Koch, Marc Sturm, and Noemie Elhadad. Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission. In Proceedings of the 21st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1721–1730, 2015. [6] Anpei Chen, Zexiang Xu, Xinyue Wei, Siyu Tang, Hao Su, and Andreas Geiger. Factor fields: A unified framework for neural fields and beyond. arXiv preprint arXiv:2302.01226, 2023. [7] Tianqi Chen and Carlos Guestrin. XGBoost: A scalable tree boosting system. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 785–794, 2016. [8] Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21(1):C1–C68, 2018. doi: 10.1111/ectj.12097. [9] Rama Cont. Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance, 1(2):223–236, 2001. [10] Jerome H. Friedman. Greedy function approximation: A gradient boosting machine. Annals of Statistics, 29(5):1189–1232, 2001. [11] João Gama, Indrė Žliobaitė, Albert Bifet, Mykola Pechenizkiy, and Abdelhamid Bouchachia. A survey on concept drift adaptation. ACM Computing Surveys, 46(4):1–37, 2014. [12] Yury Gorishniy, Ivan Rubachev, Valentin Khrulkov, and Artem Babenko. Revisiting deep learning models for tabular data. In Advances in Neural Information Processing Systems, volume 34, 2021. [13] Yury Gorishniy, Ivan Rubachev, and Artem Babenko. On embeddings for numerical features in tabular deep learning. In Advances in Neural Information Processing Systems, volume 35, 2022. [14] Léo Grinsztajn, Edouard Oyallon, and Gaël Varoquaux. Why do tree-based models still outperform deep learning on typical tabular data? In Advances in Neural Information Processing Systems, Datasets and Benchmarks Track, volume 35, 2022. [15] Shihao Gu, Bryan Kelly, and Dacheng Xiu. Empirical asset pricing via machine learning. Review of Financial Studies, 33(5):2223–2273, 2020. [16] Campbell R. Harvey, Yan Liu, and Heqing Zhu. . . . and the cross-section of expected returns. Review of Financial Studies, 29(1):5–68, 2016. [17] Noah Hollmann, Samuel Müller, Lennart Purucker, Arjun Krishnakumar, Max Körfer, Shi Bin Hoo, Robin Tibor Schirrmeister, and Frank Hutter. Accurate predictions on small data with a tabular foundation model. Nature, 637:319–326, 2025. [18] Robert A. Jacobs, Michael I. Jordan, Steven J. Nowlan, and Geoffrey E. Hinton. Adaptive mixtures of local experts. Neural Computation, 3(1):79–87, 1991. [19] Michael I. Jordan and Robert A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 6(2):181–214, 1994. [20] Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, and Tie-Yan Liu. LightGBM: A highly efficient gradient boosting decision tree. In Advances in Neural Information Processing Systems, volume 30, 2017. 12

[21] Bryan T. Kelly, Seth Pruitt, and Yinan Su. Characteristics are covariances: A unified model of risk and return. Journal of Financial Economics, 134(3):501–524, 2019. [22] Hans R. Künsch. The jackknife and the bootstrap for general stationary observations. Annals of Statistics, 17(3):1217–1241, 1989. [23] Markus Leippold, Qian Wang, and Wenyu Zhou. Machine learning in the Chinese stock market. Journal of Financial Economics, 145(2):64–82, 2022. [24] Marcos López de Prado. Advances in Financial Machine Learning. John Wiley & Sons, 2018. [25] Yin Lou, Rich Caruana, Johannes Gehrke, and Giles Hooker. Accurate intelligible models with pairwise interactions. In Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 623–631, 2013. [26] Duncan McElfresh, Sujay Khandagale, Jonathan Valverde, Vishak Prasad C, Ganesh Ramakrishnan, Micah Goldblum, and Colin White. When do neural nets outperform boosted trees on tabular data? In Advances in Neural Information Processing Systems, Datasets and Benchmarks Track, volume 36, 2023. [27] Barry L. Nelson. On control variate estimators. Computers & Operations Research, 14(3): 219–225, 1987. doi: 10.1016/0305-0548(87)90024-4. [28] Dimitris N. Politis and Joseph P. Romano. The stationary bootstrap. Journal of the American Statistical Association, 89(428):1303–1313, 1994. [29] Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, and Andrey Gulin. CatBoost: Unbiased boosting with categorical features. In Advances in Neural Information Processing Systems, volume 31, 2018. [30] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. In International Conference on Learning Representations, 2017. [31] Ravid Shwartz-Ziv and Amitai Armon. Tabular data: Deep learning is not all you need. Information Fusion, 81:84–90, 2022. [32] David H. Wolpert. Stacked generalization. Neural Networks, 5(2):241–259, 1992.

13

Appendix A

Choosing the fields: a frozen selection rule

Nothing in §2–§3 says which factor pairs should become fields. This appendix states the rule we use and the evidence that it orders candidate committees correctly out of sample. The rule consumes no label from the evaluation window and refits nothing. The rule. Let a candidate pool of ordered factor pairs be given; each pair becomes a Fold field by the construction of §2. To choose a committee of M =3: 1. Fix a scoring window strictly before evaluation (here 2020–2022, inside the 2022-12-31 freeze of §4). 2. For every candidate triple, restrict to the rows on which all three fields are simultaneously observable, so that triples are scored on a common sample. 3. On that aligned sample, compute the standard deviation of the triple’s field error, σSTD =
sd r − F̄ . 4. Rank triples by σSTD ascending and take the low end. Lower is better. 5. Freeze the choice. No triple is refitted, reweighted, or re-scored after the evaluation window opens. The statistic is deliberately crude: one number per triple, no learner, no tuning constant, and no forward return after 2022. Its only job is to order candidates. Validation design. We scored all 1,330 candidate triples by step 3 and then measured each one’s realized 2023–2026 performance using immutable strict-OOF modular-FPRC experts. Because no triple was selected or refitted at any point, the comparison below is a pure test of the ordering: if the rule is informative, performance should decline as σSTD rises, with the selector blind to everything after 2022. Table 3: Field-error STD deciles over all 1,330 candidate triples (133 per decile). D1 is the lowest STD. The selector column is scored on 2020–2022; every performance column is realized 2023– 2026. Both the STD and the single-field MSE columns are offset by a constant to expose their true spread—the entire population of triples spans 0.43 bps of STD on a base of 615 bps. selector, 2020–2022

realized, 2023–2026

Decile

STD−615 (bps)

single MSE−378k (bps2 )

complementarity (bps2 )

Sharpe

Net%

ICIR

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

0.048 0.091 0.115 0.135 0.158 0.179 0.194 0.214 0.254 0.311

439 501 524 533 564 591 608 623 663 717

149.3 159.7 153.8 137.9 140.5 143.3 141.2 132.2 122.8 107.9

2.026 2.009 1.991 1.968 1.955 1.963 1.956 1.908 1.840 1.732

17.50 17.35 17.19 17.07 16.98 17.03 17.05 16.63 16.16 15.33

5.180 5.333 5.349 5.334 5.374 5.410 5.416 5.401 5.286 5.096

+41.5

+0.294

+2.18

+0.085

D1−D10

The ordering is real. Across the full population, −σSTD correlates +0.58 with Sharpe and +0.42 with net return (Spearman, p < 10−15 ; Kendall τ = 0.41 and 0.29). A Jonckheere–Terpstra test against the ordered alternative rejects decisively for both (z = 21.9, p ≈ 2×10−106 for Sharpe; z = 15.4, p ≈ 1×10−53 for net). The trend is broad rather than strict: 8 of 9 adjacent decile steps move the right way for Sharpe and 7 of 9 for net, with D6–D7 forming a small plateau. It selects complementarity, not just strength. Low error STD correlates +0.95 with single-field strength (−MSE), so a natural reading is that it merely picks accurate members. Controlling for single-field strength, the partial Spearman correlation of −σSTD with realized performance remains 14

+0.40 (Sharpe), +0.26 (net), and +0.33 (ICIR); and −σSTD correlates +0.49 with committee complementarity, which falls monotonically from 149 to 108 bps2 across the deciles—a 28% relative decline, far larger than the hairline spread of the STD itself. The rule therefore selects on both axes FPRC needs: members accurate enough to be worth preserving, and errors different enough to leave the typed stage something to do. Table 4: Rank association between the selector −σSTD and realized 2023–2026 outcomes, over all 1,330 triples. The last column removes single-field strength (−MSE) by partial correlation. Outcome

Spearman

p

JT z (decreasing)

Partial | single strength

Sharpe Net return Validation ICIR

+0.578 +0.422 +0.040

< 10−15 < 10−15 0.14

21.87 15.38 1.71

+0.404 +0.258 +0.334

(diagnostic) single-field strength (diagnostic) complementarity

+0.950 +0.486

< 10−15 < 10−15

— —

— —

ICIR is the wrong selection gate here. The selector that orders Sharpe and net return so cleanly is nearly uninformative about validation ICIR: Spearman +0.04 (p = 0.14), only 4 of 9 adjacent steps in the expected direction, and a JT statistic that barely clears one-sided significance (z = 1.71, p = 0.043). The field committee should therefore be selected on the frozen portfolio objective, not promoted because a secondary rank statistic happens to improve. Two limits. The discriminating range is hairline—the population spans 0.43 bps of STD—so the rule reads the fourth significant figure of a quantity dominated by return noise, and any implementation must be numerically careful. The ranking survives it: the minimum probed daily unique-value ratio is 0.9967 (mean 0.9999), so ordinal ranks reproduce average-tie Spearman ranks at the reported precision. And this is a retrospective check on a frozen rule, over the same 2023–2026 window the paper evaluates on.

B

Experimental protocol

Table 5: Frozen protocol constants.

C

Item

Value

Item

Value

Market Label Fold bins per axis States per field Strong fields Selection cutoff Evaluation Strict book Median stocks/day Local expert inputs

China A-shares 5-day forward return 10 (daily rank deciles) 100 3 (heterogeneous) 2022-12-31 2023–2026, annual PIT 3,666,920 rows, 844 days 4,380 KQ + 3 per field (7 here)

Portfolio rule Phased holdings Transaction cost Time-decay half-life XGB learning rate / depth XGB max trees / early stop XGB min-child-weight / λ Purge gap Bootstrap Available book

long-only, drop bottom 20% 21 overlapping sleeves blended 10/15 bp 252 trading days 0.035 / 5 600 / 40 rounds (prior year) 2000 / 10 h+1 dates 21-day blocks, 10,000 draws 3,690,036 rows (sensitivity)

Field inventory and axis selection

Table 6: The three Fold fields. All operators are 10 × 10 daily rank-decile grids; the shared apex is turnover rank; the frozen Q set is {volatility 20d, netprofit yoy, ocf to profit, profit stability}. Field

Factor xA

Factor xB

Role

F1 momentum–reversal F2 value–momentum F3 breakout–volume

momentum 20d pb rank price ma ratio

reversal 5d momentum 20d volume change 5d

trend vs. short-reversal value vs. trend breakout vs. volume confirmation

15

Table 7: Pre-2023 Q-axis selection (selector frozen 2022-12-31): mean daily residual IC, residual ICIR, annual sign consistency, and the resulting score. No 2023–2026 label was used. Resid. IC

Resid. ICIR

Sign consist.

Score

Selected

volatility 20d netprofit yoy ocf to profit profit stability roe dt debt to assets pe rank

−0.0216 +0.0184 +0.0068 +0.0106 +0.0144 −0.0063 +0.0074

−2.31 +2.00 +1.12 +1.26 +1.02 −0.69 +0.61

1.0 1.0 1.0 0.6 0.6 0.8 0.6

2.31 2.00 1.12 0.75 0.61 0.55 0.37

✓ ✓ ✓ ✓ — — —

Operator audit details

a

b

where does Q belong?

18.69

18.71

open

19.10

18.80

local Q

closed

cross-level closure parity

local Q reopened

D

Candidate

no

18.71

18.65

yes

18.80

18.59

yes no shared P reopened

closed open shared Q

Figure 5: Capacity placement is an either/or. Net return (%) in two frozen 2×2 audits. (a) A4: opening Q locally is the best cell; adding shared Q on top of local Q reduces net (interaction −0.31 pp, CI [−0.35, −0.01]). (b) A3: reopening both cross-level edges is never the best cell.

Table 8: Selected paired contrasts from the audit chain (strict book, pp, 10,000-draw block bootstrap). Audit

Contrast

A1 A1 A2 A2 A3 A4 A4 A5

shared-Q closure over typed local-P shared closure added over local-Q-only stage disagreement magnitude D replacing Q directional contrasts C1 , C2 beside Q shared-P edge | local-Q open shared-Q effect | local-Q open local-only Q vs. predecessor (SA-LFSI) self-state vs. permuted self-state

∆

95% CI

Pr[∆ > 0]

+0.69 +3.43 −0.09 −0.14 −0.21 −0.30 +0.39 −0.38

[−0.30, 1.72] [0.66, 6.06] [−0.67, 0.40] [−0.52, 0.20] [−0.50, 0.10] [−0.73, 0.15] [−0.23, 0.95] [−0.73, 0.00]

0.91 0.99 0.31 0.19 0.10 0.10 0.86 0.03

Capacity audit. The matched direct control uses exactly the frozen FPRC-PQ total tree count in each deployment year—282, 228, 804, and 360 trees for 2023–2026, respectively—with no early stopping. It reaches 16.970% net and Sharpe 1.963 versus 19.098% and 2.089 for FPRC-PQ; all four yearly comparisons favor FPRC. The point is therefore where the same tree budget is seated, not merely how many trees are fit. 16

E

Generic model-class and exploratory controls

Table 9: Matched structural controls, full Run49 strict book (3,666,920 rows). ∆ is control minus FPRC-PQ, so negative values favor FPRC; “wins” counts control years above FPRC. The ordinary direct row reuses Run101; the remaining controls are Run137. Model

Net%

Sharpe

∆ net (pp)

Wins

FPRC-PQ Direct GBDT (Run101) Matched-capacity Direct Field concat. + unified residual Identity-free two-stage boosting Pairwise-only HGB (GA2 M-class)

19.098 17.014 16.970 16.824 17.408 16.681

2.089 1.882 1.963 1.877 1.931 1.939

— −2.083 −2.128 −2.273 −1.690 −2.417

— 0/4 0/4 0/4 0/4 0/4

The unified-residual comparison is the cleanest identity test: FPRC-PQ leads by 2.273 pp in 4/4 years, with a paired 21-day block-bootstrap interval of [+0.123, +4.453] pp. The remaining FPRC-minus-control intervals are [−0.670, +4.959] pp for matched Direct, [−0.351, +3.540] pp for identity-free two-stage boosting, and [−0.079, +4.861] pp for pairwise HGB. The latter is deliberately described as GA2 M-class because it enforces pairwise-only interactions but is not an interpret/EBM implementation. Table 10: Exploratory learned-aggregation control. Only paired deltas within the common book are comparable; absolute returns or Sharpes must not be compared with Table 9. The learned-MoE book has 2,628,660 rows. Deltas and intervals are in basis points. Family

∆ net (bp) Wins

Paired contrast

Learned MoE gate only − equal mean + closer Learned MoE gate + fresh closer − equal mean + closer

−59.3 +12.8

95% CI (bp)

2/4 [−196.8, +86.2] 2/4 [−79.6, +123.0]

Learned aggregation. The three typed local experts are frozen, and only annual PIT affine-softmax weights are learned. Gate-only is 59.3 bp below the frozen equal-mean-plus-closer system. In the mechanism-matched full-interface comparison, restoring a fresh parents-only closer leaves the learned gate only +12.8 bp ahead, with 2/4 winning years and an interval spanning zero. Learned weighting therefore does not stably improve the unit interface. Because this audit requires the narrower frozen-expert coverage intersection, its absolute performance cannot be compared with the full main table.

F

Operator-composition diagnostics: ordering, coefficients, and residual closure

Fold’s projection view and FPRC’s ordered routing motivate three structural predictions that we probed post hoc on the frozen 2023–2026 book. These are finite-capacity diagnostics on the alreadyexplored evaluation window, not preregistered promotions; we report them as structural evidence for the composition, using the same paired 21-day block-bootstrap machinery as the main text. Ordering (non-commutativity). Reversing the two stages—closing on the pooled fields first and applying the typed local relaxation only afterward (C before RPQ )—moves net return by −0.12 pp versus the frozen local-before-shared chain (CI [−1.07, 1.02]). The sign matches the ordered-closure reading (each operator should fit only the residual its predecessor leaves); the interval is wide, but taken with the significant A5 rejection of letting the closer re-read local state, the composition’s asymmetry is directional-to-significant and never neutral. The operators do not commute. Unit coefficients sit at a flat optimum. The fixed identity-erasure mean (D2) gives every field an equal offset coefficient. Relaxing this buys nothing measurable and can only add estimation 17

variance. An annually-refit common closed-form scaling α̂Y∗ —whose fitted value wanders over 0.45–1.15 across deployment years, so it is emphatically not pinned to one—moves net return by −0.01 pp (CI [−0.21, 0.18], Pr[∆ > 0] = 0.46). A two-coefficient analytic reweighting of the local and shared increments moves it by +0.05 pp (CI [−0.66, 0.84], Pr = 0.60), and its per-year solve is itself ill-conditioned. Conversely, doubling the anchor to a matched α=2 offset harms all three chains (−0.56 to −0.77 pp, higher turnover and cost). The unit interface is therefore statistically indistinguishable from the analytically-tuned coefficients while carrying none of their variance, and strictly better than over-anchoring—the fixed point in performance is flat around unit weights even though the coefficient itself is not identified at exactly one. Operational residual closure. We fit diagnostic probes to the final residual r − Sb over 2024– 2026 (the first window with enough nested out-of-fold history). Neither a typed-local probe (three field-identity learners, equal-weighted) nor a shared parents-only probe recovers any out-of-sample 2 signal: both have Roos ≤ 0 (−2.5×10−4 typed-local, −1.3×10−3 shared) with non-positive MSE balance. Within the tested local and shared hypothesis classes, the composition has exhausted the fields’ recoverable structure—an operational closure, not exact orthogonal completeness.

G

Negative-control details

Table 11: Preregistered generic meta-information controls versus frozen FPRC-PQ (strict book, pp). Control alert

Local alert C Expert-order entropy Hord Field std Field skew Map drift Dmap Drift+std+skew All metadata DTS directed surprise

Placement

Net%

∆

95% CI

Pr[∆ > 0]

local shared shared shared shared shared local+shared local

18.88 18.77 18.94 18.98 17.97 17.84 17.76 19.03

−0.22 −0.32 −0.16 −0.12 −1.13 −1.25 −1.34 −0.06

[−0.51, 0.07] [−0.71, 0.05] [−0.34, −0.02] [−0.22, −0.03] [−1.93, −0.21] [−2.07, −0.37] [−2.14, −0.43] [−0.35, 0.19]

0.07 0.05 0.01 0.01 0.01 0.003 0.002 0.31

The alert and drift channels are not information-free: Dmap correlates 0.24 with committee-level absolute error and C1alert positively with local absolute residuals. They fail because diagnosis and correction are different operations. Once field identities have been erased, a shared learner can use such metadata only for global recalibration; exposing it there also violates the fresh-residual boundary isolated by audit A5. The result supports the algebra’s placement rule, not a general claim that metadata are uninformative.

H

Yearly results and coverage sensitivity

Table 12: Yearly net return (%), strict book. 2026 covers January–June (forward stub). Model

2023

2024

2025

2026

Mean of Fold fields Shared correction only Typed local-PQ only Local-P → shared-Q FPRC-PQ

2.70 3.21 2.78 3.26 3.42

7.22 7.91 7.65 7.89 7.89

4.83 5.71 5.80 5.81 5.73

−1.24 1.18 2.06 1.75 2.06

On the available book (3,690,036 rows), FPRC-PQ earns 19.24% net. Every registered structural contrast keeps its sign and approximate magnitude, including self-state (−0.44 pp) and all-metadata (−1.32 pp). No conclusion about the residual algebra reverses between the strict and available books. 18

I

Reproduction and PIT audit ledger

Every run re-derives its frozen predecessor signal before reporting: the maximum absolute difference between rebuilt and frozen signals is 0.0 in all audits. Native-missing gates are preserved (features may be missing; the sample gate never changes), the field tables and Dmap observe labels only through t − h − 1, and the annual purge covers the five-day forward label. Q-axis coverage is 93.9% of rows. Total wall clock per audit is 2.5–7.4 minutes on four GPU workers.

J

Use of large language models

An LLM assisted with manuscript drafting (prose, LaTeX, and figure scripting) under author direction. All experimental designs, frozen protocols, and numerical results were produced by the authors’ pipeline; all numbers in the paper are copied from the frozen result ledgers, and the authors verified all claims.

19

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