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Stable and Counterfactually Robust Physical World Models from Imposed Structure and Learned Physics

Yufeng Wang, Parivesh Priye, Lu Wei, Haibin Ling

arXiv:2610.00280v1cs.LG

TL;DR

Learned world models can violate physical laws, drift over long rollouts, and answer changed-law queries using training laws. This paper imposes general physical structure while learning system-specific physics; across its studied systems, the models recover physical functions and distinguish dissipative from conservative behavior. The guarantees and conclusions have stated scope boundaries, and the evaluation relies on synthetic classical simulators.

  • Problem

    Learned simulators can violate conservation, create energy, drift or diverge beyond training horizons, and reuse training laws for changed-law queries.

    Source: §1 · p. 1

  • Method

    The model combines a learned energy and system-specific physics with a fixed antisymmetric reversible operator, a one-way dissipation port, and a supplied drive channel.

    Source: §3.2 · p. 4

  • Results

    The models distinguish conservative and dissipative worlds, recover constitutive functions including a nonlinear law’s curvature, and show that confinement alone does not prevent divergence when coupling injects net energy.

    Source: §4.1 · p. 6, §4.4 · p. 8

  • Takeaways & Limitations

    The imposed structure makes system-specific constitutive relations identifiable and accurate within the stated form, while parameter interventions require a supplied map.

    Source: §4.3 · p. 7, §5 · p. 9

    The results rely on synthetic classical simulators, and the dissipation guarantee is limited to quadratic port-channel energy.

    Source: §3.2 · p. 4, §5 · pp. 9–10

Abstract

from arXiv · show

A world model learns to forecast how a physical system evolves from recorded trajectories, yet the systems it imitates obey physical laws that are neither fully supplied nor reliably respected. The model may create energy, drift or diverge over long rollouts, and answer a changed law query using the law observed during training. We ask how much general physical structure must be hard coded into a world model, and how much system-specific physics can then be learned from data, for four properties to hold simultaneously: second law compatible dissipation, correct responses to interventions on physical parameters, stability out to one hundred times the training horizon, and robustness to disturbances. The imposed structure is general: dynamics are generated from the gradient of a learned energy through a fixed reversible operator, the energy is restricted to a confining class, a one way port can remove energy but never inject it, the drive channel is known, and the intervened parameter enters through a separable map. The model learns the energy functional, constitutive relations, dissipation rate, and couplings. Across an electromagnetic cavity, a particle in cell grid, and a shallow-water fluid, models with roughly nine thousand parameters recover constitutive functions with unit slope, separate conserving from dissipating worlds by four orders of magnitude using a single set of weights, and transfer changes in sign, magnitude, rate, and gravity to unseen values, where equal-capacity models without the same structure perform at chance or worse. A nonlinear constitutive law is recovered with its curvature preserved and predicts a held-out intervention $2$-$17\times$ better than a converged linear model.

1 INTRODUCTION

The paper asks what general physical structure a learned world model needs to learn system-specific physics while supporting dissipation, interventions, long-horizon stability, and disturbance robustness. It tests this division of supplied structure and learned content across three physical systems.

1 INTRODUCTION

  • High training-distribution fidelity does not establish physical realism: models can violate conservation, create energy, drift or diverge, and reuse training laws for changed-law queries.The introduction cites physical-realism benchmarks and failures across model scales.Source: §1 · p. 1
  • The paper tests whether imposing general physics lets models learn system-specific dynamics while supporting four behaviors: compatible dissipation, intervention responses, long-horizon stability, and disturbance robustness.The evaluation spans an electromagnetic ring cavity, a particle-in-cell grid, and a shallow-water fluid.Source: §1 · p. 2
  • The paper attributes each target behavior to a structural component and formalizes guarantees for the dissipative port and form selection.It also gives a criterion for answerable parameter interventions and tests the conclusions under increased scale, observation noise, and broader physical systems.Source: §1 · p. 2

2 RELATED WORK

Prior work imposes physical structure, identifies equations from data, and scales learned simulators, but the paper identifies gaps in how these approaches address dissipation, interventions, guarantees, and the modeller–data division of labour.

Structure-preserving learned dynamics

  • Structured dynamics models learn energy-based or reversible/irreversible dynamics, but generally fit one system at a time and do not jointly determine dissipation, guarantee the implemented discrete update, or separate supplied structure from learned content.Benchmarks also find that integrators and evaluation horizons can matter as much as model form (Zhong et al., 2021a).Source: §2 · p. 2

Identification, counterfactuals, selection, and scale

3 METHODS: REGIMES, MODELS, GUARANTEES, AND THE CRITERION

The methods divide labour between modeller-supplied structure and data-learned physics, then test each target behavior against controls that remove structural components. The evaluation spans three audited physical systems, states guarantees with explicit scope, and specifies when parameter interventions are answerable by construction.

3 METHODS: REGIMES, MODELS, GUARANTEES, AND THE CRITERION

  • The methods distinguish supplied general form from learned system-specific physics and evaluate each behavior against a control that removes a structural component.Figure 1 presents the supplied structure, learned content, supported behaviors, and refused interventions.Source: §3 · p. 3, Figure 1 · p. 3

3.1 THREE PHYSICAL TEST SYSTEMS, AUDITED BEFORE ANY TRAINING

  • Three audited systems—a chaotic ring cavity, a conserved-charge particle-in-cell grid, and a mass-conserving shallow-water fluid—are trained for 200–500 steps and evaluated for 50,000 steps.Before training, reference simulators pass conservation tests with energy drift at 10^-5 or better; the evaluation horizons equal 100× or 250× the training horizons.Source: §3.1 · pp. 3–4

3.2 THE IMPOSED FORM AND WHAT IS LEARNED

  • The modeller supplies the reversible operator, confining energy class, port, drive channel, and factored parameter map, while data determine energy, constitutive functions, dissipation rate, and couplings.The dynamics use an antisymmetric operator on the learned-energy gradient and a one-way port; coercive total energy yields compact level sets and bounded exact conservative trajectories under the stated assumption.Source: §3.2 · p. 4

3.3 TWO GUARANTEES, WITH THEIR SCOPE

  • For quadratic diagnostic energy on the port channel, the exact exponential port update contracts that energy at every step size, while the form selector retains the true in-family form with probability at least 1 − α under exchangeability.The dissipation guarantee does not cover composition with the reversible step or state-dependent rates; the coverage guarantee is marginal and does not guarantee coverage after distribution change.Source: §3.3 · p. 4, §3.2 · p. 4

3.4 CONTROLS THAT ISOLATE THE STRUCTURE, AND HOW CLAIMS ARE MEASURED

  • Controls remove one structural component at a time, alongside an equal-capacity unstructured baseline, and evaluate rollout accuracy, true-energy drift, and learned-rate error across independent initialisations.The models have approximately 8,000–12,000 parameters and train only on next-state prediction.Source: §3.4 · p. 5

3.5 WHEN IS A COUNTERFACTUAL ANSWERABLE BY CONSTRUCTION?

  • An intervention is answerable by construction when its parameter enters through a supplied map and learned content matches the truth on the rollout’s visited states.Without a restriction on parameter dependence, in-support observations cannot determine off-support responses; interventions on parameters lacking a supplied map are predicted to fail, while an oracle given that map should succeed.Source: §3.5 · p. 5

4 RESULTS

Across the tested physical systems, the results show that imposed structure supports learning dissipative and constitutive physics while enabling counterfactual transfer and long-horizon boundedness. The experiments also identify where the guarantees are limited and which structural components each behavior depends on.

4.1 ONE SET OF WEIGHTS DECIDES WHETHER A WORLD DISSIPATES AND RECOVERS ITS FUNCTIONS

  • Four orders of magnitude separate the learned conservative and dissipative rates, while the model recovers grid and fluid constitutive functions, including nonlinear curvature.The learned rate is 1.2 × 10−5 for the conservative cavity and 0.481 for the dissipative cavity; fluid M(h) and V′(h) are recovered at unit slope.Source: §4.1 · pp. 5–6, Table 1 · p. 6
  • The frozen conformal selector achieves four-way accuracy of 0.84–0.89 and true-candidate coverage of 0.73–0.90 across four generator families.Coverage below nominal 0.90 on two families occurs under generator shift, outside the exchangeable coverage guarantee.Source: §4.1 · p. 6

4.2 THE SECOND LAW HOLDS BY CONSTRUCTION, AND THE GUARANTEE HAS A PRICE

  • In fluctuating-bath tests, the structured model tracks reference thermalization while the equal-capacity baseline reaches a temperature roughly six orders of magnitude too high.The deterministic decay rate is recovered within 0.4%; the constrained port clamps negative dissipation at zero, so energy injection must enter through the explicit drive.Source: §4.2 · p. 7, Figure 2 · p. 7

4.3 INTERVENTIONS ON PHYSICAL PARAMETERS TRANSFER WHERE THE MAP IS SUPPLIED AND ARE REFUSED WHERE IT IS NOT

  • Factored models transfer unseen sign, magnitude, rate, and gravity changes when the parameter dependence is supplied; changing gravity from 1.0 to 1.5 yields error 6.8 × 10−11.The residual error depends on the state-space region visited by the new trajectory, rather than the parameter’s numerical distance from training.Source: §4.3 · p. 8
  • When an intervention’s parameter map is omitted, the factored model fails while an oracle given the map succeeds; these are implementation-level failures, not an impossibility theorem for nonlinear maps.A particle-shape width intervention gives error 111× the in-distribution value, and a pressure-exponent intervention gives approximately 109× the in-distribution error.Source: §4.3 · p. 8

4.4 LONG-HORIZON STABILITY NEEDS A CONFINING ENERGY AND AN ENERGY-CONSISTENT COUPLING

  • Over 50,000 steps—100× the training horizon—the structured model stays near 10−2 relative true-energy deviation, while the equal-capacity baseline exceeds 10^8 and the non-factored control reaches 10^4.The non-factored control diverges despite a confining energy because its independently learned particle-field coupling injects net energy.Source: §4.4 · p. 8, Figure 3 · p. 9
  • Confinement alone is insufficient: energy-consistent coupling supports stability, whereas the supplied parameter map supports counterfactual transfer.The fluid’s non-separable variant remains bounded but fails the gravity counterfactual; boundedness also does not ensure accuracy, as rare cavity episodes exceed the 5% worst-case energy criterion.Source: §4.4 · pp. 8–9

4.5 DISTURBANCES, SCALE, NOISE, AND EVERYDAY PHYSICS

  • Velocity kicks along the reachable manifold are absorbed, while coherent off-manifold field perturbations remain sensitive; conserved invariants do not amplify injected violations.Across black-box families from 10^4 to 10^8 parameters, scale did not yield reliable sign-flip counterfactuals; symbolic regression was more accurate where a clean, correctly specified dictionary existed.Source: §4.5 · p. 9

5 CONCLUSION AND LIMITATIONS

The paper concludes that modest geometric and thermodynamic structure lets data identify system-specific physics and supports reliable behavior beyond training. Its guarantees and scope remain limited, and the evidence comes from synthetic classical simulators.

5 CONCLUSION AND LIMITATIONS

  • The structured model recovers nonlinear constitutive laws, separates conservative and dissipative regimes, transfers interventions, and remains stable over long rollouts.The imposed geometric structure provides a coordinate frame for data to identify system-specific physics.Source: §5 · p. 9
  • Dissipation is guaranteed only for quadratic port-channel energy, while boundedness is proven only for conservative dynamics; potential force is identifiable only up to a Casimir constant.Brackets, channels, and kernels are user-supplied, so the work is not unsupervised law discovery.Source: §5 · p. 9
  • The results rely on synthetic classical simulators, with edge-case failures, generator shifts, and inconclusive scaling results defining operating boundaries.The stated examples include cavity energy tails and unspecified parameter interventions blocked by supplied structure.Source: §5 · pp. 9–10

AI USE STATEMENT

The authors disclose that a large language model assisted with language polishing, while they verified the paper's claims, proofs, formulations, and values and take responsibility for it.

AI USE STATEMENT

  • A large language model helped polish the manuscript; the authors verified its claims, proofs, mathematical formulations, and reported values and take full responsibility.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

The work uses synthetic physics simulations and involves no human subjects, personal data, or scraped datasets.

ETHICS STATEMENT

  • The study uses synthetic physics simulations and no human subjects, personal data, or scraped datasets.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

The authors specify the simulators, protocols, theory, per-experiment records, and verdict-integrity procedures, with deterministic data regeneration and supplementary artifacts.

REPRODUCIBILITY STATEMENT

  • Data are regenerated deterministically from seeds, and implementation, preregistration documents with amendment history, and result artifacts are provided as supplementary material.Reference simulators, calibration gates, model families, protocols, proofs, per-seed results, and the verdict-integrity protocol are specified in the cited appendices.Source: REPRODUCIBILITY STATEMENT · p. 10

OVERVIEW OF THE APPENDICES

The appendices group measurement methods, system specifications, results, scope checks, formal proofs, and reference material by purpose. The measurement program pairs each target behavior with confound-resistant metrics, positive controls, and structural controls.

OVERVIEW OF THE APPENDICES

  • Appendices A–B describe measurement and the three physical systems, while C–G report evidence by scientific question and H–N cover synthesis, scope, proofs, records, and related work.The appendices are arranged into measurement and setup, experimental evidence, synthesis and scope, and reference material.Source: OVERVIEW OF THE APPENDICES · p. 12

A.1 WHY A MEASUREMENT PROGRAM AT ALL

  • Each behavior is assessed with a confound-resistant metric, an engineered positive control, and a control model that isolates the credited structural element.The tests address trajectory error masking long-horizon energy violations, counterfactual tests passed by interpolation, and apparent stability from trivial motion.Source: §A.1 · p. 12

A.2 THE FOUR BEHAVIOURS

  • Table 3 compares the structured model with controls, each missing one structural element; flips use seed-level exact binomial tests over 10 seeds.Source: Table 3 · p. 12

A.3 DISTURBANCES, SCALE, AND NOISE: THE NUMBERS BEHIND THE MAIN-TEXT SUMMARY

Across the supplied results, the structured model transfers support-moving parameter changes and distinguishes composition from extrapolation, while additional tests map fidelity boundaries and limits. The appendices also specify how disturbances, calibration, selection guarantees, and imposed structure are evaluated.

A.3 DISTURBANCES, SCALE, AND NOISE: THE NUMBERS BEHIND THE MAIN-TEXT SUMMARY

  • A mid-rollout velocity kick is absorbed (post/pre error ratio 0.85 on the cavity, 1.14 on the grid), while a coherent field bump remains sensitive.Three strided frames reduce continuation error to 4.5 versus 22.8 from one rendered frame; under 5% training noise, the flip persists and the 10k-step stability envelope loosens from 0.045 to 0.285.Source: §A.3 · p. 13

B.3 1D SHALLOW-WATER FLUID (NON-ELECTROMAGNETIC)

  • The shallow-water tests concern the smooth, pre-shock regime; the imposed form carries across changes in conservation structure, field content, and physical domain.Its calibrated reference matches viscous decay to 10^-10 and gravity-wave speed within 0.6%.Source: §B.3 · pp. 14–15

C.1 A SUFFICIENT CONSTRUCTION AND AN UNRESTRICTED-CLASS LIMIT

  • Accurate learned content transfers through a supplied parameter map at new values, but support-limited observations cannot distinguish unrestricted functions that agree in training and differ off support.The structured fluid model uses learned mass-metric and potential-derivative functions with bounded nonlinear corrections, while irreversibility is represented by a non-negative port rate.Source: §B.4 · p. 15, §C.1 · p. 16

C.2 COVERAGE AND SEPARATION FOR THE FOUR-WAY SELECTOR

  • The selector’s finite-sample guarantee retains the true in-family candidate with probability at least 1 − α under exchangeability, but set cardinality is not a truth certificate.Coverage need not persist after conditioning on the gate or singleton selection; family coverage is reported as empirical because exchangeability under family shift is not established.Source: §C.2 · pp. 16–17

C.3 COMPOSITION AND EXTRAPOLATION DISSOCIATE THREE LEVELS OF STRUCTURE

  • For unseen mixed-charge compositions, the factored model reaches 0.979 ± 0.045 grid regime match, versus 0.896 ± 0.045 for the non-factored control and 0.469 ± 0.16 for baseline.Both structured models can compose when marginal inputs remain in support, but only the factored model passes the support-moving charge-sign test.Source: §C.3 · p. 18

C.4 SUPPORT-MOVING TRANSFERS: SIGN, MAGNITUDE, RATE, GRAVITY

  • 10/10 cavity seeds and 10/10 grid seeds reverse circulation after the charge flip; controls are below chance in the cavity and at or below chance on the grid.The cavity result persists on fresh independent draws (15/15 seeds); the learned energy has 0.8% error across tested horizons, while grid trajectory error grows out of band.Source: §C.4 · p. 18
  • At |q| = 2, the factored cavity model matches the oracle at a ratio ≈1.09, while a held-out gyration-frequency criterion was retired after oracle failures on 4/5 seeds.The grid’s flipped-charge kinematics degrade out of band (36% speed error at 200 steps), whereas the cavity holds |q| = 2 to horizon 1000.Source: §C.4 · pp. 18–19
  • Rate transfer yields 0.0017 error at σ = 0.25 and 2.9×10^-8 at ν = 0.25; gravity transfer error is 6.8 × 10^-11 at g = 1.5.For gravity, the non-separable control’s extrapolation error is 1.06 versus 3.2 × 10^-5 in-distribution; these results are scoped to the tested configurations.Source: §C.4 · p. 19

C.5 IMPLEMENTATION-LEVEL COUNTERFACTUAL STRESS TESTS

This section stress-tests where the imposed structure succeeds and remains limited, examining stability, dissipation, and physics identification across supervised, label-free, and function-learning settings. It also evaluates nonlinear constitutive prediction and BIC-based physical-form selection, while noting conservative near-degeneracy and fit-gap blind regions.

C.5 IMPLEMENTATION-LEVEL COUNTERFACTUAL STRESS TESTS

  • At refused interventions, cavity error rose 111× (oracle 1.4×), fluid error rose ∼109× (oracle fit 10−6-level), and grid error rose 7.0×, below its 10× bar.These results delimit the tested implementation, not universal unidentifiability; ambiguity was returned in 40/40 development worlds, while the quartic fit gap shifted from 42× better at β = 0 to 213× worse at β = 4.Source: §C.5 · pp. 19–20, Figure 4 · p. 20

D.1 CONDITION 1: A CONFINING LEARNED ENERGY; D.2 CONDITION 2: ENERGY-CONSISTENT COUPLING, THE MECHANISM, MEASURED

  • Confinement moved cavity rollouts from 0/5 to 5/5 bounded seeds, but identical confining energy did not prevent black-box coupling divergence; stability tracked energy-consistent coupling.The non-factored model injected ∼9× the net energy per step of the factored model; the fluid’s non-separable control stayed bounded but failed extrapolation at 10^10×.Source: §D.1 · p. 20, §D.2 · pp. 21–22

D.3 THE RESIDUAL WORST-CASE ENERGY TAIL; D.4 METRIC DISCIPLINE FOR STABILITY CLAIMS

  • The cavity’s rare energy deviations reached 5–12% despite typical 1–3% excursions, while the grid’s worst case was 0.4%; initial kinetic energy, not close approach, was the strongest cavity correlate.The tail was characterized as high-energy content strain, not a rare-event problem; stability reporting also requires rollout error, worst-case per-seed deviations, and disclosure of diverged seeds in means.Source: §D.3 · p. 22, §D.4 · p. 22

E.1 NONNEGATIVE DAMPING AND ITS SAME-CHANNEL REPRESENTATIONAL LIMIT; E.2 DECAY, TERMINAL STATE, AND EXPLICIT DRIVING; E.4 THE FLUID’S SECOND LAW: SAME STRUCTURE, DIFFERENT PHYSICS

  • The one-way port recovered cavity resistivity to 0.4%, matched field decay at 0.0038 relative error, tracked fluid wind injection exactly, and refused anti-friction by clamping its rate to zero.The sign-free port represented a genuinely divergent anti-dissipative trajectory with ∼1% energy error, which is representation fidelity rather than a stability claim; the known drive remains separate from passive damping.Source: §E.1 · p. 22, §E.2 · p. 23, §E.4 · p. 23

E.3 THE STOCHASTIC SECOND LAW: RECOVERING THE CORRECT TEMPERATURE

  • In the tested charge-0 linear bath, temperature was 0.502 ± 0.007 against 0.5, while the noise-doubled control was flagged 5/5 and the correct model 0/5.The test supports quantitative consistency in this supplied-bath setting, not unique coupling identification, detailed balance, or the invariant law of the full nonlinear simulator.Source: §E.3 · p. 23

F.1 SUPERVISED DISCOVERY: ONE CHECKPOINT, FOUR WORLDS; F.2 IDENTIFIABILITY: THE ENERGY CANNOT ABSORB THE DISSIPATION; F.4 LABEL-FREE INFERENCE OF THE MEDIUM FROM OBSERVATION

  • One shared cavity model separated conservative and dissipative rates by over four orders of magnitude and passed the reported behavioral tests; label-free estimates separated held-out regimes across five seeds.Clamping the dissipative rate produced a fit-loss minimum at the truth, while the label-free intermediate-rate correlation was 0.90 against a 0.95 gate, with residual estimator bias.Source: §F.1 · p. 24, §F.2 · p. 25, §F.4 · p. 25

F.5 DISCOVERING FUNCTIONS, NOT JUST CONSTANTS

  • Within the tested classes, grid and fluid constitutive derivatives were linear with reported slopes of 1.001 and 1.000, while a constant shift in fluid V′ left trajectories unchanged to machine precision.The fluid V′ intercept is a non-identifiable direction, so the intended target is its slope and shape; these agreements do not prove identifiability or successful optimization beyond the tested class and states.Source: §F.5 · pp. 25–26
  • For the polytropic fluid truth, learned V′ had gauge-aligned relative RMSE 0.004/0.008/0.025 across amplitudes 0.03/0.05/0.10, and its fitted quadratic coefficient was 0.78–1.08 against truth 1.Learned M was within 0.016–0.042, whereas the linear comparator recovered M = h to 10−4 but could not represent V′ curvature; the raw V′ error was largely the unidentifiable constant.Source: §F.5 · p. 26
  • At a = 0.03, the converged linear comparator’s rollout and intervention errors were 14–17× larger than the nonlinear model’s, despite similar long-horizon energy deviations.At ∆t = 0.005, intervention errors were 3.6×10^-5/1.9×10^-4/3.3×10^-3, three- to five-fold lower than at ∆t = 0.01 despite essentially unchanged constitutive error.Source: §F.5 · p. 27
  • On identical test windows, direct per-window σ-fit errors were 0.009/0.011/0.007 versus 0.051/0.058/0.070 for the amortized encoder at K = 25/50/100.At 2% observation noise, direct-fit errors rose to 0.029–0.039; these are estimator outcomes, not information-theoretic lower bounds, and the grid had an approximately 4% slope-ratio discrepancy.Source: §F.5 · p. 27

F.6 DISCOVERY RUNG 2: SELECTING THE GENERAL FORM

  • The minimum-BIC candidate matched the true form on all 40 one-degree-of-freedom runs, and the non-nested wrong form was never selected.Near σ ≲0.05, selection leaned on the complexity penalty rather than fit; the synthetic worlds had truth inside the candidate set, and σ was weakly identified near the conservative boundary and under strong drive.Source: §F.6 · pp. 27–28

F.7 THE SELECTION LOOP AND ITS FIT-GAP ALARM

  • At β ≥0.25, fit-gap ordering reversed, while smaller perturbations fell in the observed blind region; the gap is an empirical screening signal, not a test of class correctness.The workflow fits candidates and a reference, inspects the gap, then selects or investigates further; an absent alarm does not establish correctness, and candidate selection can accept an out-of-family world.Source: §F.7 · pp. 28–29

F.6 DISCOVERY RUNG 2: SELECTING THE GENERAL FORM

  • ∆BIC 2257 selected H+port in the dissipative cavity and 1054 selected H+port+drive in the driven cavity; an L1 parsimony prior restored the conservative selection.The unregularized five-seed conservative verdict selected H+port by 29 BIC, but its residual rate dissipated 99.8% of field energy over the 50k-step horizon; longer training reduced the rate below the 10^-4 evaluation-horizon bound.Source: §F.6 · p. 28, Figure 7 · p. 29

F.3 FRESH-DATA ROBUSTNESS AND THE PRECISION BOUNDARY

Fresh-data tests retain the cavity’s order-of-magnitude dissipation-rate separation but miss its strict per-value gate; the same structural program transfers to fluid physics and is tested across broader physics suites. The results also delimit what the structural guarantees establish and where selection and refusal tests remain conditional.

F.3 FRESH-DATA ROBUSTNESS AND THE PRECISION BOUNDARY

  • The cavity’s sign-flip dissociation survives 15/15 seeds (479/480 pooled), but the strict σcons < 10^-4 gate fails on 2/9 fresh fits, with a maximum of 1.07 × 10^-4.The σcons and σdiss estimates remain ∼3.7 orders apart on every draw; the worst seed implies 5.2% spurious loss over 50,000 steps, just above the 5% bound.Source: §F.3 · p. 25

G.1 THE SAME PROGRAM IN A DIFFERENT PHYSICS

  • At ∼8.7k parameters, the fluid reproduces the program’s pillars, including fit 7.8×10^-11 and gravity transfer where the non-separable control fails at 10^10×.The block also reports machine-precision mass conservation, the complete second-law suite, and a certified refusal test; it treats this as evidence that the behaviors attach to the imposed form, not electromagnetism.Source: §G.1 · p. 29

H.1 INTERPRETATION AND LIMITS OF THE MAP

  • Supplied parameter maps prescribe responses beyond training support only when learned content remains accurate on visited states; this does not rule out useful extrapolation priors in generic models.The scalar port guarantees dissipation of its quadratic diagnostic, while full-energy and stochastic-stationary behavior remain empirical outcomes.Source: §H.1 · pp. 29–30

H SYNTHESIS: WHICH STRUCTURE BUYS WHICH BEHAVIOUR, AND WHY

  • Table 5 summarizes the evaluated controls linking structural components to behaviors, while explicitly not establishing universal necessity for each component.The authors say each behavior was tested against controls differing in exactly one structural element.Source: H SYNTHESIS: WHICH STRUCTURE BUYS… · p. 29, Table 5 · p. 30

I THE STATED EXCEPTIONS, TESTED, AND A SUITE OF EVERYDAY PHYSICS

  • The authors test four stated exceptions plus an eleven-system suite selected to tile a predeclared physics taxonomy, rather than sampled from it.The aggregate claim is therefore coverage of that tiling, and the taxonomy was committed before systems were built.Source: I THE STATED EXCEPTIONS, TESTED, AND A… · p. 31

I.1 THE PASS-CARD LEDGER: ELEVEN SYSTEMS, TWELVE CARDS, THREE CARD TYPES

  • All eleven common-life systems earn their pass cards, with eight clean and four carrying exactly one measured boundary; chaotic cases reproduce climate but can lose to the baseline on short-horizon accuracy.Conservative cards meet the 5% energy-drift gate at 100× horizons, while the chaotic cards pass climate and counterfactual-climate seeds but do not promise short-horizon gains.Source: §I.1 · pp. 31–32

I.2 REFUSAL TESTS AT SCALE: THE REFUSAL QUOTA, AND WHAT A REFUSAL INSTRUMENT MUST BE

  • Eleven refusal/guard entries pass across six non-separability types, but the refusal instrument was redesigned mid-block and changed verdicts in both directions.The redesign defines refusal within-class and requires a learnability certificate, avoiding cross-class floor ratios that conflate class membership with integrator truncation.Source: §I.2 · p. 32

I.3 THE FOUR EXCEPTIONS

The exception tests cover contact, interacting baths, bare gravity, and shocks, with distinct outcomes and explicit boundaries. The frozen selector also misses its pre-registered accuracy gate, and its errors reveal limits of the alarm and reference tests.

I.3 THE FOUR EXCEPTIONS

  • The contact model recovers restitution and settling time exactly, while its architecture clamps restitution below one and cannot represent an energy-gaining violator.The guard-first protocol requires the smooth class to refuse the bouncing world before crediting the augmented class.Source: §I.3 · p. 32
  • With soft-Coulomb interactions, the bath fit estimates temperature from noise and damping, and a two-sector equipartition meter flags a sector-asymmetric violator at 66% mismatch.The magnetic-sector check also requires a workless force implementation; reference and model both use Boris rotations.Source: §I.3 · pp. 32–33
  • For bare gravity, only softening ε = 0.3 meets the energy gate; at ε = 0.01 integration becomes non-finite, so the reported 100% escape rate is not verified physical escape.At the accepted setting, the pair fit has r2 = 0.9999999, yet worst-case energy deviation is 813%; this is not evidence of a transition in physical confinement.Source: §I.3 · p. 33
  • The shock experiment tests whether a learned localized port tracks the entropy-mask front in coarse-grained dam breaks, against a certified Rusanov reference.Its pre-registered outcome allows either a localization score ≥5 or characterization of the representational limit of coarse local flux stencils.Source: §I.3 · p. 33

I.4 ONE MENU, FIFTEEN WORLDS, TWO CONTROLS

  • The frozen selector misses its ≥14/15 gate: 13/16 rows are correct, or 12/16 under the strict accepted-only budget-rule reading, while both out-of-menu controls return none-of-these.The out-of-menu worlds are an anti-dissipative system and a non-Markovian delay oscillator.Source: §I.4 · p. 33
  • The alarm’s failures stem from scope: floor-ratio alarms need dense misfit and a competent reference, while rare-event cases require guard-first tests.A separate frozen-selector confirmation reports per-family accuracy 0.84–0.89 and true-candidate coverage 0.727–0.900; only coverage is bounded by Proposition 1.Source: §I.4 · p. 33, Table 6 · p. 34

J.1 WHAT THE EXPERIMENTS SUPPORT

The experiments support recovery and prediction within supplied representations, but their guarantees and empirical outcomes have distinct boundaries. The paper identifies limits in extrapolation, scaling evidence, perception, and validation, and frames these as separate questions for future work.

J.1 WHAT THE EXPERIMENTS SUPPORT

  • Within supplied representations, the models recover useful constitutive content and fitted rates, but complete-model temperature, energy fidelity, and counterfactual prediction remain empirical tests.The fluid coefficient is reported per unit supplied gravity, not as independently discovered gravity.Source: §J.1 · p. 34

J.2 WHAT REMAINS CONDITIONAL

  • In-support observations do not identify off-support truth over unrestricted function classes, and empirical fit-gap alarms can fail or react to optimization error.Rate-estimator precision depends on protocol, while functional recovery is limited by operator nullspaces, excitation, and state support.Source: §J.2 · pp. 34–35

J.3 THE BOUNDARY MAP

  • The boundary map leaves open universal stable black-box scaling, stochastic thermodynamics beyond one (T, σ) and a charge-0 linear bath, and several physical regimes.The scaling evidence is an existence result over four families and five rungs; 2D/shock fluids, contact, and singular attraction remain future physics.Source: §J.3 · p. 35

J.4 THREATS TO VALIDITY

  • The authors report fresh-data replication for headline claims, count diverged seeds, disclose compute caps, and label unresolved competence verdicts inconclusive.Thresholds were preregistered, and corrections and the ledger are published.Source: §J.4 · p. 35

J.5 OUTLOOK

  • Future work targets broader discovery of symmetries and bracket structure, pixel-based physical-state grounding, and a decisive black-box scaling test.The paper locates a momentum boundary for single-frame perception and quantifies a short-window differencing fix.Source: §J.5 · p. 36

J.6 THREE DISTINCT BOUNDARIES

  • A static renderer depending only on position cannot distinguish arbitrary velocities at that position, though restricted support, correlations, or multiple frames can supply motion information.The paper distinguishes this mathematical limitation from the empirical failure to ground tested frozen-token pipelines.Source: §J.6 · p. 36

K.1 THE RULES

  • The validation rules require positive controls, seed-level exact binomial tests for near-binary outcomes, trained-setting reliability for counterfactual verdicts, and escalation before headline claims.A diagnostic that has never flagged anything is treated as untested rather than as evidence of success.Source: §K.1 · p. 36

K.2 THE LEDGER

The ledger records corrections, boundary cases, and changes to measurement instruments rather than hiding them. Its entries show how calibration, escalation, and revised verdicts constrain which claims survive.

K.2 THE LEDGER

  • Two cavity integrator defects and a grid self-force were caught before training by conservation tests, preventing numerical errors from being mistaken for learned-model instability or dissipation.The cavity defects were first-order splitting and tensor-view aliasing; the grid self-force came from deposition and lattice self-energy effects.Source: §K.2 · p. 36
  • Seed-level rescoring strengthened headline flips after episode-level statistics were retired, while a failed oracle gyration-frequency criterion was dropped in favor of the surviving nMSE arm.The functional-discovery estimator was also corrected from pointwise ratios near zero fields to slope regression, reported as rslope = 0.959 ± 0.017.Source: §K.2 · pp. 36–37
  • An informed mixed-data baseline still fails, and direct correlation testing refuted the proposed close-encounter explanation of the stability tail.The tested correlate was initial kinetic energy, and 67× oversampling had no effect.Source: §K.2 · p. 37
  • The fresh-data σ gate remains a boundary: σcons < 10^-4 fails on 2/9 fits despite the surviving order-of-magnitude identification.The maximum fresh-fit value is 1.07 × 10^-4.Source: §K.2 · p. 37
  • The flagship cavity selection verdict softened from 3/3 to 2/3 after escalation, with the conservative tie-break changing under a five-seed repeat.The surviving claim is seed-stable dissipative/driven selection (∆BIC ≥1054); the conservative tie-break is fit-degenerate but not horizon-degenerate.Source: §K.2 · p. 37

K.3 WHY WE PUBLISH THIS

  • The ledger presents corrections as calibration: entries include what changed, why it changed, and where the full record resides.The propositions carry boundedness and transfer claims, with their scope clauses stated inline.Source: §K.2 · p. 36, §K.3 · p. 38, L PROPOSITIONS · p. 38

L.1 ENERGY-CONSISTENT COUPLING AND BOUNDEDNESS

The section links bounded, energy-consistent dynamics to a factored coupling and explicit coercivity conditions. It also states when supplied parameter maps support transfer and where the passive-channel constraint excludes anti-damping.

L.1 ENERGY-CONSISTENT COUPLING AND BOUNDEDNESS

  • Factoring one shared learned coefficient across particle force and field current preserves energy-consistent exchange; independently learned mismatches can create phantom power.The proof sketch attributes nonzero cyclic work to independent approximation errors in force and current.Source: §L.1 · p. 38
  • Exact conservative flow conserves H, and coercive total H gives compact level sets and all-time bounded rollouts; confining Uϕ alone is insufficient.The total-energy coercivity condition is an assumption, not a consequence of confining Uϕ and T bounded below.Source: §L.1 · p. 38
  • The fluid’s energy-consistent coupling remains bounded in 5/5 runs (9.2 × 10−5), while non-factored controls show a 9× interface imbalance and divergence under identical confinement.The fluid’s non-separable control retains energy-consistent flux coupling but black-boxes only the gravity entry, failing the counterfactual test by 1010×.Source: §L.1 · p. 38
  • The boundedness scope excludes energies unbounded below, including gravitational or bare-Coulomb attraction; the stated claim is vacuum-field stability.Source: §L.1 · p. 38

L.2 SUPPLIED-MAP TRANSFER AND UNRESTRICTED EXTRAPOLATION

  • With accurate learned components on R, the supplied parameter map transfers predictions to any admissible θ′ while trajectories stay in R; unrestricted functions remain nonidentifiable off-support.Transfer on R assumes the same initial condition and uniqueness of the differential equations; numerical predictions retain integration error.Source: §L.2 · pp. 38–39
  • The supplied-map construction is sufficient, not necessary: known nonlinear maps may also transfer content, and parameter-support shift is not required for counterfactual answerability.Source: §L.2 · p. 39

L.3 RESTRICTION OF THE PASSIVE CHANNEL

  • For the isolated scalar damping channel with a positive quadratic diagnostic, σ ≥ 0 excludes anti-damping; representing sign reversal requires relaxing that constraint or changing the model.The restriction does not forbid external work, thermal energy injection, or other driven dynamics.Source: §L.3 · p. 39

M THE COMPLETE PER-EXPERIMENT RECORD

This complete per-experiment record documents successes, failures, controls, and boundary cases across physical systems, covering counterfactual transfer, stability, dissipation, and learning while distinguishing boundedness from fidelity. It also presents extended experiments and related work, and traces how learning constitutive functions and dissipation within supplied physical representations carries estimation limits into downstream predictions.

CONFINEMENT RESTORES STABILITY; THE COUPLING CONDITION APPEARS

RATE COUNTERFACTUALS AND THE SECOND LAW

STRUCTURAL ADVANTAGES SURVIVE OBSERVATION NOISE

FUNCTIONAL DISCOVERY AND REFUSAL TESTS

ONE MODEL, FOUR WORLDS (TEN SEEDS)

  • Across four worlds, one shared checkpoint achieved composition 1.0, sign flips 10/10, bounded stability 10/10, and 3.7% second-law rate error, but missed the 5% fidelity bound on 6/10 seeds.Controls without a port or with sign-free dissipation failed key tests; the informed baseline also failed, and the stability tail remained characterized rather than hidden.Source: ONE MODEL, FOUR WORLDS (TEN SEEDS) · p. 41

CAVITY AND GRID SIGN-FLIP TRANSFER AT THE SEED LEVEL

FOUR BEHAVIOURS IN THE FLUID + THE GRAVITY COUNTERFACTUAL

IDENTIFICATION, PERCEPTION, REPLICATION, AND CAPACITY BOUNDARIES

M THE COMPLETE PER-EXPERIMENT RECORD

  • The charge-flip oddness fraction was 0.994 for the structured model versus 0.33–0.51 for every trained black-box seed, with no overlap.In the quartic fit-gap experiment, reported fit error was 2.7 × 10^-3 versus 9.0 × 10^-8, and fitted curvature at x = 2 was −4.5 versus reference 13.0; bounded corrections can represent quartic behavior on compact domains, so the cause of the fitting failure remains unresolved.Source: ODD/EVEN RESPONSE DECOMPOSITION · p. 44, QUARTIC FIT-GAP EXPERIMENT · pp. 44–45
  • On the one-degree-of-freedom stress ladder, minimum BIC selected the true candidate form in 40/40 cases, while the wrong-variable dissipation form was never selected.The candidate set was {H, H+port, H+port+drive}; scope is synthetic worlds whose truth is in that set, and σ is weakly identified under strong drive and near the conservative boundary. The composed selection loop has recorded support but does not yet automatically propose new forms.Source: QUARTIC FIT-GAP EXPERIMENT · pp. 45–46, THE SELECTION LOOP AND ITS GUARD · p. 46
  • The energy-trajectory auxiliary reduced conservative σ̂ only from 0.0126 to 0.0107, while longer training reached σ̂cons = 5.1/4.1×10^-6 at 27,500 epochs.The auxiliary missed its pre-registered 10× improvement bar, whereas the longer run crossed the 10^-4 gate at 8.0–8.5k epochs; the measured decay steepened to approximately t^-2.5, so the pre-registered 1/t prediction was not confirmed. The L1 arm restored selection to 3/3 without reducing σ̂cons, and L1 plus auxiliary degraded fits 3–4×.Source: ENERGY-AUXILIARY UNIFICATION: A… · p. 46, DECOMPOSITION OF THE TRAINING PROCEDURE · pp. 46–47, THE RESIDUAL NEAR-ZERO RATE · p. 47
  • The experiment record reports PASS verdicts across eight ODE systems and four one-dimensional field systems, while the frozen selector passed 12/15 registry systems against a pre-registered ≥14/15 gate.The three registry misses—dry friction, bouncing contact, and the shallow-water dam break—were caught by card-level guard-first checks; the record assigns alarms to dense misfit, matched-structure guards to events, and budget flags to fake-conservative calls.Source: WAVE 1, THE ODE ZOO (8 SYSTEMS, FULL… · p. 47, WAVE 2, 1D FIELDS (4 SYSTEMS;… · p. 47, THE INSTRUMENT FINDINGS · p. 48

N.1 THE NEAREST NEIGHBORS, ADVERSARIALLY READ

  • D-HNN is the closest phenomenon-level precedent for a small learned dissipation on conservative data, while its unconstrained sign lacks a second-law guarantee; the comparison emphasizes combined scope, not unique individual capabilities.D-HNN evaluates unseen friction coefficients, but lacks shared-content discovery, a condition-index protocol, and the long-horizon, violator-control, and label-free context-inference program. GENERIC/metriplectic methods provide the closest structural ancestry; the stated advance is supplied-versus-learned accounting with estimation error carried into predictions.Source: §N.1 · pp. 48–49, Table 9 · p. 49

N.2 ADJACENT LINES

  • The related-work audit positions learned simulators as the scale frontier, system identification as the clean-data competitor, and context-adaptation methods as sharing the per-condition parameter protocol without physical structure or behavioral targets.It also identifies DeLaN and port-Hamiltonian system identification as robotics-side friction-learning precedents with single-system, torque-supervised scope.Source: §N.2 · p. 49

N.3 THE CENTRAL CLAIM, RESTATED

  • The paper’s central claim is that data recover useful constitutive functions and dissipation parameters within supplied physical representations, with identifiability limits and estimation error carried into downstream predictions.The dissipation guarantee is scoped to the quadratic diagnostic energy (Prop. 3), and label-free inference carries its measured rate floor into every prediction it feeds.Source: §N.3 · p. 49

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