Shared Physics Responses Recover Hidden Rankings in Neural Operator Libraries

1 Introduction

• Without the high-fidelity solution, deployment must rank competing predictions from their inputs and governing PDE; the framework targets both per-input routing and checkpoint selection for repeated use.The proxy is reference-free and uses no deployment reference solution or reference-derived labels.Source: §1 · pp. 1–2

2.1 Ranking a finite library requires less information than solution recovery

• Theorem 1 shows that projecting the hidden truth onto the candidate-difference span determines exact pair gaps, pair labels, weak ordering and tie-ruled top-1, with dimension at most N −1.This coordinate is minimal among bounded linear observations for globally exact decisions; restricted physical solution sets can require less information.Source: §2.1 · p. 4

2.2 A shared physics response recovers hidden neural-operator choices

• The proxy recovered 95.57% of exact top-1 decisions across eight Sine–Gordon libraries, while a 192-case control found it matched candidate-specific correction rankings in 191 cases.Correctly directed, input-specific responses tracked hidden-reference gap signs and magnitudes; shuffled or reversed responses did not. At 16 candidates, candidate-wise solves were 16.0 times slower for Burgers and 15.3 times slower for reaction–diffusion.Source: §2.2 · pp. 6–9
• 99.01% of cases selected a candidate within 10^-6 of best library loss, and 99.59% of non-tied pairwise preferences were recovered, versus raw residuals’ 42.66% tolerant top-1 and 67.83% pair accuracy.The benchmark crossed two equations, two architectures and two seeded ensembles under controlled input shifts.Source: §2.2 · p. 6, Figure 2 · p. 7
• Across 1,920 Burgers and reaction–diffusion cases, the shared proxy chose the exact winner in 1,899, with near-zero normalized regret when it missed.In reaction–diffusion, candidate-wise error estimators improved pairwise ranking to 82.7% but remained below the shared proxy.Source: §2.2 · pp. 7–8, Figure 3 · p. 8

2.3 Reliable ranking depends on margins, library composition and admissibility

• An arithmetic anchor pulled by two extreme reaction–diffusion candidates recovered only 67 of 240 winners, whereas a medoid recovered all 240; across 16 further libraries, both anchors won 3,800 of 3,840 cases.Neither anchor rule was uniformly better, so anchor choice must suit the candidate family.Source: §2.3 · p. 9, Figure 4 · p. 9
• Shared recovery reached 97.8% in the smallest hidden-gap decile and approached 100% as margins widened; positivity and residual backtracking raised feasible shock cases from 22 of 35 to 35 of 35.Near-ties are intrinsically fragile when proxy error is comparable to the candidate gap, and admissibility remains a separate reliability boundary.Source: Figure 4 · p. 9, §2.3 · p. 10

2.4 Accurate ranking does not require accurate solution reconstruction

• Squared-loss ranking depends only on candidate-difference directions, so full-task accuracy and decision accuracy are neither equivalent nor ordered without a margin condition.The orthogonal task-error component can be arbitrarily large without changing decisions, while arbitrarily small full-task error can reverse a near-tied choice.Source: §2.4 · pp. 10–11
• Across public PDEBench, correction reduced full-trajectory error by 9.86-fold and comparison-subspace error by 16.1-fold; in Sine–Gordon, median weighted full-trajectory error fell 1,835.7-fold.For the stated terminal squared-L2 task, q's loss was no greater than the best candidate in all 1,920 primary Burgers/reaction–diffusion cases and all 99 inputs.Source: §2.4 · pp. 10–11
• Under operator–data mismatch, the proxy recovered comparisons in all ten two-dimensional compressible-flow cases and reduced comparison-subspace error by a median factor of 22.2, but improved on the best candidate in only four.This demonstrates that ranking accuracy need not imply a better corrected prediction.Source: §2.4 · p. 11

2.5 Shared scores support routing and reusable model selection

• For the high-band spectral task, shared scoring recovered 88 of 99 choices versus 69 for the fixed terminal model, 23 for medoid centrality and 7 for raw residual, closing 86.7% of the fixed-to-oracle risk gap.The preferred checkpoint varied across inputs, isolating task dependence because the models, inputs and proxy trajectories were unchanged.Source: §2.5 · p. 11, Figure 5 · p. 12
• At B = 24, pooled shared scoring lowered normalized regret versus the stated baselines, but its advantage came from reaction–diffusion libraries, not either Burgers architecture group.Small-panel performance varied sharply; candidate-specific correction matched shared-response selection and regret in all 64 panel evaluations.Source: §2.5 · p. 11, Figure 5 · p. 12
• At B = 144, shared scoring selected a model within tolerance of the best in 122 of 128 panels and a top-two model in all 128, outperforming the stated baselines.Normalized regret was concentrated near zero.Source: §2.5 · p. 12, Figure 5 · p. 12

2.6 Strong monotonicity certifies decisions against a same-grid operator reference

• For the stated same-grid operator, every pairwise, strict top-1 and τ = 10^-6 tolerant top-1 margin exceeded its bound across 24 cases and 672 pairs; the largest bound-to-margin ratio was 0.09494.The residual-based certificate applies to the unique zero of the stated strongly monotone discretization; dataset or continuum agreement additionally requires a discrepancy bound, and failed sufficient conditions mean abstention.Source: §2.6 · pp. 12–14, Figure 6 · p. 14

3 Discussion

• Under fixed squared Hilbert loss, candidate preferences depend only on the span of candidate differences, whose dimension is at most one less than the library size.This comparison subspace identifies the fraction of an unavailable reference needed to make a deployment decision.Source: §3 · p. 15
• In the reaction-diffusion census, the shared task proxy achieved 100% non-tie pair recovery, versus 82.7% for candidate-wise variational estimators, up from 59.8%.Candidate-specific correction-magnitude proxies produced almost identical choices to the shared calculation.Source: §3 · p. 15
• One full-state linearized solve propagates a library anchor’s complete constraint defect to produce a corrected task output for scoring every candidate.Query-time routing preserves input and task context, while panel-level aggregation selects one model for repeated use; both require one post-inference response for the library.Source: §3 · p. 15
• Small candidate margins amplify proxy and algebraic errors, anchor choice depends on library composition, and shock-dominated dynamics may require damping to preserve physical admissibility.Neither the arithmetic mean nor candidate medoid was universally superior, and panel-level aggregation depends on whether a small unlabeled panel represents deployment; the retained Burgers-CNO case illustrates this vulnerability.Source: §3 · p. 15
• For strongly monotone reaction-diffusion discretizations, residuals bound the unique same-grid operator zero and certify pairwise or top-1 decisions when margins suffice.Dataset or continuum guarantees additionally require an operator-data discrepancy bound; proposed extensions include evolving or mixed libraries, non-Hilbert objectives, multifidelity data, and stress tests under severe shocks or simulator mismatch.Source: §3 · pp. 15–16

Finite libraries and decision targets

• The hidden task loss is squared Hilbert-space distance to an unavailable reference, and comparisons distinguish strict or tolerant pairwise recovery from exact or tolerant top-1 selection.The eight Burgers and reaction–diffusion libraries use non-tie pairwise and tolerant top-1 recovery; the 16-library, Sine–Gordon, and PDEBench studies use strict pairwise and exact top-1. The stated identity does not extend to nonlinear task-transfer distances.Source: Finite libraries and decision targets · p. 16

Study design and statistical units

• The primary design crossed two PDEs, two architectures, and two seeded ensembles into eight libraries, with 1,920 input–library cases evaluated using input-clustered bootstrap intervals.Each library had eight checkpoints; inputs were held out, eligibility and evaluation choices were set before reference evaluation, and resampling preserved within-input dependence among candidate pairs and matched comparisons.Source: Study design and statistical units · pp. 16–17

Shared correction and candidate scores

• A deterministic library anchor defines one shared correction and task proxy, which are used to score every candidate rather than solving a separate correction for each model.The anchor is the arithmetic candidate mean for the eight Burgers and reaction–diffusion libraries and the geometric medoid for the 16-library, Sine–Gordon, and PDEBench comparisons; linear task maps make the linearized and evaluated proxies identical.Source: Shared correction and candidate scores · pp. 17–18
• Baselines include direct residual energy, candidate-specific linearized corrections, validation loss, and candidate centrality; component timings compare one shared response with N separate responses after inference.The centrality scores use squared task-space distance to the candidate mean or medoid, without the governing equation; normalized regret divides excess loss by the best-to-worst loss span with a 10^-8 floor.Source: Shared correction and candidate scores · p. 18

Viscous Burgers

• Problem-specific defects retain enforced initial and boundary conditions: Burgers uses a time-marching tangent solve, reaction–diffusion eliminates fixed Dirichlet corrections, and Sine–Gordon linearizes its discrete velocity-Verlet map.Burgers scores the terminal state, reaction–diffusion scores the full stationary field, and Sine–Gordon scores terminal displacement and velocity.Source: Viscous Burgers · p. 19, Cubic reaction–diffusion · p. 19, Sine–Gordon · p. 20

Reaction–diffusion decision certificate

• The reaction–diffusion certificate evaluates residual-based bounds on 672 pairs, while estimator controls compare the shared proxy with raw and variational residual estimators across 1,920 cases.Certificate operators act on the 128-by-128 interior space after checking zero boundary values; approximate inverse actions are inflated by solve defects and an analytic coercivity lower bound. Controls include candidate-wise inverse applications, whereas the shared score uses one correction.Source: Reaction–diffusion decision certificate · p. 19, Candidate-wise residual-estimator… · p. 19

PDEBench two-field diffusion–reaction

• PDEBench uses the same 91-step autoregressive rollout for candidates and diagnostics, while compressible-flow probes test partial corrections under operators that do not exactly reproduce the dataset generator.The PDEBench task is terminal two-field squared L2 loss; the shock probe backtracks over fixed correction multipliers to preserve admissibility without increasing defect, and the two-dimensional probe applies reference-free admissibility and defect checks.Source: PDEBench two-field diffusion–reaction · p. 20, Compressible-flow boundary studies · pp. 20–21

Panel-level model selection

• For unlabeled panels, selection averages per-input decision scores and compares the selected checkpoint’s mean task loss with the best fixed checkpoint.Reference losses for the principal PDE studies come from independent high-resolution solutions with refinement and closure checks; PDEBench instead uses released-trajectory terminal states.Source: Panel-level model selection · p. 21, Numerical references used for evaluation · p. 21

• The code and tests for reproducing the numerical analyses, plus scripts for all six main figures, are available at https://github.com/ToughClimb/OpPropRank-open-source (commit dafb2a0).Source: Code availability · p. 22

Ethics declarations

• No ethical approval was required because the study used computational experiments and publicly available or generated data, without human participants or animals.Source: Ethics declarations · p. 22

S1 Comparison geometry: expanded statements

• The comparison setup assigns each fixed candidate y_i a squared Hilbert-space distance R_i(z)=∥z−y_i∥² from target z.Source: S1 Comparison geometry: expanded… · p. 25

S1.1 Global decision quotient and bounded-linear minimality

• For a finite library, the candidate-difference span D determines exact gaps, pair signs, complete weak orderings, and tie-ruled top-1 decisions, with dim D ≤ N−1.Any finite-dimensional bounded linear statistic that recovers these decisions globally must have output dimension at least dim D.Source: S1.1 Global decision quotient and… · p. 26
• The global quotient depends on the finite candidate library, and its top-1 guarantee is over the whole task space rather than an arbitrary physical solution manifold.A proxy error of order ρk can reverse a nonzero truth gap of the same order.Source: S1.1 Global decision quotient and… · p. 26

S1.2 Restricted target classes and exposure

• On z0+V, a bounded linear statistic suffices when its invisible directions are orthogonal to projected candidate differences; necessity holds for exposing target classes.A complete affine class z0+V has gap-, sign-, and top-1-exposure.Source: S1.2 Restricted target classes and… · p. 26

S1.3 A continuous-statistic dimension bound for exact gaps

• Any continuous statistic Φ:H→R^m that recovers exact gaps must have m≥dim D, even if its decoder has no regularity assumptions.The relative-gap map is injective on D, yielding the dimension bound.Source: S1.3 A continuous-statistic dimension… · p. 27
• Exact identifiability alone does not ensure stable comparison recovery: stability holds exactly when κ_D(T) is finite, equivalently when the comparison projection factors boundedly through T.The factorization can be constructed without a closed-range assumption.Source: S1.3 A continuous-statistic dimension… · p. 27
• With observation y=Tx+n and ∥n∥≤δ, the exact deterministic minimax amplification is characterized for unrestricted estimators and the full state class.Restricted solution manifolds, non-spherical noise, stochastic risk, or prior constraints can have a smaller minimax constant.Source: S1.3 A continuous-statistic dimension… · p. 28

S2 Stable task factorization and nonlinear decision radii

• Stable task factorization is equivalent to bounding task-response changes by residual changes, with optimal factorization norm κ*(J | L).The theorem equates J = TL for bounded T with ∥Jh∥Z ≤ κ∥Lh∥Y for every h; a factorization can be chosen with ∥T∥ = κ*(J | L).Source: S2 Stable task factorization and… · p. 28
• For scalar tasks, factorization is equivalent to solving the adjoint equation, so state identifiability may fail even when a comparison direction remains observable.With exact factorization, ηv = 0, and in a C1,1 setting the task correction is second-order accurate once a localization engine supplies a radius.Source: S2 Stable task factorization and… · p. 29

S2.1 A computable local Newton-to-decision route

• A certified local Newton radius yields a unique zero in its ball and computable pairwise, top-1, regret, and abstention decisions when the required constants and supports are available.Applying the result to a separately designated physical truth additionally requires proving that the designated branch is the local zero; residual smallness alone does not identify it.Source: S2.1 A computable local… · p. 30
• A pairwise interval strictly separated from zero certifies that comparison, and intervals favoring one candidate against every competitor certify it as top-1.The interval bounds the score-gap change using the comparison-projected error radius.Source: S2.1 A computable local… · p. 31

S2.2 A conditional finite-sample bound

• For independent identically distributed inputs and bounded candidate scores, a population best-versus-second-best gap exceeding twice the concentration radius ensures empirical selection recovers the population minimizer.The stated result uses a uniform score-mean bound over the N candidates with probability at least 1 − δ.Source: S2.2 A conditional finite-sample bound · p. 31
• The finite-sample bound does not directly cover the tested B = 24 and B = 144 settings, which require a useful common score bound and i.i.d. inputs unlike their stratified, dependent designs.The panel-selection studies instead use clustered bootstrap analysis.Source: S2.2 A conditional finite-sample bound · p. 31

S3 Periodic viscous Burgers: functional setting and proofs

• For u0∈H1(T) and f∈L2(I;L2(T)), the periodic Burgers equation has a unique strong solution with norms bounded by a finite function of ν, T, and the data norms.The existence proof uses Fourier–Galerkin approximations and the uniqueness proof an energy estimate.Source: S3 Periodic viscous Burgers:… · pp. 32–33

S3.1 Burgers correction identities, conditional localization, and sharpness

• Quadratic expansion yields exact secant and plug-in identities for Burgers, with a negative sign in the truth-linearized formula.The proof expands the operator about the candidate, solution, and midpoint.Source: S3.1 Burgers correction identities,… · pp. 33–34
• If 4αvηv < 1, the localization theorem bounds ∥u−v∥X by Rv and bounds task error for any bounded linear task.The experiments do not compute a verified inverse bound or instantiate these conditional radii; the evaluated certificate chain is in reaction–diffusion, not Burgers.Source: S3.1 Burgers correction identities,… · p. 34

S3.1.1 Self-midpoint correction

• Under uniform inverse bounds, one self-midpoint correction has cubic-order error, while repeated Newton iterations can converge faster.The cubic order is a property of one self-midpoint step, not an oracle-budget optimum.Source: S3.1.1 Self-midpoint correction · p. 34

S3.1.2 Order-sharp rank reversals

• Perturbations d1=ϵ−cϵ² and d2=ϵ+cϵ² reverse the true-error and diagnostic-score ordering; a separate fixed-decoder construction yields a cubic-gap reversal.The cubic-gap example is a sharpness result for the fixed decoder, not a PDE-wide impossibility independent of residual representation or computational budget.Source: S3.1.2 Order-sharp rank reversals · p. 35

S4.1 Support functions, robust selection and regret

• The attainable pairwise gap set equals the proxy gap plus twice the uncertainty set’s image under the candidate-difference functional, giving exact interval endpoints for robust decisions.The attainable set need not fill its interval hull when uncertainty is non-convex, but the endpoints and robust conditions remain exact.Source: S4.1 Support functions, robust… · p. 36
• A candidate is robustly optimal only when it strictly defeats every alternative throughout the uncertainty set; otherwise the strict or tolerant rule abstains.The exact worst-case regret yields a necessary-and-sufficient bound for uniform τ-regret.Source: S4.1 Support functions, robust… · p. 36
• For ball uncertainty, the sharp pairwise margin radius is 2ε∥dij∥M, and independent uncertainty sources combine by Minkowski sum without assuming unbounded components are controlled.The passage identifies physics, algebraic-solve, discretization, branch, and operator–data discrepancy as combinable sources.Source: S4.1 Support functions, robust… · p. 36

S4.2 Bregman and smooth-loss coordinates

• For differentiable Bregman-type losses, exact comparison coordinates project the dual variable onto the candidate-difference span, while general smooth losses yield only a local tangent comparison space.The local expansion has a remainder bounded by (Lij/2)∥h∥² when the gradient difference is Lij-Lipschitz; it is generally not a global exact quotient.Source: S4.2 Bregman and smooth-loss coordinates · p. 37
• Coordinate-dimension arguments require relatively open primal or dual domains, and restricted sets need an exposure condition for sign or top-1 minimality.The relevant domain is the primal target domain for the primal formula and the attainable dual domain for the dual formula.Source: S4.2 Bregman and smooth-loss coordinates · p. 37

S4.3 Anchor sensitivity, composition and multiple anchors

• Anchor changes can be damped near a regular root, but the sensitivity identities do not select a universally valid anchor or guarantee the arithmetic mean or medoid.With multiple truth-free anchors, intersecting certified pair intervals can certify pairwise preferences and top-1 selection; empty intersections expose inconsistent bounds or implementations, and this construction was not evaluated in the reported studies.Source: S4.3 Anchor sensitivity, composition… · pp. 37–38

S5 Operator extensions under finite training observations

• Finite observations permit continuous extensions that preserve all recorded observations yet shift the deployment value by any h in N⋆, with sup-norm change at most ∥h∥X.Thus finite training information alone cannot determine deployment behavior in hidden directions.Source: S5 Operator extensions under finite… · p. 38

S5.1 Pairwise adjoints and smooth losses

• For squared loss, pairwise contrast cancels quadratic prediction terms, so unknown truth enters affinely; Proposition 20 presents a direct pairwise adjoint estimator.The task uses a common-grid lift with identity or terminal trace.Source: S5.1 Pairwise adjoints and smooth losses · p. 38

Let an inexact shared correction satisfy

• One inexact linearized correction yields the shared-q pairwise point estimate for every linear-task pair, whereas the conservative adjoint envelope was too loose to select models.For general smooth losses, second derivatives contribute to the remainder, and the N −1 comparison-subspace reduction is specific to squared Hilbert distance.Source: Let an inexact shared correction satisfy · p. 39

S5.2 Decoder bias, residual gauge, and computational budget

• Directional decoder bias can reverse proxy rankings despite a true norm gap of Θ(ρ^k), even when candidate directions have equal task norm.The reversal follows when radial coefficients differ and the candidates’ radii are perturbed in opposite directions by sufficiently small cρ^k.Source: S5.2 Decoder bias, residual gauge, and… · p. 39
• High-order rank curvature depends on residual coordinates and decoder, while repeated Newton steps remove any fixed rank barrier; the empirical method therefore uses one shared linearized solve per input.The paper makes no coordinate-free optimality claim for its natural complete PDE residual.Source: S5.2 Decoder bias, residual gauge, and… · p. 40

S5.3 Polynomial constraints, nonsmooth regimes, and branches

• For polynomial constraints of degree at most d, q-point Gauss–Legendre quadrature is exact when d ≤ 2q, but unknown nodes make the higher-order construction algorithmic rather than a free certificate.The Burgers midpoint identity is the degree-two, one-node case; cubic and quartic reaction terms require two nodes.Source: S5.3 Polynomial constraints, nonsmooth… · p. 40
• After shock formation, translated steps are not locally Lipschitz into L2, and strong residual losses may be unsuitable; weak dual-norm residuals and entropy admissibility are alternatives.Zero residual also need not identify the designated solution branch, so certificates require task branch uniqueness or a branch-ambiguity radius.Source: S5.3 Polynomial constraints, nonsmooth… · p. 40

S6.1 Assembled constraint defects

• Burgers propagation includes the initial mismatch and every time-step defect, using fixed integrating-factor RK4 on a 384-point common task grid and terminal periodic physical L2 quadrature.The task is the terminal field on a periodic Fourier grid.Source: S6.1 Assembled constraint defects · p. 40
• Reaction–diffusion defects include predicted Dirichlet boundary values and the full conservative interior PDE residual, with boundary corrections imposed exactly before solving the Jacobian system.The reaction factor is λ + 3γw^2, solved by preconditioned conjugate gradients; the common grid has 128 interior points per direction and physical tensor-product quadrature including boundary nodes.Source: S6.1 Assembled constraint defects · pp. 40–41
• Sine–Gordon defects include both initial fields and every velocity-Verlet displacement and velocity step, closed by exact lower-triangular Jacobian forward substitution.The task metric is periodic physical L2 on terminal displacement plus terminal velocity with weight one, after Fourier lifting candidate fields to the common task grid.Source: S6.1 Assembled constraint defects · p. 41
• PDEBench rolls each checkpoint autoregressively to 101 frames and propagates the complete two-field trapezoidal-map defect using block-Jacobian forward substitution.The cell-centred 128×128 homogeneous-Neumann grid uses GMRES with state-dependent block-diagonal sparse-LU preconditioning, tolerances 10^-10 relative and 10^-12 absolute, and a fail-closed 100-iteration limit; the task metric is cell-area-weighted squared L2 over both terminal fields.Source: S6.1 Assembled constraint defects · p. 41
• Raw residual comparisons use explicit block, time, and quadrature product weights on the same candidates, inputs, and references as propagated physics, but propagation adds a linearized solve.Map defects already contain time-step factors; the weights approximate block integrals and are not optimized stability or dual norms, leaving sensitivity to alternative residual gauges as a separate question.Source: S6.1 Assembled constraint defects · p. 41

S6.2 Numerical-reference checks

• For the eight-library two-PDE references, coarse/fine checks and targeted second-method checks on 12 inputs per PDE found no decision-changing discrepancy.Burgers references used 768- and 1,536-point Fourier grids; reaction–diffusion used 511- and 767-point interior grids, with Newton tolerances 10^-11 relative and 10^-13 absolute and inner linear-solve tolerance 10^-12.Source: S6.2 Numerical-reference checks · p. 42
• For 16 libraries, refinement found no indeterminate pair among 107,520 or top-1 case among 3,840; Sine–Gordon had 10 unresolved pairs and three unresolved top-1 cases, counted as errors.The eight-library rule flagged three of 53,760 pairs and one of 1,920 top-1 cases, retained in reported results; Sine–Gordon used two spectral velocity-Verlet resolutions plus residual, Jacobian, closure, and 16-input DOP853 checks, while tie checks were separate from uncertainty checks.Source: S6.2 Numerical-reference checks · p. 42
• PDEBench terminal states and grids matched published trajectories; GMRES took five or six iterations per step, with maximum relative step residual 9.999 × 10^-11 and constraint closure 2.310 × 10^-10.Source: S6.2 Numerical-reference checks · p. 42

S7.1 Eight-library two-PDE comparison

• Eight cells crossed PDE, FNO/CNO architecture, and ensemble replicate, with epoch-20 and epoch-40 checkpoints from four runs per cell; each PDE had 240 balanced inputs across frequency, parameter, and combined shifts.The libraries covered Burgers and reaction–diffusion, with shifted inputs extending basis support, parameter ranges, or both beyond the training generator.Source: S7.1 Eight-library two-PDE comparison · p. 42

S7.2 Internally trained candidate contracts

• Burgers, reaction–diffusion, and Sine–Gordon candidates used 2,048 training and 256 validation samples, 40 epochs, epoch-20/40 checkpoints, and exact initial or boundary conditions.The internally trained libraries included FNO and CNO candidates; Sine–Gordon used eight libraries and enforced both initial phase channels exactly.Source: S7.2 Internally trained candidate… · p. 43

S7.3 Small-panel model selection

• Eight balanced 24-input panels per PDE used nested budgets of 3, 6, 12, and 24, with a disjoint 240-input deployment population for evaluating model selection.Inference conditioned on the eight libraries and resampled panels and deployment inputs within PDE and shift stratum.Source: S7.3 Small-panel model selection · p. 43

S7.4 Sixteen-library comparison

• Four new library replicates per PDE/architecture cell produced 16 libraries and 128 candidates, with training runs, seeds, checkpoints, training samples, and validation samples disjoint from the earlier population.The study used 480 instance-wise inputs, 2,304 selection inputs, and 1,920 evaluation inputs; the selection-panel size was 144.Source: S7.4 Sixteen-library comparison · p. 43

S7.5 Candidate-composition stress test

• 97.9883% strict pairwise and 94.5833% exact top-1 accuracy with an arithmetic-mean common state exceeded raw residual’s 71.7736% and 41.0677% on the stress population.The geometric medoid raised the affected reaction–diffusion CNO cell to 100% on both metrics; retaining the eight-candidate mean while evaluating six clustered candidates gave 54.5278% pairwise accuracy, while recomputing the shared state yielded 3,600/3,600 pair and 240/240 top-1 decisions.Source: S7.5 Candidate-composition stress test · pp. 43–44
• An outlying candidate could alter the shared decision coordinate even when excluded from a comparison; the medoid corrected the affected CNO cell, while changing no Burgers top-1 decisions.For Burgers, the medoid reduced the correct-pair count by ten among 53,760 comparisons.Source: S7.5 Candidate-composition stress test · p. 44

S7.6 Sine–Gordon comparison

• The Sine–Gordon study tested input generalization for a third PDE using eight FNO/CNO ensemble libraries and 240 new balanced inputs disjoint from earlier populations.It used the geometric-medoid common state and fixed phase-space metric; every indeterminate outcome counted as an error.Source: S7.6 Sine–Gordon comparison · p. 44

S7.7 Public PDEBench model library

• The public PDEBench library comprised three released U-Net checkpoints differing in training strategy, evaluated on identifiers 0901–0999 from the official final split.No model was fine-tuned, and the official training code assigns identifiers 0900–0999 to its validation object, so these inputs were absent from gradient training.Source: S7.7 Public PDEBench model library · p. 44

S9.2 Pooled instance-wise results · S9.4 Eight-library two-PDE results · S9.1 Evaluation sets

• In the eight-library study, exact top-1 was 1,899/1,920 for propagation versus 688/1,920 for raw residual, and propagated all-pair three-way accuracy was 91.4732%.Populations were analysed separately; the reaction–diffusion FNO cells contained most reference ties, and the eight-library pooled non-tie denominator was 49,323. Sine–Gordon retained ten pair and three top-1 refinement-indeterminate outcomes as errors; PDEBench had one uniform truth ranking across all 99 inputs.Source: Table 1 · p. 45, S9.2 Pooled instance-wise results · p. 46, Table 2 · p. 46

S9.3 Metric and mechanism sensitivity · S8 Metrics, aggregation, and uncertainty

• With the same 10^-6 tolerance on both diagnostics, propagation correctly labeled 53,517/53,760 pairs (99.5480%) and recovered 4,413/4,437 reference ties; raw residual labeled 33,456/53,760 (62.2321%) and recovered one tie.Propagation’s sign accuracy ranged from 97.7913% in the lowest margin decile to 99.3917–100% in deciles 2–10, while raw accuracy ranged from 56.5755% to 77.3133%. Mean span-normalized regret was 2.4048 × 10^-4 for propagation versus 0.274746 for raw residual. Pairwise correctness was conditioned on reference non-ties in the primary study; studies also used exact or tolerant top-1 rules as specified, and bootstrap resampling accounted for clustered inputs rather than treating pairs as independent.Source: S8 Metrics, aggregation, and uncertainty · pp. 44–45, S9.3 Metric and mechanism sensitivity · pp. 46–47

S9.5 Sixteen-library two-PDE results

• Across 16 libraries, propagated-minus-raw gains were 29.2711 percentage points [28.4673, 30.0791] pairwise and 55.4732 [51.0410, 59.9219] top-1; clustered-bootstrap means were 99.7063% and 98.9549%.The pooled count ratios were 107,205/107,520 pairwise and 3,800/3,840 top-1; strict top-1 denotes exact checkpoint agreement.Source: S9.2 Pooled instance-wise results · p. 46, S9.5 Sixteen-library two-PDE results · p. 48, Table 5 · p. 48

S9.6 Sine–Gordon cells

• In auxiliary Sine–Gordon populations, propagation reached 97.9018% pairwise and 95.0521% top-1 at the initial reference resolution, versus 51.9773% and 12.2917% for raw defect energy.After reference-grid refinement, the respective accuracies were 98.3612% and 96.4063% versus 51.6648% and 13.0729%; these auxiliary populations were not pooled with the disjoint conservative population.Source: S9.6 Sine–Gordon cells · p. 49, Table 6 · p. 49

S9.7 Public PDEBench model library

• On all 99 fixed PDEBench inputs, propagated scores matched the true autoregressive, push-forward-20, one-step ordering, while raw scores reversed it; the propagated terminal estimate was strictly better than the oracle-best checkpoint on every input.Mean truth loss was 0.004332 for the propagated estimate versus 0.010511 for the oracle-best checkpoint. The result supports the physics score on this fixed external library, not input-adaptive routing, because the excluded development input had the same candidate ordering.Source: S9.2 Pooled instance-wise results · p. 46, S9.7 Public PDEBench model library · p. 49, Table 7 · p. 49

S9.8 Selection-panel size

• Across selection-panel sizes, propagated mean NRegret fell from 0.0351 at B = 24 to 0.0049 at B = 144 and 0.0030 at B = 192, while exact-best frequency was non-monotone.In the 24-input panels, Burgers CNO ensemble 2 exceeded raw defect energy by 0.07289 mean normalized regret and selected a top-two model in only one of eight panels; neither Burgers architecture group improved on raw mean normalized regret.Source: S9.8 Selection-panel size · p. 50, Table 8 · p. 50

S9.9 Candidate-specific correction control

• Shared and candidate-specific correction selected the same checkpoint in 191/192 cases; among 89 pair-label disagreements, shared correction was correct on 48 versus 40 decisions.Reaction–diffusion was perfect for both propagated methods, with remaining errors in the two Burgers pools; the earlier eight-candidate pair control was pair-cross-fit and does not establish a one-versus-eight solve cost comparison.Source: S9.9 Candidate-specific correction… · pp. 50–51, Table 9 · p. 50

S9.10 Truth-covariant flips and direct use of q · S9.11 Component cost accounting

• Among 15,790 strict diagnostic flips, 98.3787% accompanied a truth flip, and the shared corrected output matched or improved on the best checkpoint’s task loss in all 1,920 primary cases.The shared diagnostic uses one correction solve per input/library versus one per candidate for the candidate-specific construction; measured component scaling does not determine end-to-end latency, which also depends on inference, data movement, scheduling, and serving.Source: S9.10 Truth-covariant flips and direct… · p. 51, S9.11 Component cost accounting · pp. 51–52, Table 11 · p. 51

S10.1 Fixed candidates

• For the new-input reaction–diffusion 256-to-512 transition, the four cells had 0, 1, 4, and 1 tolerance-label changes, with no top-1 changes.Every changed pair was within the bottom 0.70% of its cell’s margin distribution; these finite-sequence results do not establish a continuum rate.Source: S10.1 Fixed candidates · p. 53

S10.2 Exact remainder populations

• 65.4464% and 65.0800% pooled pair-screen coverage was achieved across the two cross-resolution populations, and interval separation exactly determined every covered fine-grid diagnostic decision.The screen measures diagnostic stability between computed grids, not hidden-reference ranking; omitting shared-diagnostic drift failed on 5,729 and 5,819 rows, respectively.Source: Table 14 · p. 54, S10.2 Exact remainder populations · p. 54

S10.3 Locality and fragility

• FNO pooled AUROC improved by 0.0662, while CNO pooled AUROC changed by −0.0342, for the first-order gap-change predictor versus the margin-only flip predictor.AUPRC rose from 0.3639 to 0.5060 for FNO and fell from 0.5724 to 0.5455 for CNO; locality’s Spearman association with AUROC increment was −0.9524 across eight cells, with locality confounded by architecture. Opposite FNO and CNO trends preclude a common empirical threshold in these data.Source: S10.3 Locality and fragility · p. 55

S10.4 Other auxiliary analyses

• Principal angles did not explain pair flips, low-mode concentration did not establish effective rank, and the sequential pilot yielded no cost reduction.The task-difference SVD placed 95.5% of energy in low modes, while the conservative adjoint-residual envelope had a median envelope-to-gap ratio of approximately 1.98 × 105; the pilot retained accuracy only after computing every expensive probe and the full solve.Source: S10.4 Other auxiliary analyses · p. 55

S11.1 Typed discrete setting

• The certificates apply to the interior restriction of a conservative, symmetric positive diffusion operator with verified zero Dirichlet boundary values, not the nonsymmetric full matrix.The task map uses a weighted adjoint, which need not be an unweighted Euclidean transpose.Source: S11.1 Typed discrete setting · p. 56

S11.2 Global operator-aligned enclosure

• The global enclosure guarantees a unique discrete solution and bounds its error using an operator-aligned metric; the constant 3/4 is sharp.The proof uses the coercive, strictly convex functional and the positive symmetric diffusion operator.Source: S11.2 Global operator-aligned enclosure · pp. 56–57

S11.3 Pairwise and top-1 decisions

• Pairwise loss preferences are certified when the approximate gap exceeds its bound, while top-1 rules certify strict or τ-tolerant choices and otherwise abstain.One residual inverse action and r comparison-basis inverse actions with the shared matrix evaluate all N(N −1)/2 bounds; cost depends on comparison rank, not just library size.Source: S11.3 Pairwise and top-1 decisions · p. 58

S11.4 Transfer from operator truth to dataset truth

• Transferring operator-truth decisions to dataset truth requires a projected discrepancy enclosure; discrepancy orthogonal to the decision subspace does not affect squared-loss decisions.No reference-free discrepancy enclosure was available for PDEBench or the exploratory flow datasets, so the proposition specifies needed information without providing it.Source: S11.4 Transfer from operator truth to… · p. 59, If only · p. 59

If only

• A projected discrepancy bound can be added to the pairwise bound to transfer its certificate, while components orthogonal to the decision subspace leave squared-loss decisions unchanged.The proposition formalizes this additional requirement; the numerical experiments did not have a reference-free discrepancy bound.Source: S11.4 Transfer from operator truth to… · p. 59, If only · p. 59

S11.5 Floating-point solve-defect control

• Residual-inflated inverse norms account for algebraic solve defects, but adding a small machine-epsilon multiple does not bound floating-point roundoff.A strict floating-point certificate would require outward rounding or another validated arithmetic implementation.Source: S11 Operator-aligned selective… · p. 56, S11.5 Floating-point solve-defect… · p. 59

S11.6 Numerical certificate margins and same-grid comparison

• The numerical study evaluated four reaction–diffusion cells with eight candidates each across 24 library–input cases, but its comparisons do not propagate numerical solver uncertainty.The high-accuracy same-grid solver residual ranged from 6.27 × 10−14 to 2.26 × 10−12; the projected fine-grid reference had operator defect 6.28 × 10−4–2.11 × 10−3 and discrete L2 discrepancy 4.13 × 10−6–9.97 × 10−6 from the numerical same-grid zero.Source: S11.6 Numerical certificate margins… · pp. 59–60, Table 15 · p. 59, Table 16 · p. 60

S12.1 One corrected trajectory, multiple deployment objectives

• Across ten correlated objectives, matched rescoring achieved 2,938/2,970 correct pairs, 979/990 top-1 decisions, and 959/990 complete orders; corrected trajectory error never exceeded the best candidate in 990 cells.For the high-band objective, it selected the true winner on 88/99 inputs, versus 69/99 for the fixed terminal score, 23/99 for objective-matched common-state rescoring, and 7/99 for raw defect energy. The same corrected trajectory is reused without another physics solve; only the two squared-L2 objectives are exact squared-Hilbert tasks, while other reductions are empirical transfer tests.Source: S12.1 One corrected trajectory,… · pp. 60–62, S12 Multi-objective transfer and proxy… · p. 60, Table 17 · p. 61

S12.2 Partial correction separates proxy and comparison mechanisms

• At α = 1, Sine–Gordon achieved 52,728/53,760 correct pairs and 1,833/1,920 exact top-1 decisions, with lower trajectory error than the oracle-best candidate in all 1,920 cases.These path results use regenerated 48-point trajectory truth and lifted terminal comparisons, unlike the conservative grid-refinement counts of 52,942/53,760 pairs and 1,835/1,920 top-1 decisions.Source: Figure 4 · p. 63, S12.2 Partial correction separates… · p. 64
• At α = 1, pairwise accuracy rose from 51.9% to 98.1% in Sine–Gordon and from 54.9% to 98.9% in PDEBench.Overshooting to α = 1.25 reduced accuracy to 96.9% and 93.5%, respectively; the candidate library and hidden truth margins stayed fixed along the path.Source: Figure 4 · p. 63, S12.2 Partial correction separates… · p. 64
• PDEBench gained comparison-conditioned accuracy (16.1-fold decision-subspace versus 9.86-fold full-trajectory contraction), while Sine–Gordon gains were primarily full-proxy effects.Exact pair-gap drift identities closed to 2.11 × 10−14 in squared-Hilbert tasks, and smaller truth margins among flipped pairs linked remaining Sine–Gordon errors to close decisions.Source: S12.2 Partial correction separates… · pp. 64–65

S13 Candidate-wise variational residual baselines

• The shared corrected proxy recovered all 42,761 non-tie pair decisions and all 1,920 tolerance-optimal choices; common-energy scoring gained 22.96 pairwise and 34.38 exact top-1 accuracy points over raw residual.The common-energy estimator also reduced mean regret by a factor of 74.9; the proxy’s three exact top-1 misses had maximum regret 4.19 × 10−9.Source: Table 18 · p. 65, S13 Candidate-wise variational… · p. 66
• The common-energy and candidate-adapted estimators chose the same top-1 candidate in all 1,920 cases and differed on only two of 53,760 pair preferences.The adapted construction added matrix setup and nearly doubled median component time without improving decisions; each variational comparator used eight residual inverse actions per case, versus one library-level linearized correction for the shared comparator.Source: S13 Candidate-wise variational… · p. 66
• Variational scores are individual error upper bounds, not pairwise certificates, so the smallest majorant need not identify the smallest true error.This comparison concerns the stationary identity-task reaction–diffusion population, and reported timings cover post-inference diagnostic components only.Source: S13 Candidate-wise variational… · p. 66

S14 Two-dimensional compressible-flow boundary probe

• Correction improved both errors in all ten two-dimensional flow cases, but the candidate-difference component improved more strongly than the perpendicular component, which improved in only four.The squared-gap identity’s maximum absolute discrepancy was 4.05 × 10−11.Source: S14 Two-dimensional compressible-flow… · p. 66, Table 19 · p. 67
• The admissibility and defect rule selected α = 1 in all ten two-dimensional cases, although full-state error was minimized at α = 1 in eight and α = 0.75 in two.The exploratory α = 1.25 worsened full error relative to α = 1 in every case, so the corrected trajectory was not uniformly preferable to returning the FNO.Source: S14 Two-dimensional compressible-flow… · p. 67
• With two candidates, the geometric-medoid objectives tied, and correction from the tie-rule-selected FNO center improved full error in 4/10 cases (median factor 0.808).Comparison error improved in 5/10 cases (median factor 1.667); the primary result was conditional on the arithmetic-mean common state, and the FNO was the hidden-reference winner on all ten inputs.Source: S14 Two-dimensional compressible-flow… · p. 67

S15 One-dimensional compressible-flow validity probe

• Backtracking improved full-state error in 33/35 cases (median factor 2.04) and comparison-subspace error in 34/35 (median factor 5.20).It selected the largest admissible multiplier whose operator defect did not exceed the common-state defect; every inadmissible full step counted as an incorrect decision.Source: Table 20 · p. 67, S15 One-dimensional compressible-flow… · pp. 67–68
• One checkpoint won on 33/35 one-dimensional inputs, while the best fixed selector reached 94.3% top-1 versus backtracking’s 91.4%.The released-truth defect was 0.102 times the common-state operator defect, quantifying but not removing operator mismatch; strong-shock conclusions require an operator-aligned generator study.Source: S15 One-dimensional compressible-flow… · p. 68