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The Cost of a Physics Prior Is Bounded by the Ablation Gap
Boris Kriuk
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 8/24/2026, 5:58:46 AM
Summary
This paper establishes a theoretical bound stating that the accuracy cost (excess risk) of enforcing a shape or physics prior on a feature subset is upper-bounded by the ablation gap of that subset. The authors demonstrate that this cost is largely a property of the free features and the validation protocol rather than the prior itself. Using an ordinal wildfire-severity task, they show that geographic coordinates act as a 'shield,' significantly reducing the ablation gap and making physics priors appear cheaper. The paper also introduces a self-calibrating noise floor for measurement resolution and a two-fit screen to reject unidentifiable experiments.
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Relation Signals (5)
Boris Kriuk → affiliatedwith → HKUST
confidence 99% · B. Kriuk is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology (HKUST)
Shape Constraint → boundedby → Ablation Gap
confidence 97% · the cost of a physics prior is bounded by the ablation gap... 0 <= P <= D
Wildfire Severity Task → uses → Macro-F1
confidence 95% · macro-F1 under spatial blocking
Geographic Coordinates → actsas → Shield
confidence 94% · Coordinates act as a shield: alone they recover 92.9% of the full model's macro-F1
Theorem 9 → proposes → Price of a shape constraint is bounded by its headroom
confidence 92% · Theorem 9 (The price of a shape constraint is bounded by its headroom)
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Abstract
Abstract:Shape-constrained and physics-informed learning reports an accuracy cost of enforcing a prior and treats it as a property of the prior. We show it is mostly a property of the free features and the validation split. Let P be the excess risk of restricting a hypothesis class to functions with a shape constraint on features S, and D the excess risk of the ablated model that ignores S. Because a function constant in x_j is both non-decreasing and non-increasing in x_j, the ablated class is contained in the constrained class, so 0 <= P <= D for every risk functional, with no convexity, smoothness, or realizability assumption. Empirically the bound is a sign test: a constrained model must never be beaten by its own ablation. We instantiate it on an ordinal wildfire-severity task (N = 26,681, K = 3) with hard monotone constraints on four meteorological drivers, coordinates left free, and a validation ladder from i.i.d. resampling to 2-degree spatial blocking. Coordinates act as a shield: alone they recover 92.9% of the full model's macro-F1 under spatial blocking, collapsing D from 0.1288 to 0.0427; the same prior costs 0.0473 shielded and 0.3470 unshielded, a ratio of 7.3 with identical physics. Because D is protocol-dependent it does not transfer: coarsening blocks from 1 to 10 degrees drives D from 0.0942 to 0.0050, leaving two configurations unidentifiable a priori. Inversions of the certified nesting bound the pipeline's additive resolution: over 318 comparisons they give a self-calibrating floor of 0.0220 macro-F1, below which no reported price is interpretable, including four cells in our own headline grid. Cost and compliance are independent: the unconstrained model violates the prior at rate 0.48-0.49 while enforcing it costs 0.0473. We give a two-fit screen that rejects unidentifiable experiments before a constrained model is trained.
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- Source: https://arxiv.org/abs/2608.21059v1
- Canonical: https://arxiv.org/abs/2608.21059v1
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The Cost of a Physics Prior Is Bounded by the Ablation Gap Boris Kriuk Thanks: 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). Thanks: Code, run configurations and raw artefacts: https://github.com/BorisKriuk/The-Cost-of-a-Physics-Prior-Is-Bounded-by-the-Ablation-Gap. 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 Δ be the excess risk of the ablated model that ignores S. Because a function constant in xjx_j is simultaneously non-decreasing and non-increasing in xjx_j, the ablated class is contained in the constrained class, so 0≤Δ0 ≤ 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 681N=26\,681, K=3K=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∘2 spatial blocking, across a nine-configuration grid. Four consequences are non-obvious. First, coordinates act as a shield: alone they recover 92.9%92.9\% of the full model’s macro-F1F_1 under spatial blocking, which collapses Δ from 0.12880.1288 to 0.04270.0427 and forces every prior on the drivers to look cheap; the same screened prior costs 0.04730.0473 shielded and 0.34700.3470 unshielded, a ratio of 7.37.3 with identical physics. Second, because Δ is protocol-dependent it is not transferable: coarsening blocks from 1∘1 to 10∘10 drives Δ from 0.09420.0942 to 0.00500.0050, 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 318318 certified comparisons yields a self-calibrating floor s^=0.0220 s=0.0220 macro-F1F_1, 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 0.480.48–0.490.49 while enforcing it costs 0.04730.0473. 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 Introduction A 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 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 “ΔF1=−0.01 F_1=-0.01” or “R2R^2 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 xjx_j. A function that does not depend on xjx_j is non-decreasing in xjx_j — weakly, but the constraint is stated weakly — and is therefore admissible. The same function is also non-increasing in xjx_j, hence admissible under the opposite prior, and admissible under convexity, concavity, Lipschitz bounds and unimodality in xjx_j. So the worst a shape constraint on S can do is force the learner to discard S (Fig. 1): (σ)⏟price of the prior on S≤Δ(S)⏟value of S given the rest. P(σ)_price of the prior on S\;≤\; (S)_value of S given the rest. (1) ℋ⟂SH SℋσH^σℋHGEO_ONLY (0.5602)MONO_sc (0.5850)MC (0.6029)Δ− -PP Fig. 1: Geometry behind Theorem 9, with the model names of our experimental zoo attached and their measured out-of-fold macro-F1F_1 under 2∘2 spatial blocking. The class ℋ⟂SH S of hypotheses that ignore the constrained features S lies inside the shape-constrained class ℋσH^σ, because a function constant in xjx_j satisfies every sign pattern on xjx_j 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 =0.0179≤Δ=0.0427P=0.0179≤ =0.0427. The bound holds for arbitrary risk functionals. We do not claim (1) is deep; its proof is three lines. We claim that its right-hand side is a quantity constrained-learning 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 ΔF1 F_1 versus the ablated baseline must be non-negative for every constrained model — a sign check on a column the pipeline already writes. I-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 (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, Δ(S)≈0 (S)≈ 0 and the constraint is automatically almost free, regardless of whether the prior is correct. In our design the shield is lat,lon\lat,lon\; it alone attains 92.9%92.9\% of the full model’s macro-F1F_1 under spatial blocking, and it makes the identical monotone prior appear 7.3×7.3× cheaper than when it is removed. 4. Bounded sign controversy (Corollary 18). The entire spread of achievable risk over all 2|S|2^|S| sign patterns, and over the choice of sign for an auxiliary channel, is bounded by the same Δ . 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 ≤s^≤ s. 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 318318 certified comparisons in our grid gives s^=0.0220 s=0.0220 macro-F1F_1. Crucially Δ^<s < s 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 =0P=0 and violation rate approaching 11, and we measure the same dissociation: the unconstrained model violates the textbook prior at rate 0.47950.4795–0.48250.4825 while the price of enforcing that prior is 0.01430.0143–0.05490.0549. 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). I-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. I Related Work I-A Monotone and shape-constrained learning Monotonicity as an inductive bias goes back to Ben-David [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. I-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 (Δ , computed over the covariates that the prescribed component touches) that turns such a statement from unfalsifiable into informative, and the regime (Δ<s < s) 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]. I-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 0.83160.8316 to 0.60290.6029 macro-F1F_1 between i.i.d. and 2∘2 -blocked splits. We add a second-order consequence: because protocol choice changes Δ , it changes the admissible range of every constraint-cost figure, so protocol sensitivity propagates into claims about inductive bias and interpretability, 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]. I Setup and Notation Let =∏j=1djX= _j=1^dX_j with each j⊆ℝX_j totally ordered, let =0,…,K−1Y=\0,…,K-1\ carry the natural order, and let D be a distribution on ×X×Y. Write V=1,…,dV=\1,…,d\ for the available features, S⊆VS V for the constrained subset, and x−Sx_-S for the coordinates outside S. Definition 1 (Risk functional). A risk is any functional R:ℋ→ℝ∪+∞R:H ∪\+∞\ with lower values preferred. For ⊆ℋG put R⋆()=inff∈R(f)R (G)= _f R(f). No structure is assumed on R. It may be an expected loss, a negated macro-F1F_1, an AUC deficit, a quantile of a per-region 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-F1F_1, is not an expected loss and is not decomposable over examples. Definition 2 (Ordinal scoring function). A hypothesis is a pair (f,dec)(f,dec) where f:→ℝf:X is a score and dec a fixed decoding of f into Y. Concretely we use the cumulative parameterization f(x)=∑k=0K−2S^k(x)f(x)= _k=0^K-2 S_k(x) with S^k(x)≈Pr[y>k∣x] S_k(x)≈ [y>k x] and S^0≥⋯≥S^K−2 S_0≥·s≥ S_K-2 enforced by running isotonization in k; the label is the argmax of the induced class probabilities. Shape constraints are imposed on each S^k S_k and hence on f, and isotonization in k preserves monotonicity in x. Definition 3 (Sign pattern and shape class). For σ∈−1,0,+1dσ∈\-1,0,+1\^d with support S(σ)=j:σj≠0S(σ)=\j: _j≠ 0\, ℋσ=f∈ℋ:∀j∈S,∀x−j,t↦σjf(x−j,t)non-decr..H^σ= \f :\ ∀ j∈ S,\ ∀ x_-j,\ t _jf(x_-j,t)\ non-decr. \. Definition 4 (S-blind class). ℋ⟂S=f∈ℋ:f(x)=f(x′)wheneverx−S=x−S′H S=\f :\ f(x)=f(x )\ whenever\ x_-S=x _-S\. Assumption 5 (Ablation closure). ℋ⟂S≠∅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). (σ) (σ) =R⋆(ℋσ)−R⋆(ℋ), =R (H^σ)-R (H), (2) Δ(S) (S) =R⋆(ℋ⟂S)−R⋆(ℋ). =R (H S)-R (H). (3) Both are non-negative by nesting [35]. Δ(S) (S) is exactly the number an applied paper reports when it writes “removing temperature costs 0.130.13 macro-F1F_1.” Definition 7 (Violation rate). For f∈ℋf , coordinate j, step hj>0h_j>0, and a distribution μ on X, let Dj(x)=f(x+hjej)−f(x−hjej)D_j(x)=f(x+h_je_j)-f(x-h_je_j) and let τ=εrange(f)τ= \,range(f) be an activity threshold. Then VR(f,j,σj)=Prx∼μ[signDj(x)≠σj||Dj(x)|>τ].VR(f;j, _j)= _x μ\! [signD_j(x)≠ _j\; |\;|D_j(x)|>τ ]. We distinguish VRpartialVR_partial, evaluated on the model’s internal score, from VRe2eVR_e2e, 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)h_j=0.05×IQR(x_j) and ε=10−3 =10^-3. IV Main Result Lemma 8 (Degenerate admissibility). For every σ∈−1,0,+1dσ∈\-1,0,+1\^d with support S, ℋ⟂S⊆ℋσH S ^σ. Proof. Let f∈ℋ⟂Sf S, j∈Sj∈ S, and fix x−jx_-j. For t≤t′t≤ t , f(x−j,t)=f(x−j,t′)f(x_-j,t)=f(x_-j,t ) since f does not depend on coordinate j∈Sj∈ S. Hence σj(f(x−j,t′)−f(x−j,t))=0≥0 _j (f(x_-j,t )-f(x_-j,t) )=0≥ 0, so t↦σjf(x−j,t)t _jf(x_-j,t) is weakly non-decreasing. As j∈Sj∈ S was arbitrary, f∈ℋσf ^σ. ∎ 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≤(σ)≤Δ(S).0\;≤\;P(σ)\;≤\; (S). (4) Equivalently R⋆(ℋσ)≤R⋆(ℋ⟂S)R (H^σ)≤ R (H S). Proof. By Lemma 8, ℋ⟂S⊆ℋσ⊆ℋH S ^σ . The infimum over a larger set is no larger, so R⋆(ℋ)≤R⋆(ℋσ)≤R⋆(ℋ⟂S)R (H)≤ R (H^σ)≤ R (H S). Subtract R⋆(ℋ)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 s s in Section VIII-H. IV-A The operational form: ablation dominance Write the metric in higher-is-better form, F1F_1, so that =F1(unconstrained)−F1(constrained)P=F_1(unconstrained)-F_1(constrained) and Δ=F1(unconstrained)−F1(ablated) =F_1(unconstrained)-F_1(ablated). Corollary 11 (Ablation dominance). Inequality (4) is equivalent to F1(constrained)≥F1(ablated),F_1(constrained)\;≥\;F_1(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)F_1(unconstrained) and no subtraction of correlated estimates. Proof. Subtract the two displayed definitions: ≤ΔP≤ iff −F1(constrained)≤−F1(ablated)-F_1(constrained)≤-F_1(ablated). ∎ Corollary 11 is the form we recommend for practice, because it is a single sign check on a difference that experiment harnesses already tabulate, and because it is numerically better behaved: it compares two quantities directly rather than differencing each against a common reference. IV-B How far the bound generalizes Definition 12. A family ⊆ℋC is S-degenerate-admissible if ℋ⟂S⊆H S . Corollary 13 (General shape constraints). If C is S-degenerate-admissible then R⋆()−R⋆(ℋ)≤Δ(S)R (C)-R (H)≤ (S). This covers, for each j∈Sj∈ S: monotonicity of either sign and isotonicity with respect to any partial order on SX_S; convexity or concavity; L-Lipschitz dependence for any L≥0L≥ 0; unimodality, quasi-convexity and bounded total variation; bounded sensitivity ‖∂f/∂xj‖∞≤κ\|∂ f/∂ x_j\|_∞≤κ; and pairwise dominance or individual-fairness constraints x⪯x′⇒f(x)≤f(x′)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 non-increasing, convex and concave, 00-Lipschitz, unimodal, of zero total variation, has vanishing partial derivative, and satisfies every dominance requirement with equality. Hence ℋ⟂S⊆H S and Theorem 9 applies verbatim. ∎ Corollary 14 (Penalized and soft constraints). Let Ω:ℋ→[0,∞) :H→[0,∞) be a penalty with Ω≡0 ≡ 0 on ℋ⟂SH S, and Rλ=R+λΩR_λ=R+λ , λ≥0λ≥ 0. Then Rλ⋆(ℋ)−R⋆(ℋ)≤Δ(S)R _λ(H)-R (H)≤ (S) for every λ≥0λ≥ 0. Proof. Rλ=R_λ=R on ℋ⟂SH S, so Rλ⋆(ℋ)≤infℋ⟂SRλ=R⋆(ℋ⟂S)R _λ(H)≤ _H SR_λ=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=1d=1, 1=0,1X_1=\0,1\, =0,1Y=\0,1\, D uniform on (0,1),(1,0)\(0,1),(1,0)\, ℋH all real functions on X, R(f)=Pr[f(x)>0≠y]R(f)= [1\f(x)>0\≠ y], σ=(+1)σ=(+1). Then R⋆(ℋ)=0R (H)=0 and (σ)=Δ(1)=12P(σ)= (\1\)= 12. Proof. f(x)=12−xf(x)= 12-x attains risk 00. Any non-decreasing f induces a non-decreasing decision rule, which must assign a label to x=1x=1 no smaller than to x=0x=0; the Bayes labels are y(0)=1,y(1)=0y(0)=1,y(1)=0, so every such rule errs on at least one of two equiprobable atoms, giving R⋆(ℋσ)=12R (H^σ)= 12. Every S-blind f is constant and errs on exactly one atom, so R⋆(ℋ⟂S)=12R (H S)= 12. ∎ Proposition 15 identifies the worst case: tightness occurs precisely when the prior is maximally anti-aligned with the mechanism. Hence a measured P close to Δ is evidence against the physics; ^≪Δ P is weak evidence for it; and Δ^≈0 ≈ 0 is evidence about the feature set, not the physics. Section VIII-C exhibits all three regimes in one corpus: ^/Δ^=0.82 P/ =0.82 unshielded, 0.370.37 shielded, and Δ^=0.0050 =0.0050 under coarse blocking. V Consequences V-A Constraint costs do not transfer across protocols Write RΠR_ for the risk induced by an evaluation protocol Π , and Π,ΔΠP_ , _ accordingly. Corollary 16 (Protocol non-transferability). For every Π , Π(σ)≤ΔΠ(S)P_ (σ)≤ _ (S). Consequently: (i) if ΔΠ′(S)≤ε _ (S)≤ then Π′(σ)≤εP_ (σ)≤ for every prior supported on S, including priors contradicting the data; (i) ΠP_ carries no information about Π′P_ beyond the joint behaviour of ΔΠ,ΔΠ′ _ , _ , and the ratio Π′/ΠP_ /P_ is unbounded above and below. Proof. Apply Theorem 9 with R=RΠR=R_ and R=RΠ′R=R_ separately; cross-validation estimators are risk functionals. (i) is immediate. For (i), Proposition 15 realizes =ΔP= and Proposition 17 realizes =0P=0 for the same σ; embedding one behaviour in Π and the other in Π′ gives arbitrary ratios. ∎ Two papers reporting “monotonicity costs 0.010.01” under random splits and “monotonicity costs 0.090.09” 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 0.04560.0456 at 1∘1 blocks and −0.0139-0.0139 at 10∘10 blocks, a sign reversal driven entirely by the split. V-B Shielding: how to make a prior look free Proposition 17 (Shielding). If there exists f†∈ℋ⟂Sf S with R(f†)≤R⋆(ℋ)+εR(f )≤ R (H)+ , then Δ(S)≤ε (S)≤ and hence (σ)≤εP(σ)≤ for every σ supported on S. Moreover Δ(S) (S) is non-increasing under enlargement of the free feature set V∖SV S, and non-decreasing under enlargement of S. Proof. The first claim is Definition 6 with Theorem 9. For the second, R⋆(ℋ)R (H) is fixed once V is fixed, while R⋆(ℋ⟂S)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 Δ(S) (S) is small and the prior is guaranteed cheap, unconditionally on whether it is correct. Reporting a small P 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\lat,lon\ and its strength is measured directly by the ablated baseline. The monotonicity in |S||S| is also observed: promoting insolation from auxiliary channel to fifth driver raises Δ from 0.12880.1288 to 0.13990.1399 under i.i.d. splitting. V-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 supσ∈ΣR⋆(ℋσ)−infσ∈ΣR⋆(ℋσ)≤Δ(S). _σ∈ R (H^σ)- _σ∈ R (H^σ)\;≤\; (S). (6) The same holds for a single auxiliary channel c∉Sc∉ S: with 0⊇+∪−C_0 _+ _- the classes obtained by leaving the sign of c free, fixing it to +1+1, and to −1-1, all three contain the c-blind class, so maxsignF1−minsignF1≤F1(0)−F1(c-blind). _signF_1- _signF_1\;≤\;F_1(C_0)-F_1($c$-blind). (7) Proof. All ℋσH^σ with σ supported on S satisfy ℋ⟂S⊆ℋσ⊆ℋH S ^σ , so their optima lie in the interval [R⋆(ℋ),R⋆(ℋ⟂S)][R (H),R (H S)] of length Δ(S) (S). For the second claim, ±C_± and 0C_0 all contain the c-blind class by Lemma 8, and ±⊆0C_± _0; hence the maximum over signs is F1(0)F_1(C_0) and the minimum is at least F1(c-blind)F_1($c$-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. V-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)(A,B) of arms is a certified comparison if the hypothesis class of A is contained in that of B, so that F1(B)≥F1(A)F_1(B)≥ F_1(A) at the population level. Its inversion is (F1(A)−F1(B))+ (F_1(A)-F_1(B) )_+. Corollary 20 (Slack lower-bounds pipeline error). Define s^=max(A,B)∈(F1(A)−F1(B))+ s= _(A,B) (F_1(A)-F_1(B) )_+ (8) over all certified comparisons in a grid K of configurations. Then s s lower-bounds the additive error of the pipeline in the reported metric, and: 1. (reporting floor) any |^c|<s^| P_c|< s is unresolvable and must not be interpreted; 2. (a priori futility) if Δ^c<s _c< s then configuration c cannot resolve the price of any prior on S, since the whole admissible interval [0,Δc][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)F_1(A)≤ F_1(B). Writing F^1(A)=F1(A)+ηA F_1(A)=F_1(A)+ _A and F^1(B)=F1(B)+ηB F_1(B)=F_1(B)+ _B, a positive observed difference gives F^1(A)−F^1(B)≤ηA−ηB≤|ηA|+|ηB| F_1(A)- F_1(B)≤ _A- _B≤| _A|+| _B|; maximize over K. Claims (1)–(3) follow by comparing [0,Δ^c][0, _c] with s s. Restricting K to ablation pairs recovers Corollary 11 and gives the narrower floor s^abl≤s s_abl≤ s. ∎ Unlike a bootstrap confidence interval, s s derives from an ordering the experiment is obliged to satisfy, so it captures finite-sample 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 00, and b2 confirms only bytewise determinism at that seed. A seed sweep would plausibly raise s s and disqualify further cells, so we regard s^=0.0220 s=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 s^=0 s=0 vacuously, so probing configurations should be included deliberately. Our grid (Table I) includes a block-size sweep spanning 1∘1 to 10∘10 and a gate-forced-open configuration that adds a free channel; the maximizing comparison turns out to lie in the headline configuration itself. Algorithm 1 Admissibility screen for a constraint-cost experiment 1: features V, constrained subset S, protocol Π , metric F1F_1, slack s s (or a grid K to estimate it) 2: fit unconstrained model on V; record F1uncF_1^unc 3: fit ablated model on V∖SV S; record F1ablF_1^abl 4: Δ^←F1unc−F1abl ← F_1^unc-F_1^abl ⊳ certified ceiling, Thm. 9 5: if Δ^<s < s then 6: report “price of any prior on S is unidentifiable under Π ” 7: stop ⊳ no constrained training required 8: end if 9: fit constrained model; record F1conF_1^con 10: if F1con<F1abl−s^F_1^con<F_1^abl- s then 11: flag solver or estimation failure ⊳ violates Cor. 11 12: end if 13: report (^,Δ^,s^)( P, , s) and the free feature set V∖SV S V-E Compliance is not price Proposition 21 (Independence of violation rate and price). For every δ>0δ>0 there exist (,ℋ,σ,μ)(D,H,σ,μ) with (σ)=0P(σ)=0 and VR(f^,j,σj)≥1−δVR( f;j, _j)≥ 1-δ for some risk minimizer f^∈argminℋR f∈ _HR. Symmetrically there exist instances with VR(f^,j,σj)=0VR( f;j, _j)=0 and (σ′)>0P(σ )>0 for a different prior on the same coordinate. Proof sketch. Take d=2d=2, y a deterministic function of x2x_2 alone, x1x_1 independent of (x2,y)(x_2,y). Every f(x)=g(x2)+ϵh(x1)f(x)=g(x_2)+ε h(x_1) attains the same risk for a decision-based metric provided ϵε 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≥ 1-δ, giving VR≥1−δVR≥ 1-δ under σ1=+1 _1=+1. Since ℋ⟂1H \1\ contains a risk-optimal hypothesis, Δ(1)=0 (\1\)=0 and Theorem 9 forces (σ)=0P(σ)=0. The symmetric statement follows by making y depend monotonically on x1x_1 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 because the unconstrained model violates the physics often. Both inferences are invalid: VRVR does not appear in (4), and the two quantities can be set independently. A high VRVR 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 Instantiation: Mapping the Theory onto a Model Zoo 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−1K-1 histogram gradient-boosted binary classifiers [30] with identical hyperparameters (300300 iterations, learning rate 0.060.06, 3131 leaves, minimum 4040 samples per leaf, L2L_2 regularization 1.01.0, no early stopping, seed 00) — 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. TABLE I: Correspondence between the theory of Section I and the fitted models. D=temp,rh,wind,precipD=\temp,rh,wind,precip\ are the meteorological drivers, G=lat,lonG=\lat,lon\ the geographic shield, c the residualized auxiliary channel (insolation). σnv=(+1,−1,+1,−1)σ^nv=(+1,-1,+1,-1) is the textbook (naive) sign pattern, σscσ^sc the fold-wise screened one. Model Features Constraint Role MAJORITY — — ℋ⟂DH D for PHYS* GEO_ONLY G none ℋ⟂DH D for MONO*, MIB* WX_ONLY D none ℋH for PHYS* MC D∪GD∪ G none ℋH for MONO* MONO_naive D∪GD∪ G σnvσ^nv on D ℋσnvH^σ^nv MONO_sc D∪GD∪ G σscσ^sc on D ℋσscH^σ^sc; c-blind arm PHYS D σnvσ^nv on D unshielded prior PHYS_sc D σscσ^sc on D unshielded prior MIB D∪G∪cD∪ G∪\c\ σscσ^sc on D 0C_0 in (7) MIB_fp D∪G∪cD∪ G∪\c\ σscσ^sc, c:+1c:+1 +C_+ MIB_fm D∪G∪cD∪ G∪\c\ σscσ^sc, c:−1c:-1 −C_- 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*) F_1( MONO*),\ F_1( MIB*) ≥F1(GEO_ONLY), ≥ F_1( GEO\_ONLY), (R1) F1(PHYS*) F_1( PHYS*) ≥F1(best constant), ≥ F_1( best constant), (R2) F1(MIB*) F_1( MIB*) ≥F1(MONO_sc), ≥ F_1( MONO\_sc), (R3) F1(MIB) F_1( MIB) ≥F1(MIB_fp),F1(MIB_fm), ≥ F_1( MIB\_fp),\ F_1( MIB\_fm), (R4) F1(MC) F_1( MC) ≥F1(MONO*), ≥ F_1( MONO*), F1(WX_ONLY) F_1( WX\_ONLY) ≥F1(PHYS*). ≥ F_1( PHYS*). (R5) Moreover maxctx signF1−minctx signF1≤F1(MIB)−F1(MONO_sc) _ctx signF_1- _ctx signF_1≤ F_1( MIB)-F_1( MONO\_sc). Proof. (R1)–(R3) are Corollary 11 applied to the corresponding rows of Table I: within the D∪GD∪ G architecture ℋ⟂DH 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 ±⊆0C_± _0 and (R5) is ℋσ⊆ℋH^σ . 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-F1F_1 the majority-class predictor is not optimal among constants — predicting a rare class can raise the macro average — so F1(MAJORITY)≤F1(best constant)F_1( MAJORITY)≤ F_1( 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∪cD∪ 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 I 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 Experimental Design The experiments are not intended to demonstrate a new method; they calibrate the quantities of Theorem 9 and measure s s on a real environmental prediction task where monotone physical priors are uncontroversial. VII-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=3K=3 levels, declared in increasing severity as Controlled burn ≺ Uncontrolled burn ≺ Wildfire, so the ordinal cumulative parameterization of Section I applies. After class filtering and completeness filtering on drivers and coordinates, N=26 681N=26\,681 records remain, with class counts [333, 2711, 23 637][333,\,2711,\,23\,637] (shares 0.01250.0125, 0.10160.1016, 0.88590.8859) and a majority-class macro-F1F_1 baseline of 0.31320.3132. Drivers are D=temp,rh,wind,precipD=\temp,rh,wind,precip\; the shield is G=lat,lonG=\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. VII-B Priors The textbook pattern is σnv=(+1,−1,+1,−1)σ^nv=(+1,-1,+1,-1) on (temp,rh,wind,precip)(temp,rh,wind,precip). The screened pattern σscσ^sc is obtained inside each training fold as the sign of the mean central finite-difference partial effect of the unconstrained MC model, with a 400400-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 σscσ^sc coincided bytewise with σnvσ^nv in 99 of 1515 training folds (2/52/5 under iid, 4/54/5 under point, 3/53/5 under spatial), and MONO_sc was consequently fitted as MONO_naive in those folds, detected by signature deduplication. VII-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≤0.90R^2≤ 0.90 and the residual relative standard deviation is ≥0.10≥ 0.10. Only one candidate is present in this corpus, insolation, and it is admitted comfortably: R2=0.2646R^2=0.2646, residual relative standard deviation 0.85760.8576. Admitted channels enter the MIB arms in residualized form, c−Ab^c-A b, with b b 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 VRe2eVR_e2e detects. The gate opened in every one of the 1515 training folds of every gate-auto configuration, a fact that turns out to neutralize our screening-leak probe (Section VIII-I). VII-D Validation ladder Three protocols in increasing severity: iid stratified 55-fold; point, grouping by exact coordinate so no location is split across folds; and spatial, grouping by blocks of a chosen angular size (429429 blocks at 2∘2 , 10801080 at 1∘1 , 115115 at 5∘5 , 4040 at 10∘10 ). Grouped folds use stratified group k-fold. This ladder is the standard remedy for autocorrelation-inflated skill [21, 22, 23]. VII-E Configuration grid Because s s in (8) is a maximum over configurations, we run the full ladder under each of the settings in Table I. 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. TABLE I: Configuration grid K. Each row is a complete run of the three-protocol ladder over all eleven arms of Table I, with 55 folds, seed 00 and B=2000B=2000 cluster bootstrap replicates. ctx, b2 and glob coincide bytewise; see Section VIII-I. Tag Setting Purpose ctx headline: 2∘2 blocks, gate auto/fold headline numbers b1 1∘1 blocks (10801080) shield strongest b2 2∘2 blocks (429429) determinism check (≡ ) b5 5∘5 blocks (115115) block sweep b10 10∘10 blocks (4040) headroom collapse; s s probe closed gate forced closed MIB*≡ _sc test drv insolation as 55th driver driver-set perturbation, |S|=5|S|=5 glob gate decided on pooled data screening-leak probe k5 K=4K=4 label granularity label-resolution perturbation The k5 configuration re-splits the majority class, yielding K=4K=4 ordered levels Controlled burn ≺ Uncontrolled burn ≺ Forest fire ≺ Natural fire with counts [333, 2711, 18 415, 5195][333,\,2711,\,18\,415,\,5195] on N=26 654N=26\,654 records and a majority baseline of 0.20430.2043. It is included because sign identifiability turns out to depend far more on label resolution than on protocol. VII-F Inference All paired comparisons use one shared set of B=2000B=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 20002000 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 0.02280.0228 at 2∘2 to 0.02460.0246 at 10∘10 . VIII Results Table I 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=0VR_own=0 for every constrained arm (C3a) and undefined for every unconstrained arm (C3b); no bootstrap degeneration (C4); MIB*≡MONO_sc MIB*≡ MONO\_sc bytewise whenever the gate is empty (C5); and complete out-of-fold coverage (C6). VIII-A Protocol ladder and the collapse of headroom Fig. 2 shows macro-F1F_1 for the key arms across the ladder. The unconstrained model falls from 0.83160.8316 (iid) through 0.83110.8311 (point) to 0.60290.6029 (spatial), a split-leakage term of +0.2287+0.2287; under 10∘10 blocks it reaches 0.52180.5218, a term of +0.3098+0.3098. The quantity that governs everything downstream is not that drop but the vertical gap between MC and GEO_ONLY, i.e. Δ^(D) (D): it falls from 0.12880.1288 under iid to 0.11650.1165 under point to 0.04270.0427 under 2∘2 spatial blocking, a factor of 3.03.0. 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. Fig. 2: Out-of-fold macro-F1F_1 across the validation ladder, headline configuration. The gap between MC and GEO_ONLY is the certified ceiling Δ^(D) (D) on the cost of any monotone prior on the drivers (0.1288→0.1165→0.04270.1288→ 0.1165→ 0.0427); the gap between MC and MONO_sc is the measured price (0.0473→0.0555→0.01790.0473→ 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+0.0572 [+0.0252,+0.0814][+0.0252,+0.0814]. VIII-B Price against certified ceiling Fig. 3 reports paired differences with 95%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∪GD∪ G must sit to the right of zero, and in the headline configuration all fifteen such cells do, with margins from +0.0029+0.0029 (MIB_fm, spatial) to +0.0941+0.0941 (MIB_fp, iid). Reading Table I as prescribed, the ratio ^/Δ P/ is the fraction of the drivers’ value that the prior destroys: 0.3670.367 for MONO_sc under iid, 0.4760.476 under point, 0.4190.419 under spatial. The corresponding unshielded ratios are 0.8210.821, 0.8220.822 and 0.6940.694. That contrast is the subject of Section VIII-C. Fig. 3: Paired macro-F1F_1 differences with 95%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. TABLE I: Headline configuration (ctx: 2∘2 blocks, gate auto/fold), all eleven arms, all three protocols. Δ is MC−GEO_ONLY MC- GEO\_ONLY for arms on D∪GD∪ G and WX_ONLY−MAJORITY WX\_ONLY- MAJORITY for arms on D alone (Remark 23). P is the negated difference against the feature-matched unconstrained reference. “dominance” is the sign test of Corollary 11 against the arm’s own ablation and must be ≥0≥ 0. † marks |^|<s^=0.0220| P|< s=0.0220, i.e. cells that carry no interpretation. ‡ marks the MIB arms, whose reference is not feature-matched (Remark 24). VRVR columns are end-to-end violation rates against the arm’s own declaration, the textbook prior, and the screened prior. Protocol Model F1F_1 (OOF) F1F_1 fold ± sd Δ P ^/Δ P/ dominance VRownVR_own VRnvVR_nv VRscVR_sc F1F_1 per class iid MAJORITY 0.3132 0.3132 ± 0.0000 — — — — n/a n/a n/a 0.000 / 0.000 / 0.940 GEO_ONLY 0.7028 0.7026 ± 0.0147 — — — — n/a n/a n/a 0.406 / 0.733 / 0.969 WX_ONLY 0.7358 0.7356 ± 0.0116 — — — — n/a 0.4849 0.4953 0.597 / 0.646 / 0.964 MC 0.8316 0.8312 ± 0.0097 0.1288 — — — n/a 0.4795 0.4957 0.700 / 0.816 / 0.979 MONO_naive 0.7815 0.7813 ± 0.0160 0.1288 0.0501 0.389 ++0.0787 0.0000 0.0000 0.1572 0.605 / 0.766 / 0.973 MONO_sc 0.7843 0.7843 ± 0.0095 0.1288 0.0473 0.367 ++0.0815 0.0000 0.2068 0.0000 0.602 / 0.776 / 0.974 PHYS 0.3842 0.3842 ± 0.0100 0.4226 0.3516 0.832 ++0.0710 0.0000 0.0000 0.0852 0.000 / 0.211 / 0.941 PHYS_sc 0.3888 0.3885 ± 0.0138 0.4226 0.3470 0.821 ++0.0756 0.0000 0.3855 0.0000 0.000 / 0.225 / 0.942 MIB 0.7888 0.7881 ± 0.0219 0.1288 0.0428‡ 0.332 ++0.0860 0.0000 0.2280 0.0000 0.586 / 0.803 / 0.978 MIB_fp 0.7968 0.7969 ± 0.0095 0.1288 0.0347‡ 0.269 ++0.0941 0.0000 0.2245 0.0000 0.616 / 0.797 / 0.977 MIB_fm 0.7809 0.7808 ± 0.0161 0.1288 0.0507‡ 0.394 ++0.0781 0.0000 0.2035 0.0000 0.593 / 0.776 / 0.974 point MAJORITY 0.3132 0.3132 ± 0.0004 — — — — n/a n/a n/a 0.000 / 0.000 / 0.940 GEO_ONLY 0.7146 0.7143 ± 0.0203 — — — — n/a n/a n/a 0.433 / 0.741 / 0.970 WX_ONLY 0.7476 0.7471 ± 0.0132 — — — — n/a 0.4922 0.4928 0.610 / 0.667 / 0.966 MC 0.8311 0.8309 ± 0.0196 0.1165 — — — n/a 0.4799 0.4868 0.691 / 0.823 / 0.980 MONO_naive 0.7761 0.7755 ± 0.0175 0.1165 0.0549 0.471 ++0.0616 0.0000 0.0000 0.0494 0.586 / 0.769 / 0.973 MONO_sc 0.7755 0.7750 ± 0.0184 0.1165 0.0555 0.476 ++0.0609 0.0000 0.0747 0.0000 0.581 / 0.771 / 0.974 PHYS 0.3888 0.3888 ± 0.0086 0.4344 0.3588 0.826 ++0.0756 0.0000 0.0000 0.0326 0.000 / 0.225 / 0.942 PHYS_sc 0.3907 0.3907 ± 0.0087 0.4344 0.3569 0.822 ++0.0775 0.0000 0.1661 0.0000 0.000 / 0.230 / 0.942 MIB 0.8070 0.8066 ± 0.0098 0.1165 0.0241‡ 0.207 ++0.0924 0.0000 0.0880 0.0000 0.637 / 0.806 / 0.978 MIB_fp 0.8023 0.8021 ± 0.0056 0.1165 0.0288‡ 0.247 ++0.0877 0.0000 0.0912 0.0000 0.634 / 0.795 / 0.977 MIB_fm 0.7790 0.7786 ± 0.0115 0.1165 0.0521‡ 0.447 ++0.0644 0.0000 0.0826 0.0000 0.592 / 0.772 / 0.974 spatial MAJORITY 0.3132 0.3109 ± 0.0069 — — — — n/a n/a n/a 0.000 / 0.000 / 0.940 GEO_ONLY 0.5602 0.5544 ± 0.0617 — — — — n/a n/a n/a 0.082 / 0.640 / 0.958 WX_ONLY 0.5030 0.5045 ± 0.0628 — — — — n/a 0.4904 0.4888 0.130 / 0.433 / 0.946 MC 0.6029 0.6014 ± 0.0721 0.0427 — — — n/a 0.4825 0.4968 0.208 / 0.640 / 0.961 MONO_naive 0.5886 0.5883 ± 0.0635 0.0427 0.0143† 0.335 ++0.0284 0.0000 0.0000 0.1904 0.194 / 0.613 / 0.959 MONO_sc 0.5850 0.5856 ± 0.0639 0.0427 0.0179† 0.419 ++0.0248 0.0000 0.2650 0.0000 0.183 / 0.613 / 0.959 PHYS 0.3679 0.3643 ± 0.0271 0.1898 0.1351 0.712 ++0.0547 0.0000 0.0000 0.0638 0.006 / 0.161 / 0.937 PHYS_sc 0.3713 0.3692 ± 0.0326 0.1898 0.1317 0.694 ++0.0581 0.0000 0.3462 0.0000 0.000 / 0.174 / 0.939 MIB 0.6125 0.6105 ± 0.0765 0.0427 −-0.0096†‡ −-0.225 ++0.0523 0.0000 0.2885 0.0000 0.192 / 0.681 / 0.964 MIB_fp 0.6159 0.6102 ± 0.0886 0.0427 −-0.0130†‡ −-0.305 ++0.0557 0.0000 0.2863 0.0000 0.204 / 0.679 / 0.964 MIB_fm 0.5630 0.5626 ± 0.0516 0.0427 0.0399‡ 0.934 ++0.0029 0.0000 0.2494 0.0000 0.127 / 0.604 / 0.958 VIII-C Shielding, measured The shield’s strength is F1(GEO_ONLY)=0.7028F_1( GEO\_ONLY)=0.7028 under iid against a majority baseline of 0.31320.3132, and 0.56020.5602 under spatial blocking. Expressed as a fraction of the full model it rises with protocol severity: coordinates alone recover 84.5%84.5\% of MC under iid and 92.9%92.9\% under blocking. Under blocking they also beat the four drivers together by +0.0572+0.0572 [+0.0252,+0.0814][+0.0252,+0.0814]. The consequence for constraint accounting is stark. The same screened monotone prior costs 0.0473 0.0473 shielded: MONO_sc versus MC, :\ \ MONO\_sc versus MC, 0.3470 0.3470 unshielded: PHYS_sc versus WX_ONLY, :\ PHYS\_sc versus WX\_ONLY, a ratio of 7.347.34 under iid; the ratio is 6.436.43 under point and 7.367.36 under spatial, and 7.027.02 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 Δ : the normalized price ^/Δ P/ also more than doubles, from 0.3670.367 shielded to 0.8210.821 unshielded. Under Proposition 15 the unshielded ratio of 0.820.82 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 to weaken the coordinate shield and so raise Δ^(D) (D); instead F1(GEO_ONLY)F_1( GEO\_ONLY) moves only from 0.55920.5592 to 0.51680.5168 while F1(MC)F_1( MC) collapses from 0.65340.6534 to 0.52180.5218 (Table IV, spatial rows), so Δ^(D):0.0942→ 0.0427→ 0.0184→ 0.0050. (D): 0.0942\;→\;0.0427\;→\;0.0184\;→\;0.0050. Coordinates transfer across coarse block boundaries far better than the drivers do: F1(WX_ONLY)F_1( WX\_ONLY) falls from 0.56860.5686 to 0.44100.4410 over the same sweep. What Proposition 17 guarantees is monotonicity of Δ 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: 0.04560.0456, 0.01790.0179, 0.02290.0229, −0.0139-0.0139 for MONO_sc. At 10∘10 the ceiling 0.00500.0050 is a fifth of s s, so the last two entries are not measurements of anything. TABLE IV: Complete out-of-fold macro-F1F_1 across the grid. ctx is the headline configuration; b2 and glob reproduce it bytewise and are not repeated. Rows marked ∗ have an empty exogeneity gate, so MIB*≡MONO_sc MIB*≡ MONO\_sc by Lemma 8 (verified bytewise). Δ^G=MC−GEO_ONLY _G= MC- GEO\_ONLY is the certified ceiling for the shielded arms and Δ^M=WX_ONLY−MAJORITY _M= WX\_ONLY- MAJORITY for the unshielded arms. Because block size also sets the bootstrap clustering, the iid and point rows of b1/b5/b10 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 Δ^G _G / Δ^M _M ctx iid 0.3132 0.7028 0.7358 0.8316 0.7815 0.7843 0.3842 0.3888 0.7888 0.7968 0.7809 0.1288 / 0.4226 point 0.3132 0.7146 0.7476 0.8311 0.7761 0.7755 0.3888 0.3907 0.8070 0.8023 0.7790 0.1165 / 0.4344 spatial 0.3132 0.5602 0.5030 0.6029 0.5886 0.5850 0.3679 0.3713 0.6125 0.6159 0.5630 0.0427 / 0.1898 b1 iid 0.3132 0.7028 0.7358 0.8316 0.7815 0.7843 0.3842 0.3888 0.7888 0.7968 0.7809 0.1288 / 0.4226 point 0.3132 0.7146 0.7476 0.8311 0.7761 0.7755 0.3888 0.3907 0.8070 0.8023 0.7790 0.1165 / 0.4344 spatial 0.3132 0.5592 0.5686 0.6534 0.6061 0.6078 0.3711 0.3996 0.6182 0.6381 0.5982 0.0942 / 0.2554 b5 iid 0.3132 0.7028 0.7358 0.8316 0.7815 0.7843 0.3842 0.3888 0.7888 0.7968 0.7809 0.1288 / 0.4226 point 0.3132 0.7146 0.7476 0.8311 0.7761 0.7755 0.3888 0.3907 0.8070 0.8023 0.7790 0.1165 / 0.4344 spatial 0.3132 0.5670 0.4975 0.5854 0.5846 0.5625 0.3443 0.3865 0.5696 0.5847 0.5516 0.0184 / 0.1844 b10 iid 0.3132 0.7028 0.7358 0.8316 0.7815 0.7843 0.3842 0.3888 0.7888 0.7968 0.7809 0.1288 / 0.4226 point 0.3132 0.7146 0.7476 0.8311 0.7761 0.7755 0.3888 0.3907 0.8070 0.8023 0.7790 0.1165 / 0.4344 spatial 0.3132 0.5168 0.4410 0.5218 0.5419 0.5357 0.3439 0.3569 0.5488 0.5411 0.5460 0.0050 / 0.1279 closed∗ iid 0.3132 0.7028 0.7358 0.8316 0.7815 0.7843 0.3842 0.3888 0.7843 0.7843 0.7843 0.1288 / 0.4226 point 0.3132 0.7146 0.7476 0.8311 0.7761 0.7755 0.3888 0.3907 0.7755 0.7755 0.7755 0.1165 / 0.4344 spatial 0.3132 0.5602 0.5030 0.6029 0.5886 0.5850 0.3679 0.3713 0.5850 0.5850 0.5850 0.0427 / 0.1898 drv∗ iid 0.3132 0.7028 0.7807 0.8427 0.7956 0.7996 0.4208 0.4229 0.7996 0.7996 0.7996 0.1399 / 0.4675 point 0.3132 0.7146 0.7850 0.8443 0.7965 0.7991 0.4282 0.4293 0.7991 0.7991 0.7991 0.1297 / 0.4718 spatial 0.3132 0.5602 0.5371 0.6207 0.6018 0.6136 0.4133 0.4263 0.6136 0.6136 0.6136 0.0606 / 0.2239 k5 (K=4K=4) iid 0.2043 0.6701 0.6763 0.7904 0.7361 0.7361 0.2979 0.2979 0.7549 0.7448 0.7385 0.1203 / 0.4720 point 0.2043 0.6684 0.6887 0.7987 0.7288 0.7288 0.2956 0.2956 0.7616 0.7428 0.7355 0.1303 / 0.4844 spatial 0.2043 0.4608 0.4112 0.5262 0.5058 0.4912 0.2935 0.2912 0.5148 0.5071 0.5074 0.0655 / 0.2069 VIII-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 0.12880.1288 to 0.13990.1399 under iid and from 0.42260.4226 to 0.46750.4675 for the unshielded arms, while the price rises less (0.0473→0.04310.0473→ 0.0431 shielded, 0.3470→0.35780.3470→ 0.3578 unshielded), so the normalized price falls to 0.3080.308 shielded and 0.7650.765 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 15/1515/15 folds, agrees with the textbook prior in 15/1515/15, and its bootstrap interval excludes zero in 12/1512/15 — against 00–6/156/15 for the four meteorological drivers. Exogeneity (R2=0.2646R^2=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=4K=4 (k5) leaves the ceiling essentially unchanged (0.12030.1203, 0.13030.1303, 0.06550.0655 across the ladder) but raises the price sharply: ^/Δ P/ for MONO_sc becomes 0.4510.451, 0.5370.537 and 0.5340.534 versus 0.3670.367, 0.4760.476 and 0.4190.419 at K=3K=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). VIII-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 5353 of 6060 driver-folds (88.3%88.3\%) and the interval excludes zero in only 1313 of 6060 (21.7%21.7\%). Temperature and relative humidity agree in 15/1515/15 folds each; wind in 14/1514/15; precipitation in only 9/159/15, with the screened sign coming out positive — against the textbook −1-1 — in 66 of 1515 folds while being identified in just 22. 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 bounded by Δ^=0.0427 =0.0427 under spatial blocking and 0.12880.1288 under iid, regardless of how it is resolved — and by 0.00500.0050 at 10∘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%95\% clustered interval excludes zero, and the clusters are the spatial blocks. Refining blocks therefore manufactures identification: temperature is identified in 9/159/15 folds at 1∘1 (10801080 clusters), 5/155/15 at 2∘2 (429429), 4/154/15 at 5∘5 (115115) and 6/156/15 at 10∘10 (4040). 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=4K=4 every driver’s mean effect grows by a factor of 22–4040 and precipitation becomes identified in 15/1515/15 folds with mean effect −0.01144-0.01144 against −0.00028-0.00028 at K=3K=3. Fig. 4: Fold-wise identification of driver signs, headline configuration: mean central-difference partial effect of the unconstrained model with 95%95\% clustered intervals, three protocols × five folds per driver. Circles mark folds where the interval excludes zero (1313 of 6060); 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 66 of 1515 folds. A non-identified sign implies little headroom and hence, by Corollary 18, little accuracy at stake in choosing it. TABLE V: Sign screening across 1515 training folds (33 protocols × 55 folds) per configuration. “agree” counts folds where the screened sign matches the textbook prior; “ident.” counts folds whose 95%95\% clustered interval excludes zero; “n+n_+” counts folds with a positive screened sign; “eff.” is the mean partial effect on E[y]E[y]. Textbook signs are temp+temp+, rh−rh-, wind+wind+, precip−precip-, solar+solar+. Config Driver agree ident. n+n_+ eff. ctx (headline) temp 15/15 5/15 15/15 +0.00126+0.00126 rh 15/15 6/15 0/15 −0.00129-0.00129 wind 14/15 0/15 14/15 +0.00146+0.00146 precip 9/15 2/15 6/15 −0.00028-0.00028 b1 (1∘1 ) temp 15/15 9/15 15/15 +0.00129+0.00129 rh 15/15 8/15 0/15 −0.00130-0.00130 wind 14/15 3/15 14/15 +0.00153+0.00153 precip 9/15 4/15 6/15 −0.00048-0.00048 b5 (5∘5 ) temp 15/15 4/15 15/15 +0.00124+0.00124 rh 14/15 2/15 1/15 −0.00128-0.00128 wind 14/15 0/15 14/15 +0.00135+0.00135 precip 8/15 2/15 7/15 −0.00022-0.00022 b10 (10∘10 ) temp 15/15 6/15 15/15 +0.00121+0.00121 rh 15/15 2/15 0/15 −0.00133-0.00133 wind 15/15 0/15 15/15 +0.00123+0.00123 precip 8/15 1/15 7/15 −0.00032-0.00032 drv (|S|=5|S|=5) temp 15/15 2/15 15/15 +0.00049+0.00049 rh 15/15 1/15 0/15 −0.00085-0.00085 wind 14/15 1/15 14/15 +0.00105+0.00105 precip 8/15 1/15 7/15 +0.00013+0.00013 solar 15/15 12/15 15/15 +0.00166+0.00166 k5 (K=4K=4) temp 14/15 6/15 14/15 +0.00336+0.00336 rh 15/15 12/15 0/15 −0.00274-0.00274 wind 15/15 13/15 15/15 +0.00501+0.00501 precip 15/15 15/15 0/15 −0.01144-0.01144 VIII-F Compliance is not price, empirically Constrained arms satisfy their own declaration exactly, VRown=0.0000VR_own=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 I, 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 0.47950.4795 (iid), 0.47990.4799 (point) and 0.48250.4825 (spatial), computed over 40 87340\,873–45 07145\,071 active partial-effect evaluations; against the screened prior the rates are 0.49570.4957, 0.48680.4868 and 0.49680.4968. Per driver under blocking they are 0.4320.432 (temp), 0.4920.492 (rh), 0.5480.548 (wind) and 0.4630.463 (precip), i.e. indistinguishable from the 0.50.5 of a coin flip (Fig. 5). The measured price of enforcing that same prior is 0.01430.0143–0.05550.0555. A rate of 0.480.48 would ordinarily be reported as “the model has not learned the physics”; a price of 0.020.02 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 0.20680.2068 (iid) and 0.26500.2650 (spatial) — it must, since its own declaration differs from the textbook one in 66 of 1515 folds — while MONO_naive violates the screened prior at 0.15720.1572 and 0.19040.1904. PHYS_sc, which has no shield to hide behind, disagrees with the textbook prior most strongly of all, at 0.38550.3855 and 0.34620.3462. Those numbers quantify how far apart the two priors are in deployment, and they are the only VRVR 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 VRe2eVR_e2e against the declared prior would reveal that the certificate does not survive preprocessing. No such leakage was observed: VRown=0.0000VR_own=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. Fig. 5: Violation rate of the unconstrained MC model against the textbook and the screened prior, per driver, under 2∘2 spatial blocking (overall 0.48250.4825 naive, 0.49680.4968 screened, over 45 07145\,071 active evaluations). Every bar sits near the dashed 0.50.5 line, indicating a partial effect whose sign is essentially arbitrary in the unconstrained fit. The only visible gap is precip, whose screened rate (0.5270.527) exceeds its naive rate (0.4630.463) 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 0.01790.0179 in the same cell. Fig. 6: Certificate versus deployed pipeline, headline configuration (c=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 0.00000.0000 — 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 0.00000.0000 there because the textbook prior is their own declaration. VIII-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 gate decisively (R2=0.2646R^2=0.2646 against the threshold 0.900.90; residual relative standard deviation 0.85760.8576 against the threshold 0.100.10) and was admitted in all 1515 training folds of every gate-auto 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 0.00000.0000 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)F_1( MIB)-F_1( MONO\_sc). Table VI reports both quantities. The bound holds in 88 of the 1515 gate-open cells and is violated in 77, by up to 0.02950.0295. The violations are not mysterious: each one decomposes exactly into the two elementary nesting inversions of Proposition 22 that produce it, excess= \;=\; (F1(MIB_fp)−F1(MIB))+⏟R4 (F_1( MIB\_fp)-F_1( MIB) )_+_R4 + +\; (F1(MONO_sc)−F1(MIB_fm))+⏟R3, (F_1( MONO\_sc)-F_1( MIB\_fm) )_+_R3, to four decimal places in all seven cells — for instance 0.0295=0.0199+0.00960.0295=0.0199+0.0096 at b1/spatial and 0.0254=0.0034+0.02200.0254=0.0034+0.0220 at ctx/spatial. Each summand is an inversion of a certified ordering and is therefore already accounted for in s s; the composite excess must not be added to s s 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)=0.0045F_1( MIB)-F_1( MONO\_sc)=0.0045 under iid and 0.02750.0275 under blocking, against s^=0.0220 s=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. TABLE VI: Auxiliary-channel arms in the five gate-open configurations. “spread” is max−min - over MIB,MIB_fp,MIB_fm\ MIB, MIB\_fp, MIB\_fm\; “bound” is F1(MIB)−F1(MONO_sc)F_1( MIB)-F_1( MONO\_sc) as required by (7). Cells where the spread exceeds the bound are marked ×; the excess equals the sum of the R4 and R3 inversions listed in Table VII. Gate-closed configurations (closed, drv) have spread == bound =0.0000=0.0000 exactly and are omitted. Config Protocol spread bound excess ctx iid 0.0160 0.0045 × 0.0114 ctx point 0.0280 0.0315 — ctx spatial 0.0529 0.0275 × 0.0254 b1 iid 0.0160 0.0045 × 0.0114 b1 point 0.0280 0.0315 — b1 spatial 0.0399 0.0104 × 0.0295 b5 iid 0.0160 0.0045 × 0.0114 b5 point 0.0280 0.0315 — b5 spatial 0.0331 0.0071 × 0.0260 b10 iid 0.0160 0.0045 × 0.0114 b10 point 0.0280 0.0315 — b10 spatial 0.0077 0.0131 — k5 iid 0.0164 0.0188 — k5 point 0.0261 0.0328 — k5 spatial 0.0077 0.0236 — VIII-H Calibrating the slack We enumerate every certified comparison of Proposition 22 over the seven numerically distinct configurations and three protocols: 8787 distinct R1 tests (after collapsing the 1818 alias cells of the gate-closed runs), 4242 R2, 6363 R3, 4242 R4 and 8484 R5, for 318318 orderings that the pipeline is obliged to satisfy. Eighteen are inverted; Table VII lists all of them. Hence s^=0.0220macro-F1, s=0.0220\ macro-F_1, (9) attained at ctx/spatial/MIB_fm versus MONO_sc, i.e. in the headline configuration itself. Restricting to ablation-dominance pairs alone gives the narrower floor s^abl=0.0154 s_abl=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+0.0307 to +0.1150+0.1150. For comparison, the fold standard deviation at the maximizing cell is 0.05160.0516 and the largest anywhere in the grid is 0.08890.0889 (drv/spatial/MONO_naive); the widest cluster bootstrap half-width among the constrained-arm dominance intervals is 0.07930.0793 (b5/spatial/MONO_sc versus GEO_ONLY). So s s is 0.43×0.43× the local fold spread and 0.28×0.28× the widest interval — but 0.87×0.87× the half-width of the very quantity it invalidates, the spatial MONO_sc price (0.02540.0254). The point of s s 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∘10 blocks the constrained MONO_naive beats its own unconstrained superclass MC by 0.02020.0202 with interval [−0.0128,+0.0529][-0.0128,+0.0529] — a violation the bootstrap cannot see, in a cell where Δ^=0.0050 =0.0050 and nothing was measurable to begin with. Consequences, stated as Corollary 20 prescribes: 1. Reporting floor. Four of the twenty-one price cells in Table I satisfy |^|<s^| P|< s and carry no interpretation: MONO_naive (0.01430.0143), MONO_sc (0.01790.0179), MIB (−0.0096-0.0096) and MIB_fp (−0.0130-0.0130), 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 × protocol) cells have Δ^<s < s: b5/spatial (Δ^=0.0184 =0.0184) and b10/spatial (Δ^=0.0050 =0.0050). Their ten shielded-arm price cells were uninformative before any constrained model was trained, so Algorithm 1 would have terminated them after two unconstrained fits. A third, ctx/spatial with Δ^=0.0427 =0.0427, clears the floor by only a factor of 1.91.9. 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 ≥0P≥ 0 does not transfer to estimates. The largest such effect in our grid, b10/spatial/MIB beating MC by 0.02700.0270 with interval [+0.0071,+0.0496][+0.0071,+0.0496] and p=0.003p=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 s s 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. TABLE VII: Complete slack ledger: all 1818 inverted orderings among the 318318 certified comparisons of the grid. “relation” refers to Proposition 22. s^=0.0220 s=0.0220 is the maximum of the last column. Intervals are the paired 95%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 —. Configurations b2 and glob reproduce ctx bytewise and would triple rows #1, #5, #9, #10; they are not listed. # Config Protocol Comparison (subclass ⪯ superclass) relation F1F_1 sub F1F_1 super inversion 1 ctx iid MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.7809 0.7843 0.0034 2 b1 iid MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.7809 0.7843 0.0034 3 b5 iid MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.7809 0.7843 0.0034 4 b10 iid MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.7809 0.7843 0.0034 5 ctx iid MIB_fp⪯MIB MIB\_fp MIB R4 0.7968 0.7888 0.0080 6 b1 iid MIB_fp⪯MIB MIB\_fp MIB R4 0.7968 0.7888 0.0080 7 b5 iid MIB_fp⪯MIB MIB\_fp MIB R4 0.7968 0.7888 0.0080 8 b10 iid MIB_fp⪯MIB MIB\_fp MIB R4 0.7968 0.7888 0.0080 9 ctx spatial MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.5630 0.5850 0.0220 10 ctx spatial MIB_fp⪯MIB MIB\_fp MIB R4 0.6159 0.6125 0.0034 11 b1 spatial MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.5982 0.6078 0.0096 12 b1 spatial MIB_fp⪯MIB MIB\_fp MIB R4 0.6381 0.6182 0.0199 13 b5 spatial MONO_sc⪯GEO_ONLY MONO\_sc GEO\_ONLY R1 0.5625 0.5670 0.0045 14 b5 spatial MIB_fm⪯GEO_ONLY MIB\_fm GEO\_ONLY R1 0.5516 0.5670 0.0154 15 b5 spatial MIB_fm⪯MONO_sc MIB\_fm MONO\_sc R3 0.5516 0.5625 0.0109 16 b5 spatial MIB_fp⪯MIB MIB\_fp MIB R4 0.5847 0.5696 0.0151 17 b10 spatial MONO_naive⪯MC MONO\_naive MC R5 0.5419 0.5218 0.0202 18 b10 spatial MONO_sc⪯MC MONO\_sc MC R5 0.5357 0.5218 0.0139 VIII-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 ΔF1(MONO_sc)=+0.0000,ΔF1(MIB)=+0.0000, F_1( MONO\_sc)=+0.0000, F_1( MIB)=+0.0000, exactly, in all three protocols; glob reproduces ctx bytewise on every arm. Two reasons combine. First, the gate is unanimous: insolation was admitted in 15/1515/15 training folds, so pooling the gate decision changes nothing. Second, in this implementation the pooled-screening flag governs only the gate; the sign pattern σscσ^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 3333 model-protocol cells, confirming that the harness is deterministic under seed 00 and that no result in this paper depends on run-to-run variation in fitting. IX Discussion IX-A What should be reported Any claim about the accuracy cost of a domain constraint should be accompanied by the triple (^,Δ^,s^)( P, , s) and an explicit statement of the free feature set. The first number alone is uninterpretable: our headline price of 0.04730.0473 becomes 0.34700.3470 when the shield is removed and −0.0139-0.0139 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], Δ 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 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∘10 to 1∘1 moves temperature from 6/156/15 to 9/159/15 identified folds without touching the data. Second, report the label resolution, since moving from K=3K=3 to K=4K=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. IX-B On reporting our own negative cells Four of our twenty-one headline price cells fall below s s 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 s s, which is what a resolution-limited measurement of a small quantity is expected to look like. IX-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 arm that recovers 92.9%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=0VR_own=0 end-to-end), which is a property of the artefact, not of its macro-F1F_1. 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 Conclusion 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. 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