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Detecting and Discriminating Operator Misspecification in Hybrid PDE-Parameter Learning: a Reference-Free Instrument, with Discrimination Bounded In Sample

Eric Fock

arXiv:2608.16925v1cs.LGmath.NAstat.ME

TL;DR

Hybrid PDE-parameter estimators can fit observations accurately while using a wrong operator, and existing diagnostics do not distinguish operator misspecification from parameter non-identifiability. The paper introduces a reference-free, single-fit instrument whose two readings separate these failures across three deployable designs, remaining quiet under correct specification and firing under both misspecifications.

  • Problem

    A missing single-fit, oracle-free reading leaves practitioners unable to distinguish a wrong operator from a merely unidentifiable parameter.

  • Method

    The paper uses an information-matrix specification statistic with plug-in scale and a complementary rank reading to discriminate operator misspecification from non-identifiability.

  • Results

    The instrument stays quiet under correct specification but fires in every replicate under two misspecifications, while two readings separate the failures across three deployable designs.

  • Takeaways & Limitations

    Practitioners should not accept a recovered physical coefficient from in-domain fit accuracy alone, but should combine out-of-domain evaluation, specification testing, and non-identifiability checks.

  • Takeaways & Limitations

    The guarantee is established only for the exactly solvable operator at the studied noise levels, leaving breadth across other physics and measured data for future work.

Abstract

from arXiv · show

We build an instrument that reads, from a single fit and with no oracle, whether the operator a hybrid PDE-parameter estimator postulates is wrong-and separates that from a merely unidentifiable parameter. On one self-adjoint parabolic inverse problem, an information-matrix statistic with plug-in scale and per-seed parameter has median 0.19 under correct specification, rejection rate $0.033$ against a pre-registered ceiling of $0.10$, and rises to $224$ and $85$ under two misspecifications, firing in every replicate. On a correctly specified but non-identifiable design it stays mute-$0.050$ at $n=200$, Clopper-Pearson $[0.024, 0.090]$-while a rank statistic collapses to zero at a pre-registered boundary $c_5^*=2.15\times10^{-3}.$ Two readings of one fit therefore separate the two failures across the three designs a deployable test reaches. That separation is the contribution; detection alone is a crowded flank. In sample it is a bound, out of sample a direction. It is needed because the usual accuracy check is blind: the misspecified estimator's in-domain RMSE is $2.7\times 10^{-2}$, below the observation noise for $σ\geq 0.05,$ while the coefficient is wrong by $29.7\%$ at zero noise, $31.2\%$ at the loudest. Nor is the failure architectural: a one-parameter curve fit, a bare parameter and multilayer perceptrons of $49$ and $241$ parameters converge to the same pseudo-true, matched in closed form to $0.07\%,$ whereas a physics-informed network, with its composite objective, converges to a disjoint one. We report where the instrument is blind, a pre-registered negative where a neural estimator loses to Tikhonov-regularized inversion at recovery, and the hypothesis under which its guarantee holds but a trained network violates it.

I. INTRODUCTION

The paper introduces a reference-free, single-fit instrument that detects operator misspecification and distinguishes it from parameter non-identifiability. It shows why standard residual and accuracy checks can miss materially wrong coefficients, while the observed plateau reflects specification and objective rather than architecture.

  • Diagnostic behavior: Under correct specification αerr scales linearly with noise, with measured slopes 1.007 and 1.000, whereas MSE scales quadratically at 2.015 and 2.004.Misspecification sends both slopes to zero, making a flat slope a signature only when accompanied by its stated qualifier.
  • Motivation: 2.7 × 10−2 in-domain RMSE remains below measurement noise for σ ≥0.05, while diffusivity is wrong by 29.7% at zero noise.The misspecified estimator can therefore appear accurate inside the fitted window despite a materially incorrect coefficient.
  • Contribution: A one-parameter curve fit, scalar parameter, and 49- and 241-parameter MLPs converge to the same wrong pseudo-true value, whereas a PINN converges elsewhere.White’s closed-form pseudo-true parameter matches to 0.07%, showing the plateau tracks the loss class and specification rather than network architecture.
  • Contribution: A single-fit information-matrix test uses no oracle or held-out correct model to assess whether the operator is wrong.Under correct specification, its median is 0.19 with rejection 0.033 against a pre-registered ceiling of 0.10; under two misspecifications it rises to 224 and 85.
  • Contribution: On a correctly specified but non-identifiable design, IN stays mute at rejection 0.067 while the Fisher-rank statistic collapses to zero.This supplies the second reading needed to distinguish operator misspecification from a parameter that the data cannot constrain.

II. RELATED WORK

Prior work explains physics-informed and hybrid PDE learning through optimization, spectral, and representation failures, while misspecification and identifiability are usually treated as separate problems. This paper distinguishes operator misspecification from parameter reachability by using two readings of one fit, while separating both from out-of-distribution failure.

  • Learning PDE coefficients and failure: Physics-informed learning recovers coefficients through PDE-residual penalties, with documented failures from ill-conditioning, gradient-flow pathologies, and spectral shifts.These works motivate established explanations for failure in physics-informed PDE learning.
  • What the network is allowed to represent: Spectral bias explains the ordering of collapse severity, but the tested flexible head produces spurious low-frequency fields that spectral bias alone could represent.The paper therefore treats spectral bias as insufficient to explain the observed collapse.
  • Hybrid estimators: Hybrid estimators combine first-principles structure with learned components; the χ-architecture preserves the postulated form and learns only coefficients through χ = ∂F/∂θ.This places the paper’s estimator within a longstanding semi-parametric and neural-identification lineage.
  • Misspecification: Under misspecification, M-estimators converge to pseudo-true parameters, while existing discrepancy and uncertainty methods correct or bound the discrepancy rather than treating the pseudo-true value as signal.The paper instead uses the computable, architecture-invariant pseudo-true parameter within its stated scope.
  • Misspecification detection: Existing information-matrix tests can be blind to misspecified first-order moments, whereas this work adds an orthogonal reading of the same fit to distinguish operator error from coefficient reachability.The cited limitation arises when nuisance substitution makes the information forms share a probability limit; the complementary question is whether the coefficient was reachable at all.
  • Identifiability and adjacent failure modes: Identifiability analysis diagnoses whether parameters are recoverable under a potentially correct operator, while out-of-distribution detection concerns sampled support rather than a changed data-generating law.Benchmark splits that vary physical parameters while keeping the generating operator fixed therefore address a different axis from operator misspecification.

III. SETUP · A. Problem and data

The paper sets up a one-dimensional heat-equation inverse problem and identifies the decay rate against a nominal coefficient. It defines operator misspecification as imposing the wrong PDE form, independent of learned-component capacity or probabilistic-model errors.

  • III. SETUP: Table II collects the notation and separates three pairs of symbols that readers might conflate.This notation is part of the setup’s implementation details.
  • A. Problem and data: The setup uses the one-dimensional heat equation on [0, L] × [0, T] with homogeneous Dirichlet conditions.Its solution is represented as a modal superposition.
  • A. Problem and data: The identification target is the rate ¯αn from equation (1).The target is reported against the nominal α0 = 0.01.
  • A. Problem and data: Operator misspecification means that the PDE form imposed on the estimator is wrong.This definition applies even when the learned component has arbitrarily large capacity.
  • A. Problem and data: Capacity misspecification instead refers to a target outside the hypothesis space.The paper distinguishes this from its operator-focused use of misspecification.
  • A. Problem and data: Probabilistic-model misspecification refers to a mis-stated likelihood or prior.The paper treats this as distinct from an incorrectly specified PDE operator.

B. Specification scenarios · C. Estimators

The study probes well-specified, single-misspecified, and double-misspecified operator structures, while comparing scalar, modal, field, bare-parameter, and neural estimators. Results show that the observed plateau follows the postulated ansatz rather than estimator family, with the correctly structured modal estimator remaining free of it.

  • B. Specification scenarios: Well-specified configuration 32cf4a5526cb uses modes [5], coefficients [1.0, 0.3, 0.1], and one shared diffusivity, making the scalar model correct at α∗ = 0.010000.The configurations are content-hashed to probe the postulated operator structure.
  • B. Specification scenarios: Single-misspecified configuration 7935b218c150 assigns diffusivities {1:0.010, 3:0.005, 5:0.020}, making the scalar model wrong while the modal model remains correct at α∗ = 0.007033.The same modes and coefficients are retained across the comparison.
  • B. Specification scenarios: The notation deliberately separates indexed h_i from unindexed h, scalar information from profiled I_55, and two distinct δ quantities.These distinctions prevent conflating per-point Hessian contributions, miss-rating amplitude, information statistics, data misspecification, and loss perturbation.
  • B. Specification scenarios: α∗ = 0.007778 characterizes the double regime, where the modal head learns three of four modes but is structurally wrong and has no scalar pseudo-true parameter.The estimand changes in nature because the double regime is head-dependent and yields per-mode rates.
  • C. Estimators: The estimator set includes Hyb scalar with 1 bare parameter, MLP scalar with 49 parameters, MLP scalar big with 241 parameters, Hyb modal with 3 bare parameters, and a PINN contract fixed at 4 layers and 8578 parameters.Hyb modal uses one diffusivity per spectral mode, whereas the MLP estimators emit the same scalar coefficient through multilayer perceptrons.
  • C. Estimators: The plateau tracks the ansatz rather than estimator family: it appears for scalar and field heads but is absent for the correctly structured modal head.This pattern is reported across the estimator-and-specification ablation cascade.
  • C. Estimators: Architecture contracts verify layer presence or absence and parameter-count bounds, including 0 linear layers for Hyb scalar and Hyb modal and a true MLP for MLP scalar.The PINN architecture is frozen at 4 layers and 8578 parameters under the stated certification contract.
  • C. Estimators: Identification error is defined as αerr = |α̂ − α0|/α0 and reported at zero noise as a median over n = 30 against nominal α0.This table establishes the metric and evaluation setting for estimator comparison.

D. Baseline fidelity: the PINN is not weakened … A. The zero of the instrument

The paper validates the PINN as a faithful discriminant and uses architecture-free controls to show that the misspecified plateau is not caused by network capacity, optimization, or under-training. Under correct specification, the instrument reaches a near-zero baseline while prediction and identification errors remain separately interpretable.

  • D. Baseline fidelity: the PINN is not weakened: 1.9 × 10−3 recovery at zero noise shows that the reference PINN reproduces the original work’s sub-percent result under correct specification.The study varies capacity and includes a per-mode variant to separate capacity from specification.
  • E. The architecture-free control: At σ = 0, cf and Hyb scalar converge to the same minimizer, White’s α∗, making cf an architecture-free control rather than a competitor.This supports attributing the plateau to the misspecified least-squares projection rather than to network architecture.
  • A. The zero of the instrument: 2.2 × 10−8 identification error and out-of-domain MSE below 10−15 establish the correct-specification zero for the χ-architecture.All 30 seeds were bit-identical, with identification error below the float32 epsilon of 1.2 × 10−7.
  • A. The zero of the instrument: Median IN = 0.193 and rejection rate 0.033 (1/30) stay below the pre-registered ceiling of 0.10 under correct specification.This information-matrix baseline is the zero against which the misspecified cases are read.
  • A. The zero of the instrument: 0.60% and 0.70% median error remain within a factor 1.7 of the 0.42% Cramér–Rao bound at σ = 0.02.The result indicates that the 30% plateau is not explained by estimator imprecision; PINN small does not pass this reading.
  • A. The zero of the instrument: Exactly 0 stored MSE for all 30 seeds, reported as < 10−15, results from bit-identical reconstruction on the exact analytical basis.The single-precision machine floor on this grid is approximately 3 × 10−15.
  • A. The zero of the instrument: 36.5% versus 29.7% identification error shows that co-estimating coefficients lowers prediction error while worsening parameter recovery.This separates fit quality from identification quality.
  • A. The zero of the instrument: 3.7 × 10−8 in-domain PINN error and cf’s matching pseudo-true value close the under-training and architecture objections.The PINN fits its own window closely, while cf reaches the same pseudo-true without a network or training.

B. A plateau appears … E. It is not the architecture

The misspecified estimator reaches a noise-insensitive plateau at a wrong diffusivity, matching a closed-form pseudo-true parameter rather than the nominal coefficient. This plateau is shared across least-squares architectures but not by a PINN, showing that loss class—not capacity or architecture—drives the failure.

  • B. A plateau appears: 29.7% at σ = 0 and 29.8% at σ = 0.02 quantify the scalar estimator’s error against nominal α0 = 0.01 under a missing modal component.Its out-of-domain MSE is 1.44 × 10−3.
  • C. It is not noise: 0.0049 is the scalar estimator’s out-of-domain MSE log–log slope over σ ∈[0, 0.2], 409 times below the correctly specified modal estimator’s 2.004.The 95% CI is [0.0019, 0.010], and identification error remains 29.7% at σ = 0 and 29.8% at σ = 0.02.
  • D. It is not chance: White’s pseudo-true: 0.010001 / 0.007033 / 0.007777 are the closed-form pseudo-true values, versus empirical medians 0.010000 / 0.007030 / 0.007772.The corresponding gaps are 0.01% / 0.05% / 0.07%, establishing the plateau as a quasimaximum-likelihood pseudo-true.
  • E. It is not the architecture: 0.007033 is shared by a 1-parameter curve fit, a 1-parameter bare model, and MLPs with 49 and 241 parameters within least squares.Their overlapping bootstrap intervals span [0.0070178, 0.0070408].
  • E. It is not the architecture: [0.0069141, 0.0069567] is the PINN’s disjoint interval, separated from the least-squares cluster by 6.11 × 10−5.Its composite data-misfit-plus-PDE-residual objective defines a different M-estimation problem and pseudo-true.
  • E. It is not the architecture: Identifiability of the parameterization does not ensure correctness of the postulated operator form: constrained and unconstrained heads share the same wrong α∗.The passage frames this as the limit of architectural constraints, not a remedy for misspecification.
  • E. It is not the architecture: The paper takes the plateau as a loss-class property but leaves the analytic characterization of the composite pseudo-true to future theory.This explicitly limits the claim while using the PINN’s departure to show least-squares invariance is not tautological.
  • E. It is not the architecture: Capacity does not help because the error is not approximation error; therefore in-domain validation and model scaling cannot detect or repair operator misspecification.No architecture within a loss class escapes that class’s pseudo-true.

F. It is reversible · V. A REFERENCE-FREE TEST ON A SINGLE FIT

Restoring the missing structure makes the modal-head indicators return toward machine precision, while a reference-free, single-fit statistic evaluates specification using only fitted quantities. Its certified scope is deliberately bounded: it discriminates selected failures and supports mode localization, but does not discover physics outside the postulated basis.

  • F. It is reversible: The modal head recovers the true per-mode rates out of sample, with errors as low as 2.2×10−6% and MSE of 1.4 × 10−14 under single misspecification.The return to zero is established for the modal head through per-mode recovery and out-of-domain MSE, not through an information-matrix statistic.
  • V. A REFERENCE-FREE TEST ON A SINGLE FIT: The statistic compares fitted residual-score and Hessian information, whose discrepancy has expectation zero when the operator is correctly specified.The curvature term ri∂2 contributes only vanishes in expectation when the operator is right.
  • V. A REFERENCE-FREE TEST ON A SINGLE FIT: Everything is evaluated at the fitted pair (α̂, σ̂2), so the reading requires no true field or oracle.The sensitivity is the same χ injected into backpropagation, and the statistic retains the residual-curvature term.
  • V. A REFERENCE-FREE TEST ON A SINGLE FIT: Each seed estimates its own α̂ and σ̂2, while a 95th-percentile parametric bootstrap supplies one fitted-pair threshold per regime.The bootstrap uses Nboot = 2000 and does not share quantities between seeds.
  • V. A REFERENCE-FREE TEST ON A SINGLE FIT: 0.10 is the out-of-sample well-specified rejection rate, exactly at the pre-registered ceiling; refitting shifts the threshold from 2.408 to 2.484 while leaving rejection at 0.033.The χ2(1) calibration failed with Kolmogorov–Smirnov p = 0.000, and the certified convention is the frozen bootstrap.
  • V. A REFERENCE-FREE TEST ON A SINGLE FIT: 100% of four permutations identify the missing mode for σ ≤0.05 using sine-basis residual projections, declining to 99.2% at σ = 0.10 and 81.7% at σ = 0.20.The peak coefficient’s signal-to-noise ratio exceeds 3, with severity ordered by the missing mode’s energy.
  • V. A REFERENCE-FREE TEST ON A SINGLE FIT: The projection can locate a missing coordinate within the postulated sine basis, but cannot discover missing physics whose functional form lies outside that span.This reading is supporting rather than certified: it carries no frozen verdict and discloses two pre-registration exceptions.

VI. DISCRIMINATING A WRONG OPERATOR FROM AN UNIDENTIFIABLE ONE

Two readings of one fit distinguish a correctly specified but unidentifiable operator from a wrong but identifiable one across the three deployable designs. The result is bounded in sample, directional out of sample, and limited to the studied parabolic system.

  • Empirical discrimination: 0.067 rejection under non-identifiability stays below the ceiling while the rank statistic collapses to 0.000; under misspecification, the test fires at 1.00 and rank stays large.The non-identifiability interval is Clopper–Pearson [0.01, 0.22].
  • Empirical discrimination: Two readings of one fit separate correct-identifiable, correct-unidentifiable, and wrong-identifiable designs without an oracle or held-out correct model.The wrong-operator, unidentifiable corner is only an analytic reading using the true field, not a deployable result here.
  • Threshold and regularity: 2.15 × 10^-3 is the alternative threshold obtained by anchoring T5(c5=0.1) at 2159 in the correctly specified row.T5 grows with N and 1/σ, so the threshold is not design-invariant.
  • Empirical discrimination: 0.050 rejection in the extended non-identifiable design has Clopper–Pearson [0.024, 0.090], while the modal rank reading rejects at 0.030.Both results use σ = 0.02 and 200 replicates; the deployable interval’s upper edge remains below the ceiling.
  • Limits: 0.075 out-of-sample rejection satisfies the point-estimate ceiling, but its interval [0.043, 0.121] does not, making discrimination a bound in sample and a direction out of sample.The evidence concerns one parabolic system misspecified by an omitted mode and does not establish transfer to another operator.

VII. WHY THE INSTRUMENT IS NEEDED: THE IN-DOMAIN ACCURACY READING IS BLIND HERE

In-domain accuracy is blind to this misspecification: predictive error remains below observation noise while the recovered coefficient is about 30% wrong. Residual structure and out-of-domain behavior motivate a second, reference-free reading, but the evidence is bounded to this exactly solvable operator and measured noise levels.

  • Why in-domain accuracy fails: 2.73 × 10−2 RMSE persists across noise levels, while the misspecified model’s coefficient is wrong by 29.7–31.2%.For σ ≥0.05, predictive error is smaller than observation noise, so accuracy checks cannot flag the misspecification.
  • Why a second reading is needed: 100% of replicates detect either misspecification with variance-ratio and lag-one-autocorrelation checks, but neither distinguishes operator error from parameter unreachability.Their correct-specification firing rates are 0.033 and 0.067, respectively.
  • Reference-free readings: 1.92 and 1.64 out-of-domain MSE ratios remain stable to about 1% across the noise sweep, providing a reference-free regime fingerprint.The ratios correspond to the single and double misspecifications and require only evaluating the estimator outside its fitting interval.
  • Limits of accuracy comparisons: 3.7 × 103 separates misspecified from well-specified in-domain MSE at σ = 0.02, but exploiting that contrast requires knowing the correct operator’s error floor.This makes the information present in principle but unavailable to a reference-free accuracy check.
  • Scope: 30% coefficient error remains undetectable by in-domain accuracy on the exactly-solvable operator at the measured noise levels, with broader transferability left open.The stated evidence establishes blindness in this setting, not across other physics or measured data.

VIII. THE COST OF A FREE NETWORK

Freeing more of the state to the network monotonically worsens out-of-sample identification, while correctly structuring the modal estimator can substantially outperform corrector-based alternatives. The evidence supports that advantage only up to σ≈0.3 and shows that correcting a misspecified operator can degrade performance under noise.

  • Flexibility cost: 7% out-of-sample identification error occurs with a free state and spatial field α(x), versus 1.5% for a free-state PINN and 0.004% with no nuisance.Degradation is monotone in the amount of unconstrained network.
  • Flexibility cost: 30/30 seeds produce a spatially varying field across all four missing-mode permutations, including low-frequency modes, so collapse is not caused by high-frequency representation limits.Severity is ordered by the missing mode’s energy.
  • Baseline comparison: 1.41% error for the correctly structured modal head versus 16.5% for the corrector at σ = 0.02 establishes a large advantage (Cliff’s δ = −0.958, p = 3.7 × 10−9).The paired comparison remains significant under Holm correction across the reported noise sweep, subject to the stated evidence boundary.
  • Evidence boundary: 0.54 median error ratio at σ = 0.4 and 0.45 at σ = 0.5 show the structured estimator remains less than half the corrector’s error at the noisiest tested point.At σ = 0.4, the corrected p is 0.053 versus 0.008 uncorrected; the authors do not claim the advantage beyond σ = 0.3.
  • Corrector asymmetry: 1.10, 1.19, 1.43, 1.41, and 1.86 are the corrector-to-uncorrected-PINN median ratios from σ = 0.1 to 0.5, indicating correction worsens faster under noise.This sweep is reported as supporting evidence without a frozen verdict artefact.

IX. LIMITS, AND RESULTS THAT WENT THE OTHER WAY

The section bounds the diagnostic: it loses recovery contests, relies on a freeze-before-testing condition for networks, and has a documented blind spot. Its guarantees and scope therefore remain conditional on the tested operator, data regime, and operating hypothesis.

  • Pre-registered negative on recovery: 4.03, 2.45 and 1.79 times worse: the network loses to oracle-tuned Tikhonov recovery on the structured configuration at σ = 0.01, 0.02, 0.05.It performs better on the smooth control by 46% and 41%, without rescuing the recovery claim.
  • The hypothesis behind the specificity guarantee: ∂αuθ ≡0: the specificity guarantee imports hypothesis H2, which analytic-field and least-squares surrogates satisfy but a trained network violates through state absorption.The operational remedy is fit, freeze, test: compute IN(y, ˆα) with the trained state and its α-derivatives held analytically fixed.
  • A blind spot of the detector: 0.067 ([0.008, 0.221]): at h = 0.180, deployed in-sample detection falls far below 0.90, while the analytic convention gives 0.000 ([0.000, 0.116]).The complete grid, out-of-sample block, and decision rule were frozen before the certified run; the point was not an artifact.
  • Mechanism: two pre-registered hypotheses refuted: κ = 2.27 (median over 90 cells, range 2.08–2.69): anisotropy misses the pre-registered κ ≥10 criterion, and the modal-field flexibility hypothesis is not validated.Correct specification has κ = 2.57; under perfect isotropy κ already reads 2, so κ ≥10 requires fivefold greater mean-direction stiffness.
  • Scope: Scope: results cover a one-dimensional self-adjoint parabolic operator, omitted-mode or mis-rating misspecification, and absolute Gaussian observation noise.Transfer to other operators or data not generated by the model class remains unsettled.

X. REPRODUCIBILITY AND PROVENANCE · XI. CONCLUSION

The paper establishes a reproducible, reference-free instrument that distinguishes operator misspecification from non-identifiability using one fit, while documenting certified provenance and explicit limits on its guarantee. Its conclusion pairs deployable diagnostic evidence with reproducibility controls, negative results, and an operational recommendation.

  • X. REPRODUCIBILITY AND PROVENANCE: Every decisional threshold was pre-registered before its run, while certified figures have SHA-256 artefacts and independent second computation paths.Table IX maps claims to artefacts, including run-level raw/ tables and the certified register.
  • X. REPRODUCIBILITY AND PROVENANCE: Preregistrations, certified artefacts, digests, and second computation paths are deposited at OSF, with analysis and certification code slated for versioned Zenodo archiving.The deposited materials include integrity checks enforcing the protocol.
  • X. REPRODUCIBILITY AND PROVENANCE: Artefact names remain pinned to their digests, and ood denotes evaluation on t ∈[1, 2] after training on t ∈[0, 1], not out-of-distribution learning.The convention preserves the certified chain and clarifies the paper’s terminology.
  • XI. CONCLUSION: The instrument answers whether postulated physics is wrong from an existing fit: its statistic is mute under correct specification at 0.19 with rejection 0.033, but fires every replicate under misspecifications at 224 and 85.The deployable reading uses one fit, plug-in scale, and a per-seed parameter; the analytic reading agrees.
  • XI. CONCLUSION: Two readings from one fit discriminate across three deployable designs, while the fourth corner is presented only as an analytic reading.The two readings are supported by separate artefacts and are not interchangeable because the analytic χ2 calibration differs from the deployable bootstrap thresholds.
  • XI. CONCLUSION: One and 241 parameters converge to the same wrong value, predicted in closed form to 0.07%, whereas a network with a different objective converges elsewhere.The plateau belongs to the loss class rather than architecture, so increasing capacity cannot remove the failure.
  • XI. CONCLUSION: The study reports a neural-recovery loss to Tikhonov inversion, detector blindness beyond a noise level, a violated specificity hypothesis, and two refuted mechanisms.These bounds qualify the diagnostic: it is established, but its mechanism is not.
  • XI. CONCLUSION: Practitioners should evaluate the estimator out of domain, test specification on the same fit, and check non-identifiability before modifying the model.Extending the guarantee to arbitrary trained estimators by removing H2 is identified as the next result.

DECLARATIONS

The declarations disclose two pre-registration exceptions while preserving the unchanged decision rule and reporting failed claims without post hoc threshold adjustment. The author reports no conflict of interest and no specific grant funding.

  • Pre-registration exceptions: Two pre-registration exceptions are disclosed: the information-matrix size criterion was authored contemporaneously and failed, while the spectral localizer used different regimes and seed blocks.The spectral localizer’s decision rule was unchanged, and the failed criterion is reported in section V.
  • Negative results: All pre-registered claims that failed are reported in section IX, with no threshold adjusted after seeing a result.
  • Conflict of interest: The author reports no conflict of interest to disclose.
  • Funding: The work received no specific grant from any funding agency.
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