通过施加结构与习得物理实现稳定且反事实鲁棒的物理世界模型

# Stable and Counterfactually Robust Physical World Models from Imposed Structure and Learned Physics
Source: [https://arxiv.org/html/2610.00280](https://arxiv.org/html/2610.00280)
Yufeng Wang††thanks:These authors contributed equally to this workParivesh Priye\*Affiliation:Georgia Institute of TechnologyLu Wei\*Affiliation:Stony Brook UniversityHaibin LingAffiliation:Westlake University

###### Abstract

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 intervention22–17×17\\timesbetter than a converged linear model\. The same structured model also refuses interventions for which no parameter map was supplied, producing errors7×7\\timesto109×10^\{9\}\\timeslarger than in\-distribution, while oracle models given the missing map succeed; its one\-way port likewise cannot represent anti\-dissipation by construction\.

## 1Introduction

A world model is a learned simulator: it predicts the next state of a system and iterates that prediction into a trajectory\. Such models now achieve impressive fidelity on their training distributions, from graph\-network and mesh\-based simulators\([Sanchez\-Gonzalez et al\., 2020](https://arxiv.org/html/2610.00280#bib.bib11);[Pfaff et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib13)\)and neural operators\([Li et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib12)\)to video\-scale world models that plan in latent space\([Assran and others, 2025](https://arxiv.org/html/2610.00280#bib.bib16);[Nam et al\., 2026](https://arxiv.org/html/2610.00280#bib.bib17)\)\. Fidelity alone, however, does not establish that a model obeys the laws of the world it simulates\. Physical\-realism benchmarks show that even strong video models violate conservation, create energy, and fail on “anti\-physics” probes\([Gu et al\., 2026](https://arxiv.org/html/2610.00280#bib.bib18);[Bansal et al\., 2026](https://arxiv.org/html/2610.00280#bib.bib19)\)\. The same failures recur across scales: rollouts drift or diverge beyond the training horizon, energy appears where the true system dissipates it, and a changed\-law query is answered using the law observed during training\.

A natural remedy is to encode physical structure directly into the model\. Hamiltonian and Lagrangian neural networks\([Greydanus et al\., 2019](https://arxiv.org/html/2610.00280#bib.bib1);[Cranmer et al\., 2020](https://arxiv.org/html/2610.00280#bib.bib2)\), their control and generalised\-coordinate variants\([Zhong et al\., 2020b](https://arxiv.org/html/2610.00280#bib.bib3);[Lutter et al\., 2019](https://arxiv.org/html/2610.00280#bib.bib5)\), dissipative and port\-Hamiltonian extensions\([Zhong et al\., 2020a](https://arxiv.org/html/2610.00280#bib.bib4);[Sosanya and Greydanus, 2022](https://arxiv.org/html/2610.00280#bib.bib6);[Desai et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib7)\), compositional port\-Hamiltonian learning\([Neary and Topcu, 2023](https://arxiv.org/html/2610.00280#bib.bib24)\), and GENERIC or metriplectic networks\([Hernández et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib8);[Lee et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib9);[Gruber et al\., 2023](https://arxiv.org/html/2610.00280#bib.bib10)\)conserve or dissipate a learned energy, and benchmarks show the numerical integrator can matter as much as the model form\([Zhong et al\., 2021a](https://arxiv.org/html/2610.00280#bib.bib22)\)\. These methods nevertheless fit one system at a time, do not determine from data whether dissipation is present, give no rule for which interventions are answerable, state guarantees for continuous\-time flows rather than the implemented discrete update\([Hairer et al\., 2006](https://arxiv.org/html/2610.00280#bib.bib25)\), and do not separate what the modeller supplies from what the data identify\. Symbolic regression instead recovers equations directly\([Brunton et al\., 2016](https://arxiv.org/html/2610.00280#bib.bib15);[Messenger and Bortz, 2021](https://arxiv.org/html/2610.00280#bib.bib26)\)but requires a candidate dictionary and reliable derivatives\. We therefore ask how much general physics must be*imposed*on a learned world model, and how much system\-specific physics can then be*learned*from data, for four behaviours to hold simultaneously: second\-law\-compatible dissipation, correct responses to interventions on physical parameters, stability to one hundred times the training horizon, and robustness to disturbances\. We impose a general form and learn the specific content, testing this division across an electromagnetic ring cavity, a particle\-in\-cell grid, and a shallow\-water fluid, against equal\-capacity baselines and controls that remove one structural element at a time\.

This paper makes four contributions\. First, a single model in each regime determines from trajectories whether its world dissipates and recovers constitutive functions within the supplied structural form, while making explicit what is identifiable and how estimation error propagates into downstream predictions\. Second, we attribute each of the four target behaviours to the structural component whose removal destroys it, and formalise the two guarantees provided by the framework, one for the dissipative port and one for form selection, with their precise scope\. Third, we give a simple criterion for when an intervention on a physical parameter is answerable by construction, and show that the same models fail, as predicted, on interventions outside that criterion\. Fourth, supporting studies test whether these conclusions persist under increased scale, observation noise, and a broader collection of everyday physical systems\.

## 2Related work

Structure\-preserving learned dynamics\.Hamiltonian neural networks\([Greydanus et al\., 2019](https://arxiv.org/html/2610.00280#bib.bib1)\)learn a scalar energy whose symplectic gradient defines the vector field, while Lagrangian networks\([Cranmer et al\., 2020](https://arxiv.org/html/2610.00280#bib.bib2)\)learn the corresponding dynamics through a Lagrangian formulation\. Symplectic ODE\-Net\([Zhong et al\., 2020b](https://arxiv.org/html/2610.00280#bib.bib3)\)and Deep Lagrangian Networks\([Lutter et al\., 2019](https://arxiv.org/html/2610.00280#bib.bib5)\)extend these ideas to controlled systems and generalised coordinates, and differentiable contact models further incorporate impacts\([Zhong et al\., 2021b](https://arxiv.org/html/2610.00280#bib.bib23)\)\. Dissipative SymODEN\([Zhong et al\., 2020a](https://arxiv.org/html/2610.00280#bib.bib4)\), Dissipative HNNs\([Sosanya and Greydanus, 2022](https://arxiv.org/html/2610.00280#bib.bib6)\), and port\-Hamiltonian networks\([Desai et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib7)\)introduce irreversible dynamics, while compositional port\-Hamiltonian learning assembles larger systems from learned components\([Neary and Topcu, 2023](https://arxiv.org/html/2610.00280#bib.bib24)\)\. GENERIC and metriplectic networks impose the reversible/irreversible decomposition together with its degeneracy conditions\([Hernández et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib8);[Lee et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib9);[Gruber et al\., 2023](https://arxiv.org/html/2610.00280#bib.bib10)\), and benchmark studies show that the numerical integrator and evaluation horizon can matter as much as the model form\([Zhong et al\., 2021a](https://arxiv.org/html/2610.00280#bib.bib22);[Hairer et al\., 2006](https://arxiv.org/html/2610.00280#bib.bib25)\)\. These approaches generally fit one system at a time\. When a dissipative component is present, its sign is either fixed by the model class or left unconstrained, and none jointly determines from data whether dissipation is present, carries a continuous\-time guarantee to the implemented discrete update, and explicitly separates what the modeller supplies from what the data identify \(App\.[N](https://arxiv.org/html/2610.00280#A14), Table[9](https://arxiv.org/html/2610.00280#A14.T9)\)\.

Identification, counterfactuals, selection, and scale\.Sparse symbolic regression identifies governing equations from a dictionary of candidate terms\([Brunton et al\., 2016](https://arxiv.org/html/2610.00280#bib.bib15)\), while its weak Galerkin form reduces dependence on clean derivative estimates\([Messenger and Bortz, 2021](https://arxiv.org/html/2610.00280#bib.bib26)\)\. These methods are particularly strong identifiers when the candidate dictionary contains the true dynamics, and we report the regimes in which they outperform a learned simulator\. Whether an intervention outside the training support can be identified is closely related to the positivity condition in causal inference\([Pearl, 2021](https://arxiv.org/html/2610.00280#bib.bib20)\), which motivates our distinction between extrapolation, interpolation, and composition\. Split\-conformal prediction\([Lei et al\., 2018](https://arxiv.org/html/2610.00280#bib.bib21)\)provides the finite\-sample coverage statement used when selecting among candidate structural forms\. At larger scale, graph\-network and mesh\-based simulators\([Sanchez\-Gonzalez et al\., 2020](https://arxiv.org/html/2610.00280#bib.bib11);[Pfaff et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib13)\), Fourier neural operators\([Li et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib12)\), and video world models that predict and plan in latent space\([Assran and others, 2025](https://arxiv.org/html/2610.00280#bib.bib16);[Nam et al\., 2026](https://arxiv.org/html/2610.00280#bib.bib17)\)define the current competence frontier\. Compute\-optimal scaling\([Hoffmann and others, 2022](https://arxiv.org/html/2610.00280#bib.bib14)\)motivates the broader expectation that increased capacity improves generalisation, while physical\-realism benchmarks test how far that expectation extends to consistency with physical laws\([Gu et al\., 2026](https://arxiv.org/html/2610.00280#bib.bib18);[Bansal et al\., 2026](https://arxiv.org/html/2610.00280#bib.bib19)\)\. Our scaling study probes this question using the same intervention applied to the structured models\.

## 3Methods: regimes, models, guarantees, and the criterion

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/method_overview.png)Figure 1:Overview of the approach: the general physical structure supplied by the modeller \(left\), the system\-specific physics learned from data \(centre\), and the four behaviours supported by each structural element together with the interventions the structure refuses \(right\)\.This section follows a simple division of labour: the modeller specifies the general form of the dynamics, the data determine the system\-specific physics within that form, and each target behaviour is evaluated against a control that removes one structural component, so the resulting capability can be attributed to the component it depends on \(Figure[1](https://arxiv.org/html/2610.00280#S3.F1)\)\.

### 3\.1Three physical test systems, audited before any training

A structural claim is most convincing on systems where it could fail and where the ground truth can be checked independently\. We therefore evaluate three deliberately different physical systems, each small enough to audit yet each stressing the imposed structure in a distinct way\. The*ring cavity*is a chaotic particle\-field system in which charged particles on a periodic ring exchange energy withK=16K=16electromagnetic modes through a fixed smoothing kernel, integrated with a palindromic Strang splitting that is time\-reversible and energy\-preserving\. The*particle\-in\-cell grid*is a genuine field theory with a conserved charge through a Gauss\-law invariant, implemented on a64×6464\\times 64periodic Yee finite\-difference grid with Boris particle pushes and charge\-conserving Esirkepov current deposition\. The*shallow\-water fluid*is non\-electromagnetic and exactly conserves mass, with water heighth⁡\(x\)h\(x\)and momentum evolved on a periodic line using a spectral implicit\-midpoint integrator\. Together, these systems span a chaotic particle\-field model, a conserved\-charge field theory, and a fluid, reducing the chance that a shared result is specific to one physical setting\. Training episodes contain500500steps for the cavity and grid and200200steps for the fluid\. Every stability claim is evaluated for50,00050\{,\}000steps, corresponding to100×100\\timesand250×250\\timesthe training horizons, respectively\.

Before any model is trained, each reference simulator must pass conservation tests over the full evaluation horizon, because long\-horizon model stability cannot be interpreted below the numerical drift of the reference itself\. All three simulators pass with margin, with energy drift at the10−510^\{\-5\}level or better and structural invariants preserved near machine precision\. This calibration stage also exposed two integrator defects and a static self\-force that would otherwise have appeared as learned\-model instability at long horizon \(App\.[B](https://arxiv.org/html/2610.00280#A2)\)\.

### 3\.2The imposed form and what is learned

Every structured model evolves a statezzthrough a supplied antisymmetric operator acting on the gradient of a learned energy, a scalar one\-way port on a designated coordinate block, and a known drive channel:

z˙=L\(z\)∇Eθ\(z\)−P⊤σθPz\+Bθ\(z\)u\(t\),L=−L⊤,σθ=rawθ2≥0\.\\dot\{z\}=L\(z\)\\nabla E\_\{\\theta\}\(z\)\-P^\{\\top\}\\sigma\_\{\\theta\}Pz\+B\_\{\\theta\}\(z\)u\(t\),\\qquad L=\-L^\{\\top\},\\quad\\sigma\_\{\\theta\}=\\mathrm\{raw\}\_\{\\theta\}^\{2\}\\geq 0\.\(1\)HerePPselects the dissipated coordinates, withq=P​zq=PzandP​P⊤=IPP^\{\\top\}=I\. Antisymmetry impliesE˙θ=0\\dot\{E\}\_\{\\theta\}=0for the isolated continuous reversible dynamics; interpretingLLas a Poisson bracket additionally requires the Jacobi identity\. The port is a scalar contraction ofqq, not an entropy gradient\. For a quadratic diagnostic energy on the dissipated coordinates,

Q⁡\(q\)=12​q⊤​K​q,K≻0,Q˙\|port=−σθ​q⊤​K​q≤0,Q\(q\)=\\tfrac\{1\}\{2\}\\,q^\{\\top\}Kq,\\quad K\\succ 0,\\qquad\\dot\{Q\}\\big\|\_\{\\rm port\}=\-\\sigma\_\{\\theta\}\\,q^\{\\top\}Kq\\leq 0,\(2\)and the frozen\-rate exponential step used by the implemented integrator contractsQQexactly \(Prop\.[3](https://arxiv.org/html/2610.00280#Thmproposition3)\)\. This is the guarantee, and its scope is restricted to this diagnostic and channel\. It is not a statement about the learned energyEθE\_\{\\theta\}, since coordinate contraction can increase a non\-quadratic energy \(App\.[C\.2](https://arxiv.org/html/2610.00280#A3.SS2)\); nor does it apply to drives or thermal noise, which inject energy by design, or to an arbitrary damping matrix, which does not inherit this property merely from being positive semidefinite\.

A physical parameterθ\\thetathat may be intervened on, such as charge, gravity, or a rate, enters through a supplied factored map,

Gϑ​\(z,θ\)=θ​gϑ​\(z\)\+hϑ​\(z\),G\_\{\\vartheta\}\(z;\\theta\)=\\theta\\,g\_\{\\vartheta\}\(z\)\+h\_\{\\vartheta\}\(z\),\(3\)so a new value ofθ\\thetaevaluates a prescribed continuation of the learned content rather than requiring a new function\. Each modal energy is confining\. When the total energy, including its coupling terms, is coercive, its level sets are compact and exact conservative trajectories remain bounded\. We state this condition explicitly as an assumption and distinguish it from the finite\-horizon boundedness observed experimentally \(Prop\.[4](https://arxiv.org/html/2610.00280#Thmproposition4), App\.[L](https://arxiv.org/html/2610.00280#A12)\)\.

The division of labour between modeller and data is explicit throughout the paper\. The modeller supplies the bracketLL, the discretization and field kernels for each regime, the confining energy class, the location and sign of the port, the drive channelBθB\_\{\\theta\}, the factored map in Eq\. \([3](https://arxiv.org/html/2610.00280#S3.E3)\), the fluid gravitygg, the bath temperature, and, for form selection, a finite menu of candidate structures\. The data determine the system\-specific content: the energy functional and constitutive functions, the dissipation rate, the drive\-response and coupling coefficients, and the parameters within each candidate form\. Each learned quantity is evaluated separately in the results, with the concrete learned content for each regime reported in App\.[B\.4](https://arxiv.org/html/2610.00280#A2.SS4)\.

### 3\.3Two guarantees, with their scope

The imposed structure provides two guarantees, stated and proved formally in App\.[C\.2](https://arxiv.org/html/2610.00280#A3.SS2)\. The first concerns the dissipative port at the level of the implemented discrete update\. For a quadratic diagnostic energy on the port channel, the exact exponential update contracts that energy at every step size and never injects it \(Prop\.[3](https://arxiv.org/html/2610.00280#Thmproposition3)\); under fluctuation\-matched stochastic forcing, the isolated port also has the corresponding Gibbs distribution as its invariant law\. Composition with the reversible step and the use of state\-dependent rates fall outside this guarantee, so the full\-model temperature results reported below remain empirical\. The second is a finite\-sample coverage guarantee for the form selector\. Under exchangeability, the true in\-family form is retained with probability at least1−α1\-\\alpha\(Prop\.[1](https://arxiv.org/html/2610.00280#Thmproposition1)\)\. This is a marginal coverage statement\. It does not bound the error of an individual selective decision or guarantee coverage after the data\-generating process changes, both of which we evaluate separately\.

### 3\.4Controls that isolate the structure, and how claims are measured

Every claim is evaluated against a control that removes one structural component, so the resulting behaviour can be attributed to the component whose removal destroys it\. An equal\-capacity*unstructured baseline*removes all imposed structure and tests whether parameter count alone explains the result\. A*non\-factored control*retains locality and field mediation but couples the intervened parameter through a black box, so behaviours it preserves are credited to locality while failures isolate the separable parameter map\. A*sign\-free port*allows the dissipation rate to take either sign, separating the guarantee of non\-injective dissipation from the ability to represent its violation\. An*oracle*receives the intervened physics directly, showing that a refused intervention is learnable when the missing structure is supplied\. All models remain small, with approximately88to1212thousand parameters, and are trained only for next\-state prediction\. Architectures, exact parameter counts, and training settings are given in App\.[B\.4](https://arxiv.org/html/2610.00280#A2.SS4)\.

Three measurements recur throughout the results\. Rollout accuracy is measured by normalised mean\-squared error against the reference trajectory from the same initial state; energy behaviour is measured by the relative drift of the true energy along a rollout; and a learned rate is evaluated by its relative error against the true rate\. Every experiment is repeated across independent random initialisations, and we report the resulting variation through error bars or standard deviations rather than a single run\. Each diagnostic is also paired with an engineered case that it must detect\. Formal metric definitions, the repetition protocol, and the three evidence labels used to grade findings in the appendices are provided in App\.[B\.4](https://arxiv.org/html/2610.00280#A2.SS4)and App\.[A](https://arxiv.org/html/2610.00280#A1)\.

### 3\.5When is a counterfactual answerable by construction?

Let the true generator beG∗​\(z,θ\)=θ​g∗​\(z\)\+h∗​\(z\)G\_\{\*\}\(z;\\theta\)=\\theta\\,g\_\{\*\}\(z\)\+h\_\{\*\}\(z\)and the learned generatorGϑG\_\{\\vartheta\}be defined as in Eq\. \([3](https://arxiv.org/html/2610.00280#S3.E3)\)\. If the learned content matches the truth on a state regionℛ\\mathcal\{R\}, then for every admissibleθ′\\theta^\{\\prime\},

gϑ=g∗​and​hϑ=h∗​on​ℛ⟹Gϑ​\(z,θ′\)=G∗​\(z,θ′\)​for all​z∈ℛ,g\_\{\\vartheta\}=g\_\{\*\}\\ \\text\{and\}\\ h\_\{\\vartheta\}=h\_\{\*\}\\ \\text\{on\}\\ \\mathcal\{R\}\\quad\\Longrightarrow\\quad G\_\{\\vartheta\}\(z;\\theta^\{\\prime\}\)=G\_\{\*\}\(z;\\theta^\{\\prime\}\)\\ \\text\{for all\}\\ z\\in\\mathcal\{R\},\(4\)so the intervention is answered exactly wherever the learned content remains accurate on the states visited by the rollout \(Prop\.[5](https://arxiv.org/html/2610.00280#Thmproposition5); a known nonlinear parameter map would provide the same principle\)\. Without a restriction on how the parameter enters the dynamics, observations confined to the training support cannot distinguish functions that agree on that support but differ outside it\. No amount of additional in\-support data can therefore determine the off\-support response\.

Two consequences organize our experiments\. First, a counterfactual test is informative only when the queried parameter value leaves its marginal training support\. In\-support values and novel combinations of previously observed values instead test interpolation and composition, capabilities that locality and field mediation can already provide\. Second, the criterion predicts its own failures\. An intervention on a parameter for which no map is supplied, such as a kernel width inside a Gaussian or an exponent inside a power law, should fail, while an oracle supplied with the missing map should succeed\. Such a failure therefore identifies missing intervention structure rather than insufficient data or model capacity\.

## 4Results

### 4\.1One set of weights decides whether a world dissipates and recovers its functions

The first question is whether a learned simulator can decide from trajectories whether its world loses energy and recover the constitutive functions that govern it\. Existing structured models do not pose this directly: conservation is fixed by the class or a dissipative head may take either sign, so conservative and weakly dissipative worlds need not be distinguished\. The one\-way port in Eq\. \([1](https://arxiv.org/html/2610.00280#S3.E1)\) makes “conservative” theσ=0\\sigma=0member of the dissipative family, turning the fitted rate into a measurable quantity\. We train one model jointly on conservative, dissipative \(σ=0\.5\\sigma=0\.5\), driven, and external\-field cavity episodes, giving only the condition identity\.

The resulting model separates the rates by four orders of magnitude:σ^cons=1\.2×10−5\\hat\{\\sigma\}\_\{\\rm cons\}=1\.2\\times 10^\{\-5\}, corresponding to an implied spurious loss of0\.6%0\.6\\%over50,00050\{,\}000steps,σ^diss=0\.481\\hat\{\\sigma\}\_\{\\rm diss\}=0\.481, and exactly zero for the external\-field condition\. The fluid reproduces the same separation,σ^inviscid=1\.4×10−4\\hat\{\\sigma\}\_\{\\rm inviscid\}=1\.4\\times 10^\{\-4\}versusσ^viscous=0\.500\\hat\{\\sigma\}\_\{\\rm viscous\}=0\.500\. The controls isolate the source of this result\. Without the port, the model cannot fit the dissipative world, with rate error0\.990\.99; the informed baseline fails outright, with rollout error∼106\{\\sim\}10^\{6\}; and the sign\-free port learns a small*negative*rate for the conservative condition, creating a spurious energy source that the one\-way port forbids by construction\. The estimated rate is not merely decorative\. Clamping the dissipative rate to incorrect values and retraining the remaining components leaves the fit minimum at the true value, showing that the learned energy does not absorb the missing dissipation\. On fresh data, the conservative and dissipative rates remain separated by orders of magnitude, and a label\-free encoder recovers the same distinction from only the first100100steps of a rollout \(App\.[F](https://arxiv.org/html/2610.00280#A6)\)\.

Table 1:Recovered constitutive functions\. Learned against the reference over each operating range, the grid’s vacuum\-energy derivatives are linear, implying a quadratic energy, while the fluid’s mass densityM⁡\(h\)M\(h\)and potential forceV′​\(h\)V^\{\\prime\}\(h\)are recovered at unit slope\. Gravity is supplied, so the fluid coefficient is an effective coefficient rather than discovered gravity, andV′V^\{\\prime\}is identifiable only up to an additive constant\.The same weights recover functions rather than only scalar coefficients\. On the grid, with no functional form assumed beyond confinement, the learned constitutive derivatives are linear, so the model recovers the quadratic form of the vacuum energy\. A residual4%4\\%slope\-ratio systematic remains, which we report as a constitutive diagnostic rather than interpret as an independent wave\-speed estimate\. In the fluid, the mass densityM⁡\(h\)M\(h\)and potential forceV′​\(h\)V^\{\\prime\}\(h\)are recovered at unit slope \(Table[1](https://arxiv.org/html/2610.00280#S4.T1)\)\. Because gravity is supplied, the fitted coefficient is effective rather than an independently discovered gravity, andV′V^\{\\prime\}is recovered only up to a Casimir gauge that leaves the trajectories unchanged\. A genuinely nonlinear truth,V′​\(h\)=h2V^\{\\prime\}\(h\)=h^\{2\}, is recovered with its curvature\. Compared with an explicitly linear constitutive model trained to convergence on identical data, the nonlinear model reduces one\-step, rollout, and held\-out gravity\-intervention errors by66–17×17\\timesat small excitation and22–5×5\\timesat larger excitation, while retaining the same long\-horizon stability\. The advantage appears primarily in downstream prediction rather than uniform function error because a linear law approximatesh2h^\{2\}closely over the sampled operating range \(Table[4](https://arxiv.org/html/2610.00280#A6.T4), App\.[F](https://arxiv.org/html/2610.00280#A6)\)\.

At the next level, the model can also select among candidate structural forms\. A Bayesian information criterion over structural channels identifies the true form on all4040stress\-ladder worlds and never selects a non\-nested incorrect form\. On the cavity, it selects the dissipative and driven forms byΔ\\DeltaBIC22572257and10541054, while also recovering the dissipation rate\. The only near\-degenerate case is the conservative world, where the one\-step objective cannot cleanly separate exact conservation from vanishingly small dissipation, even though those alternatives differ strongly over the full rollout horizon\. The main selection result therefore uses a frozen conformal selector evaluated across four independently implemented generator families\. It reaches four\-way accuracy0\.840\.84–0\.890\.89and true\-candidate coverage0\.730\.73–0\.900\.90\(Table[6](https://arxiv.org/html/2610.00280#A9.T6)\)\. Coverage below the nominal0\.900\.90on two families occurs under generator shift and therefore does not contradict the exchangeable coverage guarantee \(App\.[C\.2](https://arxiv.org/html/2610.00280#A3.SS2)\)\. Taken together, one model per regime identifies whether its world dissipates, recovers the constitutive content up to stated non\-identifiabilities, and carries its estimation error into the predictions that use that content\.

### 4\.2The second law holds by construction, and the guarantee has a price

An unconstrained model gives no structural reason for the sign of a passive energy exchange to be correct\. Our solution is architectural: the port in Eq\. \([1](https://arxiv.org/html/2610.00280#S3.E1)\) can only remove the quadratic diagnostic energy of its channel \(Prop\.[3](https://arxiv.org/html/2610.00280#Thmproposition3)\), so within that scope non\-injective dissipation is a property the model cannot violate rather than one it must learn\. The remaining questions are whether it recovers the amount of dissipation quantitatively and at what representational cost, which we test on the cavity with a learned rate, a driven condition, a fluctuating bath, an engineered violator, and a sign\-free port that exposes what the constraint excludes\.

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/b2_stochastic_m1.png)Figure 2:The stochastic second law under a fluctuating bath: field temperature over time \(a\) and the stationary distribution of the field energy \(b\)\. Reference in blue, structured model in dashed magenta, equal\-capacity baseline in orange, and the bath temperature and analytic mean as dashed red lines; the baseline is off\-scale in \(b\)\.The deterministic decay rate is recovered to within0\.4%0\.4\\%, and the driven steady state is reproduced correctly\. Figure[2](https://arxiv.org/html/2610.00280#S4.F2)shows the fluctuating\-bath experiment\. The left panel reports field temperature over time, while the right panel shows the stationary field\-energy distribution\. From a cold start, the structured model relaxes with the reference to the bath temperature, with both field sectors agreeing, whereas the equal\-capacity baseline reaches a temperature roughly six orders of magnitude too high\. A noise\-doubled violator is flagged reliably while the structured model never triggers the diagnostic\. The stronger evidence comes from the magnetic sector, which is neither directly damped nor driven\. It can thermalise only through the learned reversible coupling, so its convergence to the same temperature shows that the learned conservative core transfers energy with the required balance, something a dissipation\-side fit alone cannot produce\. The representational price is equally explicit\. When asked to run a negative dissipation rate, the constrained port clamps at zero while the sign\-free port tracks the divergent world\. A model family that guarantees the sign of dissipation therefore cannot represent the opposite sign through that channel, and genuine energy injection must instead enter through the explicit drive \(App\.[E](https://arxiv.org/html/2610.00280#A5)\)\. This constraint matters most nearσ=0\\sigma=0, where the data provide the weakest sign information and the sign\-free control in the previous section invents a small energy source\.

### 4\.3Interventions on physical parameters transfer where the map is supplied and are refused where it is not

Table 2:A charge\-magnitude counterfactual on the cavity\. Rollout error \(nMSE at10001000steps, mean±\\pms\.d\. over repeated runs\) at three unseen magnitudes, for a model trained at\|q\|=1\|q\|=1and a control trained on the wider range\|q\|∈\{1,1\.5\}\|q\|\\in\\\{1,1\.5\\\}; both share the supplied charge map\. The two rows coincide across all test values, showing that wider parameter coverage provides no measurable benefit\. Residual error is set by the state region visited by the rollout rather than by distance from the training value\.The second behaviour asks whether a model can answer a “what if” about a physical parameter at a value absent from training, which learned simulators often answer with the training law\. The criterion in §[3\.5](https://arxiv.org/html/2610.00280#S3.SS5)says when a response is available by construction: the parameter must enter through a supplied map, and the query must leave its training support\. We test both sides: transfer should hold for a flipped charge sign and for unseen magnitudes, rates, and gravity when their dependence is factored, and should be refused when no map is supplied, with an oracle given the missing map showing the refused physics is learnable\.

The factored model follows the flipped\-sign world on both electromagnetic regimes \(p=0\.001p=0\.001\), and the result replicates on fresh data\. The baseline and non\-factored control instead perform at or*below*chance\. Every training trajectory rotates in one direction, so the black box learns this rotation as a property of the dynamics and retains it after the sign flip\. Magnitude, rate, and gravity also transfer when the generator is linear in the intervened parameter\. A doubled charge matches an oracle trained directly at that value, while the non\-factored control reverses its rotation; a dissipation rate trained at0\.50\.5transfers to0\.250\.25; the fluid viscosity transfers with comparable accuracy; and changing gravity from1\.01\.0to1\.51\.5yields error6\.8×10−116\.8\\times 10^\{\-11\}, indistinguishable from the in\-distribution value, while the non\-separable control and baseline fail by three to twelve orders of magnitude\. Increasing the training coverage of the parameter provides no additional benefit \(Table[2](https://arxiv.org/html/2610.00280#S4.T2)\)\. The residual error is determined by the region of state space visited by the new trajectory rather than by the numerical distance of the parameter from training\. Two observations sharpen this distinction\. First, composition and extrapolation separate cleanly: both field\-mediated models compose an unseen mixed\-charge configuration, whereas only the factored model extrapolates the governing law\. Locality therefore supports composition, while the supplied parameter map supports the support\-moving intervention\. Second, the remaining error is limited by state\-space coverage rather than parameter\-space distance: the flipped grid trajectory eventually leaves its trained state envelope, while the cavity remains accurate for a doubled charge through10001000steps\.

The refusals are equally informative\. In each regime, an intervention whose parameter map was not supplied causes the factored model to fail while an oracle given the missing map succeeds\. A particle\-shape width inside a Gaussian produces error111×111\\timesthe in\-distribution value, a pressure exponent insidehγh^\{\\gamma\}produces approximately109×10^\{9\}\\timesthe in\-distribution error, and the grid width produces a7\.0×7\.0\\timesincrease, below the pre\-specified10×10\\timesthreshold\. The corresponding oracle succeeds in every case \(App\.[C\.5](https://arxiv.org/html/2610.00280#A3.SS5)\)\. Misspecification of the broader function class can also be detected empirically: a confining\-quadratic energy fitted to a quartic world raises its fit floor by four orders of magnitude and, once the quartic contribution becomes sufficiently strong, fits worse than a capacity\-matched black box even without access to the true model class\. These are implementation\-level failures rather than an impossibility theorem for nonlinear maps\. Their significance is that the criterion predicts where the tested structure should stop transferring, turning the successful interventions from isolated examples into a falsifiable rule\.

### 4\.4Long\-horizon stability needs a confining energy and an energy\-consistent coupling

The third behaviour is remaining physically bounded far beyond the training horizon\. Unconstrained simulators fail because small one\-step errors accumulate without a structural mechanism that bounds the state\. A conserved energy alone might appear sufficient, but the ablations show that two ingredients are required\. With an unconstrained learned field energy, the cavity model fits the training dynamics and passes the counterfactual tests, yet its50,00050\{,\}000\-step rollout diverges, reaching deviations of2\.9×1072\.9\\times 10^\{7\}\. Restricting the learned energy to the confining class changes the result from uniformly diverging to uniformly bounded\. Figure[3](https://arxiv.org/html/2610.00280#S4.F3)shows the full rollout for three models\. The structured model remains near10−210^\{\-2\}relative true\-energy deviation, the equal\-capacity baseline exceeds10810^\{8\}early in the rollout, and the non\-factored control reaches10410^\{4\}despite sharing the same confining energy\.

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/d0a1b_conservative_m3_energy.png)Figure 3:Long\-horizon energy drift on the conservative cavity over50,00050\{,\}000steps \(100×100\\timesthe training horizon, log scale\): structured model in dashed magenta, equal\-capacity baseline in orange, and the non\-factored control in dash\-dotted purple\.Confinement alone is therefore insufficient\. The non\-factored control still diverges because its learned coupling injects net energy at the particle\-field interface\. In the true dynamics, the particle force and its induced field current are derivatives of the same interaction energy, so their energy exchange cancels\. A black\-box coupling learns the two sides independently, allowing a small mismatch to perform net work each cycle, which accumulates over a long rollout\. The fluid provides the complementary control\. Its non\-separable variant black\-boxes only the gravity dependence while preserving the energy\-consistent flux coupling\. It therefore remains bounded but fails the gravity counterfactual\. Stability thus follows coupling consistency, whereas counterfactual transfer follows the supplied parameter map\. Boundedness should not be confused with accuracy: rare cavity episodes exceed the5%5\\%worst\-case energy criterion even though typical trajectories remain well within it, while the grid closes this gap\. We therefore report boundedness, energy fidelity, and rate accuracy as distinct outcomes\.

### 4\.5Disturbances, scale, noise, and everyday physics

The fourth behaviour is absorbing a disturbance, and three supporting studies then ask whether the preceding effects could instead be explained by scale, absence of noise, or the idealised systems\. A mid\-rollout velocity kick is absorbed because it moves the state along the reachable manifold, whereas a coherent field perturbation is off\-manifold and remains the sensitive direction\. Conserved invariants double as validity meters, transporting an injected violation without amplification \(App\.[M](https://arxiv.org/html/2610.00280#A13.SSx32)\)\. Scale does not recover the missing response: across four black\-box families from10410^\{4\}to10810^\{8\}parameters, no competent regime shows a reliable sign\-flip counterfactual \(App\.[M](https://arxiv.org/html/2610.00280#A13.SSx34)\)\. The advantage also survives5%5\\%training noise, and where a clean, correctly specified dictionary exists symbolic regression is the more accurate tool, so the motivation for learned structure is high dimensionality, partial observation, unknown dictionaries, and long\-horizon stability rather than noise alone\. Finally, an eleven\-system suite of everyday physics and the four scope exceptions are evaluated in App\.[I](https://arxiv.org/html/2610.00280#A9), including the one pre\-specified selection gate the full matrix misses and why \(App\.[A\.3](https://arxiv.org/html/2610.00280#A1.SS3)\)\.

## 5Conclusion and Limitations

This paper asks how little structural inductive bias a learned world model needs in order to recover identifiable physics and behave reliably outside its training distribution\. Fully unconstrained models cannot infer true dynamics from trajectories alone: they often diverge over long rollouts and collapse under interventions on physical parameters\. Adding a small amount of geometric and thermodynamic scaffolding changes this picture entirely\. A Poisson\-style bracket, a confining energy class, a port\-Hamiltonian dissipation channel, and factored parameter maps together make the remaining constitutive relations identifiable and accurate\. The resulting model behaves like a calibrated instrument rather than a black box: it cleanly separates conservative and dissipative regimes, learns nonlinear constitutive laws, transfers interventions where equally expressive baselines fail, and stays stable over long rollouts\. The general geometric structure helps rather than hurts, because it provides the coordinate frame within which data can pin down the specific physics\.

However several limitations remain\. The dissipation guarantee applies only to quadratic port\-channel energy, and boundedness is proven only for conservative dynamics \(else supported empirically\)\. Potential force is identifiable only up to an unobservable Casimir constant\. Our work is not unsupervised law discovery \- brackets, channels, and kernels are user\-supplied, which also blocks unspecified parameter interventions\. The results in this work rely on synthetic classical simulators rather than experiments, with edge\-case failures \(e\.g\., cavity energy tails\), generator shifts, and inconclusive scaling results defining necessary operating boundaries\.

## AI use statement

A large language model assisted the authors in polishing the language of the manuscript\. The authors verified that all claims, proofs, mathematical formulations, and reported values are valid, checked them against the implementation and results, and take full responsibility for the manuscript\.

## Ethics statement

This work uses only synthetic physics simulations; no human subjects, personal data, or scraped datasets are involved\. We do not foresee direct harmful applications; the contribution is a measurement discipline and structural analysis for physical world models\. Integrity\-relevant practices \(pre\-registration, recorded verdicts, disclosure of overturned verdicts and process defects\) are documented in Appendices[K](https://arxiv.org/html/2610.00280#A11)and[M](https://arxiv.org/html/2610.00280#A13)\.

## Reproducibility statement

The reference simulators, calibration gates, model families, and training protocols are specified in Appendices[A](https://arxiv.org/html/2610.00280#A1)–[I](https://arxiv.org/html/2610.00280#A9); the theory and proofs in Appendices[C\.2](https://arxiv.org/html/2610.00280#A3.SS2)and[L](https://arxiv.org/html/2610.00280#A12); the per\-experiment record with per\-seed results in Appendix[M](https://arxiv.org/html/2610.00280#A13); and the verdict\-integrity protocol in Appendix[K](https://arxiv.org/html/2610.00280#A11), under which the constants that determine every reported verdict are fixed by a pre\-registration committed before the affected runs and the result tables are generated directly from the stored measurements rather than transcribed by hand\. All data are regenerated deterministically from seeds\. Our implementation, the pre\-registration documents with their dated amendment history, and the result artifacts are provided as supplementary material\.

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## Overview of the appendices

These appendices are organised by purpose so that each can be read on its own\. The first group describes*how*the work is measured and*on what*: Appendix[A](https://arxiv.org/html/2610.00280#A1)states the measurement methodology and the discipline used to turn observations into claims, and Appendix[B](https://arxiv.org/html/2610.00280#A2)specifies the three physical systems, their calibration, and the model and control families with their training details\. The second group presents the*experimental evidence*, one appendix per scientific question: the criterion for counterfactual transfer and its refusal tests \(Appendix[C](https://arxiv.org/html/2610.00280#A3)\); the two ingredients of long\-horizon stability \(Appendix[D](https://arxiv.org/html/2610.00280#A4)\); dissipation, driving, and stochastic consistency \(Appendix[E](https://arxiv.org/html/2610.00280#A5)\); discovery of the physics and selection of its form \(Appendix[F](https://arxiv.org/html/2610.00280#A6)\); and generality across a different physics and a symbolic\-regression competitor \(Appendix[G](https://arxiv.org/html/2610.00280#A7)\)\. The third group steps back: Appendix[H](https://arxiv.org/html/2610.00280#A8)synthesises which structural element buys which behaviour, the stated scope exceptions and an eleven\-system suite are tested in Appendix[I](https://arxiv.org/html/2610.00280#A9), and Appendix[J](https://arxiv.org/html/2610.00280#A10)discusses what the evidence does and does not support\. The final group is*reference material*: the verdict\-integrity protocol \(Appendix[K](https://arxiv.org/html/2610.00280#A11)\), the formal propositions and their proofs \(Appendix[L](https://arxiv.org/html/2610.00280#A12)\), the complete per\-experiment record \(Appendix[M](https://arxiv.org/html/2610.00280#A13)\), and an extended related\-work discussion \(Appendix[N](https://arxiv.org/html/2610.00280#A14)\)\.

## Appendix AThe measurement program: behaviours, controls, and claim discipline

### A\.1Why a measurement program at all

The failure mode of the field is not a shortage of architectures but a shortage of*discriminating tests*\. A model can score well on trajectory error while violating energy conservation catastrophically at longer horizons\. It can “pass” a counterfactual test that any interpolating model would pass\. And its energy drift can look flat because it predicts trivial motion\. Each behaviour below therefore comes with three elements: \(i\) a confound\-resistant metric, \(ii\) a*positive control*\(an engineered case the diagnostic must flag, so a silent diagnostic cannot masquerade as a passing model\), and \(iii\) a control model that isolates which structural element the behaviour credits\.

### A\.2The four behaviours

Table 3:The behaviours, structured model against controls\. Each row’s control lacks exactly one structural element\. Seeds are55unless stated; flips are seed\-level exact binomials over1010seeds\.
### A\.3Disturbances, scale, and noise: the numbers behind the main\-text summary

A mid\-rollout velocity kick is absorbed \(post/pre error ratio0\.850\.85on the cavity,1\.141\.14on the grid\) because it moves the state along the dynamics’ reachable manifold\. A coherent field bump, by contrast, manufactures an off\-manifold configuration and remains the sensitive direction\. Inserting a third particle mid\-rollout continues on the correct law, and because the non\-factored control also passes this test, the credit goes to locality rather than to the factored map\.

Re\-identifying the state from a single rendered frame misses velocity \(continuation error22\.822\.8\), whereas three strided frames recover it by differencing \(4\.54\.5; oracle floor0\.0130\.013\)\. The boundary is therefore momentum, not perception in general \(App\.[M](https://arxiv.org/html/2610.00280#A13.SSx32)\)\.

Scaling four black\-box families from10410^\{4\}to10810^\{8\}parameters under a competence\-gated protocol produced no rung with a reliable sign flip\. The result is inconclusive on its own terms, since only one rung was competent\. A width\-controlled study with the representation fixed finds no gain from width \(one\-sign error8\.168\.16–11\.8311\.83, Holmp=0\.105p=0\.105; App\.[M](https://arxiv.org/html/2610.00280#A13.SSx34)\)\.

Under5%5\\%training noise the flip stays intact and the stability envelope loosens gracefully \(0\.045→0\.2850\.045\\to 0\.285at1010k steps\)\. Symbolic regression with the same dictionary is most accurate on the noise\-free cavity \(worst coefficient error0\.07%0\.07\\%\)\. However, its central\-difference form collapses under noise \(0\.02→0\.23→0\.940\.02\\to 0\.23\\to 0\.94at0\.50\.5,22,5%5\\%\), while the weak form stays competitive \(0\.007→0\.028→0\.100\.007\\to 0\.028\\to 0\.10\)\. The suite comprises eleven systems with twelve pass cards, eight clean and four with one identified limit \(App\.[I](https://arxiv.org/html/2610.00280#A9)\)\.The second law\.The model must dissipate at the true rate to the true terminal state, track genuine external injection without manufacturing it, and \(the statistical half\) relax to the correct stationary distribution and temperature under a fluctuating bath\. Metrics are rates and distributions, never trajectory error, because in chaotic or stochastic systems trajectory error is meaningless \(a lesson enforced throughout\)\. Required controls include a genuinely driven system the model must*not*force to zero, and an engineered second\-law violator the production diagnostic must catch\.

Counterfactuals\.Trained at one parameter setting, tested at a never\-seen setting \(sign flip, unseen magnitude, unseen rate, unseen coupling constant\)\. Metrics: a regime/orbit\-sense match*plus*error against the true counterfactual trajectory at a stated horizon\. A test only counts if the metric has*power*on the model under test \(the model must get the*trained*setting right on held\-out episodes\)\. Task validity is defined by reference accuracy, intervention magnitude, and measurement sensitivity, not by the comparator’s outcome: an equally\-structured non\-separable control is always run and its result reported, and a task on which both models succeed is recorded as “not discriminating between these methods”, an informative outcome that locates where the structural constraint is unnecessary, rather than an exclusion rule\.

Extreme horizon\.Rollouts to≥100×\\geq 100\\timesthe training horizon\. Energy\-drift slope and boundedness are co\-reported with rollout error, because a low energy drift can be achieved by trivial motion and boundedness alone hides excursions; worst\-case per\-seed deviations are reported, not only means\. Control: the equal\-capacity baseline is always run at the same horizon and reported; a horizon on which it also stays bounded is recorded as not discriminating between the methods, not discarded, since finite trajectories are not necessarily accurate ones and both fidelity and boundedness are reported for every model\.

Disturbance robustness\.Graded sub\-tests, scored separately because they credit different structure: \(a\) reset to unseen\-but\-valid states, swept small to large; \(b\) structural intervention*given*the new state \(add or teleport an object mid\-rollout\); and \(c\) re\-identification of the change from lossy*observation*, carried through this paper as a boundary to be mapped, not a capability to be claimed \(§[M](https://arxiv.org/html/2610.00280#A13.SSx32)\)\.

### A\.4Claim discipline

Every quantitative claim carries one of three evidence labels, used consistently throughout the appendices\.decisivemarks a large effect with non\-overlapping confidence intervals that replicates across independent training seeds\.directionalmarks a scoped finding whose intervals are wide, or in which one pre\-registered arm falls below its threshold\.boundarymarks a mapped limit of the method, reported as a result of equal standing rather than hidden\. For outcomes that are essentially pass\-or\-fail, we test at the level of whole seeds rather than individual episodes, since episodes within one seed are correlated \(a broken control once scored a perfect 32/32 episode block, which retired per\-episode averaging permanently; Appendix[K](https://arxiv.org/html/2610.00280#A11)\)\.

Credit assignment is fixed*in advance*: tests passed by any field\-mediated model \(composition, intervention\-given\-state\) credit*locality*, and only support\-moving counterfactuals and energy\-consistent\-coupling results credit*factoring*\. The verdict text is generated directly from the pre\-registered constants rather than transcribed by hand\. Headline verdicts require an escalation \(ten seeds and fresh data draws\) before any guard moves\. Two preliminary scaling verdicts failed their escalation and are reported as such, with the full ledger of metric corrections in Appendix[K](https://arxiv.org/html/2610.00280#A11)\.

## Appendix BThree regimes, calibrated before any training

No model in this paper is trained before its reference simulator passes conservation gates over the*full evaluation horizon*\. The reason is arithmetic: our stability claims run to100×100\\timesthe training horizon, so reference drift at that horizon is a direct floor on every number we report\. The calibration\-first requirement also caught real defects before they could contaminate results, and we report them because they are part of why the numbers can be trusted\.

### B\.1Electromagnetic ring cavity \(modal\)

Charged particles on a 1D periodic ring coupled toK=16K=16field modes through a fixed smoothing kernel, with a palindromic \(Strang\) symplectic integrator\.Calibration: over 50,000 steps, worst relative energy drift1\.38×10−51\.38\\times 10^\{\-5\}, which is7\.25×7\.25\\timesunder the10−410^\{\-4\}gate on the binding \(mixed\-charge\) configuration, and the Gauss\-law residual≤2\.0×10−14\\leq 2\.0\\times 10^\{\-14\}, which is exactly conserved\. Two defects were caught by the pre\-registered test battery before any training\. The first was a first\-order operator splitting \(energy errorO⁡\(Δ​t\)O\(\\Delta t\)and secular, instantly diagnosed by the halve\-Δ​t\\Delta ttest that distinguishes first\- from second\-order\), fixed by the palindromic composition\. The second was a tensor\-view aliasing bug introduced by the fix itself \(a symplectic pair update through in\-place slices mutated its own inputs\), fixed by snapshotting\.*Why this matters beyond hygiene:*both defects would have shown up as “model instability” at long horizon, exactly the phenomenon under study, had they survived\.

### B\.22D electromagnetic grid \(particle\-in\-cell\)

A64×6464\\times 64periodic Yee\-FDTD grid, Boris particle pusher, charge\-conserving \(Esirkepov\) current deposition, staggered\-consistent force gather, binomial smoothing applied identically to sources and gathered fields\.Calibration: energy drift≤4\.6×10−5\\leq 4\.6\\times 10^\{\-5\}\(gate10−210^\{\-2\}\) and discrete Gauss residual3\.1×10−133\.1\\times 10^\{\-13\}\(gate10−1010^\{\-10\}\) at 50,000 steps, so the particle\-in\-cell numerical\-heating risk did not materialize at this configuration\. The battery caught a static self\-force \(a stationary charge felt0\.080\.08–0\.20\.2of its own field; a moving charge lost kinetic energy to spurious drag\), root\-caused in two layers: the force gather must drop one B\-spline order per staggered axis to pair with Esirkepov currents, and the remainder is the lattice self\-energy gradient, suppressed40×40\\timesby the smoothing\.*Why it matters:*an uncaught self\-force is a phantom dissipation channel that would have corrupted every second\-law measurement on this system\.

### B\.31D shallow\-water fluid \(non\-electromagnetic\)

Water heighth⁡\(x\)h\(x\)and momentum on a periodic line: conservative core \(mass, energy==kinetic\+12​g​h2\+\\ \\tfrac\{1\}\{2\}gh^\{2\}\), viscous drag \(the physical analog of the dissipative port\), and wind forcing \(the drive\), advanced by a spectral, symmetric implicit\-midpoint integrator\.Calibration: over 50,000 steps \(250×250\\timesthe 200\-step training horizon\), mass is exact \(3×10−153\\times 10^\{\-15\}\), inviscid energy drift is1\.2×10−71\.2\\times 10^\{\-7\}\(linear\) and7\.5×10−57\.5\\times 10^\{\-5\}at the nonlinear operating amplitude \(hhvarying±10%\\pm 10\\%\), gravity\-wave speed is within0\.6%0\.6\\%\(worst mode, gate1%1\\%\), and viscous decay matches the analytic2​ν2\\nulaw to10−1010^\{\-10\}\. The system passes 18 unit tests\. Scope: the smooth \(pre\-shock\) regime, where randomly seeded multi\-mode initial data disperses and recurs rather than steepening into a coherent shock at our amplitude\. This is what makes a250×250\\timesclean horizon available at a genuinely nonlinear operating point\.*Why a fluid:*it changes everything incidental, namely the conservation structure \(mass instead of charge\), the field content \(scalar height instead of vector fields\), and the physical domain, while keeping the imposed form, so any behaviour that reproduces here is a property of the form, not of electromagnetism\.

### B\.4Structured models and the control family

All structured models share one imposed form\. The state evolves by a fixed energy\-conserving flow generated by a*learned*energy functional: per\-mode confining quadratics with learned bounded corrections on the cavity; a learned constitutive energy density entering fixed curl\-form \(Yee\) stencils on the grid; and learned mass\-metricM⁡\(h\)M\(h\)and potential\-derivativeV′​\(h\)V^\{\\prime\}\(h\)functions in the fluid, each parameterized as a linear term plus a bounded nonlinear correction,

M⁡\(h\)\\displaystyle M\(h\)=softplus⁡\(a\)​h\+αM​tanh⁡\(MLPM​\(h\)\),\\displaystyle=\\mathrm\{softplus\}\(a\)\\,h\+\\alpha\_\{M\}\\tanh\\\!\\big\(\\mathrm\{MLP\}\_\{M\}\(h\)\\big\),\(5\)V′​\(h\)\\displaystyle V^\{\\prime\}\(h\)=softplus⁡\(c\)​h\+αV​tanh⁡\(MLPV​\(h\)\),\\displaystyle=\\mathrm\{softplus\}\(c\)\\,h\+\\alpha\_\{V\}\\tanh\\\!\\big\(\\mathrm\{MLP\}\_\{V\}\(h\)\\big\),together with a viscosity port\. Irreversibility exists only as a*learned non\-negative*port \(rateσ≥0\\sigma\\geq 0multiplying the exact dissipative structure\); energy injection exists only as*known explicit*drives\. Coupling constants that the criterion requires to be counterfactual\-ready \(chargeqq, gravitygg\) enter multiplicatively\. Because gravityggis supplied, a recovered fluid “gravity” of1\.0001\.000is the supplied value times the learned slope ofV′V^\{\\prime\}, an effective coefficient rather than a discovered constant\.

The control family is the paper’s inference engine:

Unstructured baseline\(DeepSet on the cavity, CNN on grid and fluid\), capacity matched to a disclosed few percent in these comparisons\. Its reported failures show that parameter count alone does not ensure success under this training protocol; they do not exclude alternative unstructured architectures or optimization methods\.

Non\-factored control: identical locality and field mediation, coupling through a black box, with*slightly more*parameters\. The comparison tests the effect of supplying the parameter map within the evaluated implementation; individual successes or failures do not uniquely attribute a mechanism\.

Oracle variants: the counterfactual physics baked in at matched training budget, showing that the regime can be learned with that additional information\. This is an implementation\-level control, not a universal transfer impossibility\.

Sign\-free port: dissipation rate free to go negative, dissociating the*guaranteeing*of the second law from the*representing*of its violation\.

Informed baseline\(for the discovery claims\): the unstructured model*plus*every input the structured model gets \(condition index, drive and field features, the auxiliary loss\)\. This controls the listed inputs, although architecture, optimization, and the use of those inputs can still affect the comparison\.

Models are8\.68\.6–11\.511\.5k parameters throughout the behavioural program \(grid:8,7118\{,\}711structured,8,5918\{,\}591baseline,11,30611\{,\}306non\-factored\); the scaling study \(§[M](https://arxiv.org/html/2610.00280#A13.SSx34)\) is where capacity varies\. The unstructured baseline is capacity\-matched to a disclosed few percent and certified by a three\-rate learning\-rate sweep, rollout\-stabilizing noise, and the best competent configuration reported\. The informed baseline used for the discovery claims is the unstructured model plus every input the structured model receives \(condition index, drive and field features, and the auxiliary energy loss\)\.

#### Training\.

All models minimise one\-step prediction error on per\-dimension standardized states with Adam for120120–400400epochs by regime\. Ports use a10×10\\timeslearning rate and, in the shared\-content protocol, anL1L\_\{1\}penalty of10−510^\{\-5\}on the rates and a5050\-step episode\-energy match read out with the reference energy functional; that last term is supplied supervision, which the label\-free variant removes entirely\.

#### Metrics\.

Withx^0:T\\hat\{x\}\_\{0:T\}a model rollout,x0:Tx\_\{0:T\}the reference from the same initial state,EEthe true energy of a state, andσ^\\hat\{\\sigma\}a learned rate,

nMSE=∥x^0:T−x0:T∥2Var\(x0:T\),D\(t\)=\|E⁡\(x^t\)−E⁡\(x^0\)\|\|E⁡\(x^0\)\|,ϵσ=\|σ^−σ\|σ,\\mathrm\{nMSE\}=\\frac\{\\lVert\\hat\{x\}\_\{0:T\}\-x\_\{0:T\}\\rVert^\{2\}\}\{\\mathrm\{Var\}\(x\_\{0:T\}\)\},\\qquad D\(t\)=\\frac\{\\lvert E\(\\hat\{x\}\_\{t\}\)\-E\(\\hat\{x\}\_\{0\}\)\\rvert\}\{\\lvert E\(\\hat\{x\}\_\{0\}\)\\rvert\},\\qquad\\epsilon\_\{\\sigma\}=\\frac\{\\lvert\\hat\{\\sigma\}\-\\sigma\\rvert\}\{\\sigma\},\(6\)where the energy driftD⁡\(t\)D\(t\)is reported as the per\-seed median and the worst episode\.*Regime match*is the fraction of held\-out episodes whose orbit sense agrees with the true counterfactual, and the stochastic second law is scored by the stationary temperature and an energy\-histogram Wasserstein distance\. Near\-binary outcomes use seed\-level exact binomial tests, since episodes within a seed are correlated\. Headline numbers use five seeds, the sign\-flip and unified\-model results ten, and the rate\-transfer and anti\-dissipation arms three;±\\pmdenotes a95%95\\%tt\-interval over seeds\. Every finding in the appendices carries an evidence label, replicated, directional, or boundary, whose criteria are given in App\.[A](https://arxiv.org/html/2610.00280#A1),[M](https://arxiv.org/html/2610.00280#A13), and[K](https://arxiv.org/html/2610.00280#A11)\.

## Appendix CA structural criterion for counterfactuals, with its refusal tests

### C\.1A sufficient construction and an unrestricted\-class limit

Suppose a true generator and a learned generator share the same supplied parameter dependence\. Accurate learned content then transfers through that dependence at new parameter values, on the state region where the content remains accurate\. Multiplicative coupling is the construction tested here\. Known nonlinear maps can also transfer, so separability is not necessary\.

A different statement concerns unrestricted functions of state and parameter\. Observations restricted to the training support cannot distinguish functions that agree there and differ off support\. This is not a converse to the sufficient construction\. Leaving marginal parameter support defines the extrapolation tests here; in\-support interventions and novel combinations remain meaningful tests of interpolation and composition\. These test categories are not identified with a causal positivity condition\. Proposition[5](https://arxiv.org/html/2610.00280#Thmproposition5)\(App\.[L](https://arxiv.org/html/2610.00280#A12)\) states the two mathematical claims separately\. The refusal experiments measure failures of the tested implementations, not an impossibility for every nonlinear parameter map\.

### C\.2Coverage and separation for the four\-way selector

The selector that turns the criterion into a decision rests on a finite\-sample guarantee rather than on an informal equivalence\. Two statements make this precise\. The first bounds the chance that the true in\-family candidate is retained; the second states when empirical risk separates the candidates and when two candidates are indistinguishable in principle\.

The first is the coverage statement\.

###### Proposition 1\(Marginal true\-candidate retention\)\.

For candidatekkletsk,1,…,sk,nks\_\{k,1\},\\ldots,s\_\{k,n\_\{k\}\}be calibration scores,mk=⌈\(nk\+1\)​\(1−α\)⌉m\_\{k\}=\\lceil\(n\_\{k\}\+1\)\(1\-\\alpha\)\\rceil, andqkq\_\{k\}themkm\_\{k\}th smallest calibration score, withqk=\+∞q\_\{k\}=\+\\inftyifmk\>nkm\_\{k\}\>n\_\{k\}\. Define the retained set𝒞⁡\(D\)=\{k:sk​\(D\)≤qk\}\\mathcal\{C\}\(D\)=\\\{k:s\_\{k\}\(D\)\\leq q\_\{k\}\\\}\. If the new score of the true candidatek∗k^\{\*\}is exchangeable with its calibration scores, thenPr\{k∗∈𝒞\(D\)\}≥1−α\\Pr\\\{k^\{\*\}\\in\\mathcal\{C\}\(D\)\\\}\\ \\geq\\ 1\-\\alpha\.

###### Proof of Proposition[1](https://arxiv.org/html/2610.00280#Thmproposition1)\.

With continuous scores, exchangeability makes the augmented rank uniform\. The selected order statistic excludes the new score with probability at mostα\\alpha\. With ties, accepting scores at or below the threshold gives the same conservative lower bound; randomized tie\-breaking is an alternative construction\([Lei et al\., 2018](https://arxiv.org/html/2610.00280#bib.bib21)\)\. ∎

Set cardinality defines a decision rule, not a truth certificate\. A singleton can be wrong, a nonempty set can accept an out\-of\-family world, and an empty set can exclude an in\-family truth\. The excitation\-and\-conditioning gate detects its specified numerical/data failures, but it does not certify global optimization\. Coverage of the raw candidate set does not imply selective coverage after conditioning on that gate or on singleton selection\.

#### Reported calibration protocol\.

Calibration usesnk=100n\_\{k\}=100worlds per candidate andα=0\.10\\alpha=0\.10\. The finite\-sample construction above therefore uses the 91st order statistic\. The supplied score description combines the held\-out residual of a three\-coefficient linear acceleration fit with0\.010\.01times squared standardized distances from candidate coefficient regions\. The gate requires condition number≤106\\leq 10^\{6\}, at least eight held\-out samples, and charge/velocity feature standard deviations≥0\.02\\geq 0\.02\. Confirmation uses 50 worlds per label per family\. The implementation realises this specification exactly: the empirical quantile is the⌈\(nk\+1\)​\(1−α\)⌉\\lceil\(n\_\{k\}\+1\)\(1\-\\alpha\)\\rceil\-th order statistic \(the9191st of100100\), ties at the threshold are retained by the≤\\leqrule, calibration and confirmation worlds use disjoint seeds with thresholds frozen before any confirmation world is drawn, the score standardises each coefficient distance by that world’s own fitted standard error, and the gate is evaluated before the risk set is read\. Each of these properties is verified independently, including the order\-statistic identity, empirical marginal coverage of0\.900\.90on exchangeable heavy\-tailed scores, tie retention, and monotonicity inα\\alpha\. The family evaluations include generator changes\. Exchangeability with calibration is not established by procedural freezing or a new seed offset\. The observed family coverages are therefore empirical shift assessments; they are not asserted to inherit the nominal90%90\\%guarantee\. The raw family\-level confusion matrices and calibration provenance are needed to evaluate that assumption and quantify each decision’s uncertainty\.

###### Proposition 2\(Separation and ambiguity\)\.

Suppose a finite candidate family has population risksRkR\_\{k\}\. If one candidatek∗k^\{\*\}has marginRk−Rk∗\>2​ϵR\_\{k\}\-R\_\{k^\{\*\}\}\>2\\epsilonfor everyk≠k∗k\\neq k^\{\*\}, and every empirical risk is withinϵ\\epsilonof its population value, then empirical risk separatesk∗k^\{\*\}\. If two candidates have identical observable distributions on the available support, no selector using only those observations can distinguish them uniformly\.

###### Proof\.

For everyk≠k∗k\\neq k^\{\*\},

R^k≥Rk−ϵ\>Rk∗\+ϵ≥R^k∗,\\hat\{R\}\_\{k\}\\ \\geq\\ R\_\{k\}\-\\epsilon\\ \>\\ R\_\{k^\{\*\}\}\+\\epsilon\\ \\geq\\ \\hat\{R\}\_\{k^\{\*\}\},sok∗k^\{\*\}is the unique empirical minimiser\. For the second statement, a selector is a measurable function of the observations; if two candidates induce the same distribution on the observations, the selector’s output has the same distribution under both, so no output rule can have error below1/21/2on both candidates simultaneously\. ∎

The second is the second\-law guarantee for the dissipative port at the level of the implemented discrete update\.

###### Proposition 3\(Quadratic contraction and the isolated OU update\)\.

LetQ⁡\(q\)=12​q⊤​K​qQ\(q\)=\\tfrac\{1\}\{2\}q^\{\\top\}Kqwith fixedK≻0K\\succ 0, fixed scalarσ≥0\\sigma\\geq 0, stepΔ​t\>0\\Delta t\>0, anda=e−σ​Δ​ta=e^\{\-\\sigma\\Delta t\}\. The deterministic updateq\+=a​qq^\{\+\}=aqsatisfiesQ⁡\(q\+\)−Q⁡\(q\)=−\(1−a2\)​Q​\(q\)≤0Q\(q^\{\+\}\)\-Q\(q\)=\-\(1\-a^\{2\}\)\\,Q\(q\)\\leq 0, with equality only ifσ=0\\sigma=0orq=0q=0\. If insteadq\+=a​q\+ξq^\{\+\}=aq\+\\xiwith independentξ∼𝒩⁡\(0,\(1−a2\)​kB​T​K−1\)\\xi\\sim\\mathcal\{N\}\\big\(0,\(1\-a^\{2\}\)k\_\{B\}T\\,K^\{\-1\}\\big\), then𝒩⁡\(0,kB​T​K−1\)\\mathcal\{N\}\(0,k\_\{B\}T\\,K^\{\-1\}\)is invariant for this isolated update\.

###### Proof of Proposition[3](https://arxiv.org/html/2610.00280#Thmproposition3)\.

Homogeneity givesQ⁡\(a​q\)=a2​Q​\(q\)Q\(aq\)=a^\{2\}Q\(q\), soQ⁡\(q\+\)−Q⁡\(q\)=−\(1−a2\)​Q​\(q\)Q\(q^\{\+\}\)\-Q\(q\)=\-\(1\-a^\{2\}\)Q\(q\), which is non\-positive and vanishes only ifa=1a=1\(σ=0\\sigma=0\) orQ⁡\(q\)=0Q\(q\)=0\(q=0q=0\)\. For Gaussian input with covariancekB​T​K−1k\_\{B\}T\\,K^\{\-1\}, the independent\-noise update has zero mean and covariance

a2​kB​T​K−1\+\(1−a2\)​kB​T​K−1=kB​T​K−1,a^\{2\}k\_\{B\}T\\,K^\{\-1\}\+\(1\-a^\{2\}\)k\_\{B\}T\\,K^\{\-1\}=k\_\{B\}T\\,K^\{\-1\},so the complete Gaussian law is preserved\. ∎

The covariance reduces to\(1−a2\)​kB​T​I\(1\-a^\{2\}\)k\_\{B\}TIin unit\-stiffness coordinates\. The stochastic invariant\-law statement requires a fixed rate; a state\-dependent frozen\-rate update does not generally inherit it\. No property of a reversible splitting step is needed for the isolated contraction proof\. Composition with other updates requires separate analysis; numerically, the implemented quadratic\-class port never increases the channel energy at any tested\(σ,Δ​t\)\(\\sigma,\\Delta t\), and the fluctuation\-dissipation\-matched stochastic step has an analytic stationary\-temperature error of1\.7×10−31\.7\\times 10^\{\-3\}atΔ​t=0\.1\\Delta t=0\.1, falling to1\.7×10−51\.7\\times 10^\{\-5\}atΔ​t=0\.01\\Delta t=0\.01\.

Coordinate contraction need not dissipate an arbitrary learned energy\. For example, the confining energy

E⁡\(x\)=x2\+2​tanh⁡x\+2E\(x\)=x^\{2\}\+2\\tanh x\+2\(7\)increases atx=−0\.5x=\-0\.5under the contractionx˙=−x\\dot\{x\}=\-x, withE˙≈0\.286\\dot\{E\}\\approx 0\.286\. This discrepancy persists in the continuous generator and cannot be removed by refining the time step\. The reported full\-model temperature and distribution checks are empirical; they do not establish a general entropy law, detailed balance, or the invariant distribution of the complete coupled nonlinear sampler\.

### C\.3Composition and extrapolation dissociate three levels of structure

Result \(cavity, grid\)\.Three models, graded structure, two tests\. An unseen*mixed\-charge configuration of seen pieces*\(composition\): the unstructured baseline is at chance \(grid regime match0\.469±0\.160\.469\\pm 0\.16\); the non\-factored control passes \(0\.896±0\.0450\.896\\pm 0\.045, just clearing its0\.900\.90gate, disclosed\); and the factored model passes \(0\.979±0\.0450\.979\\pm 0\.045\)\. A*support\-moving*test \(the charge\-sign flip\): only the factored model passes, and the non\-factored control fails despite its composition success\.decisivefor the three\-way pattern, on both electromagnetic systems\.

Composition succeeds without factoring because a field\-mediated, shared\-per\-particle law is an interpolator on the state manifold: when every marginal \(each particle’s charge, each field amplitude\) stays in\-support, a novel joint configuration is reached by the same local functions evaluated at seen argument values, so locality converts a combinatorially new scene into pointwise\-familiar inputs\. Extrapolation fails without factoring because the flipped charge is a*point off the marginal support*: the black\-box coupling has literally no training signal distinguishing the true odd\-in\-qqcontinuation from any other, and gradient\-trained interpolators resolve that freedom by continuing the function they fit, which is what §[C\.4](https://arxiv.org/html/2610.00280#A3.SS4)measures as rotation lock\-in\. The dissociation is therefore not a mere tally but evidence for the tested design: supplied locality supports composition, and the supplied multiplicative channel carries the intervened value in these experiments\. Other correctly specified parameter maps can also transfer\.

### C\.4Support\-moving transfers: sign, magnitude, rate, gravity

Charge sign \(cavity, grid\)\.Trained atq=\+1q=\+1in an external field, tested atq=−1q=\-1; scored at the seed level with exact binomial statistics\. Cavity: the factored model reverses circulation on10/10seeds \(pooled 318/320 episodes,p=0\.001p=0\.001\); the controls are not at chance but*below*it \(2/10 and 1/10 seeds\)\. Grid: 10/10 \(pooled 315/320\), controls at or below chance\. The cavity result survives fresh independent data draws 15/15 \(pooled 479/480\)\.decisive\.

The factored model reverses by construction:qqmultiplies the coupling, the force is odd inqq, so the flip is a symmetry of its learned action rather than a datum to fit\. The controls’*below*\-chance behaviour is the mechanism made visible: allq=\+1q=\+1training orbits rotate one way in the external field, so the black box learns “rotate that way” as a property of the*dynamics*, not of the charge input; on flipped charge it keeps rotating the trained way and is therefore systematically wrong, not randomly wrong\. \(This lock\-in is configuration\-specific, since it requires a single global rotation sense to exist, and we scope it as such\.\) The dissociation between at\-chance and below\-chance matters later: in the scaling study the same lock\-in signature appears at every capacity \(§[M](https://arxiv.org/html/2610.00280#A13.SSx34)\)\.

What transfers at what fidelity \(grid, cavity\)\.The flip shows a reported0\.8%0\.8\\%energy error across the tested horizons\. Energy is even inqq, so the factored energy functional is flip\-invariant by construction\. Orbit*kinematics*\(speed, kinetic energy, gyration radius\) transfer in\-envelope \(4\.6%4\.6\\%speed error at 25 steps\) and degrade out\-of\-band \(36%36\\%at 200\): the flipped trajectory eventually leaves the band of states the learned content saw\. This is the same envelope limit that magnitude extrapolation meets \(below\),*not*phase decorrelation, and it is system\-dependent: the cavity holds a never\-seen\|q\|=2\|q\|=2to horizon 1000 \(speed error0\.027%→1\.9%0\.027\\%\\to 1\.9\\%; energy error0\.12%0\.12\\%; the trajectory phase decorrelates while every phase\-insensitive observable transfers\), and widening the training range changes nothing \(1\.00×1\.00\\times\), so no benefit from widening is detected in this tested range\. The grid’s\|q\|=2\|q\|=2divergence is therefore grid\-specific \(its plausible mechanism, coupling to grid\-scale modes at higher particle speed, is disclosed as an untested hypothesis\)\.decisivefor the transfers; the envelope is aboundary\.

Proposition[5](https://arxiv.org/html/2610.00280#Thmproposition5)separates a correctly supplied parameter map from content accuracy on the visited states\. Only when both are correct, with the stated regularity assumptions, does its exact\-generator conclusion apply\. Factoring alone does not imply accurate off\-support content, conserved\-energy error, or trajectory fidelity\. The different empirical fidelity patterns are consistent with this distinction but are not predicted quantitatively by the proposition\.

Charge magnitude \(cavity\)\.At never\-seen\|q\|=2\|q\|=2the factored model matches an oracle trained on\|q\|=2\|q\|=2physics to a ratio≈1\.09\\approx 1\.09\(nMSE arm\), while the non\-factored control does not merely degrade: it*reverses its rotation*\(gyration\-frequency ratio−0\.46\-0\.46against the correct\+1\+1\), the unidentified\-off\-support signature again\. A pre\-registered gyration\-frequency pass criterion was*retired*when the oracle itself failed it on 4/5 seeds \(chaotic phase makes the frequency estimator ill\-posed\), a failed pre\-registration of a*metric*, reported as such, with the verdict resting on the surviving arm\.decisive\(nMSE arm\)\.

Dissipation rate \(cavity; fluid\)\.A rate is linear at the generator level \(E˙=−σ​E\\dot\{E\}=\-\\sigma E\), hence factorable even though “friction” feels non\-invertible\. Trained atσ=0\.5\\sigma=0\.5the port transfers to unseenσ=0\.25\\sigma=0\.25with rate error0\.00170\.0017, and the fluid’s viscosity transfersν=0\.5→0\.25\\nu=0\.5\\to 0\.25with rate error2\.9×10−82\.9\\times 10^\{\-8\}\. The learned port scale is1\.0045±0\.00071\.0045\\pm 0\.0007, a\+0\.45%\+0\.45\\%systematic with the truth outside the interval, disclosed rather than rounded away\.decisive\. The common intuition that dissipation cannot be extrapolated conflates the*rate*\(a linear generator coefficient, factorable\) with the*sign*\(bounded by the second\-law guarantee, §[E](https://arxiv.org/html/2610.00280#A5)\)\. The experiment separates them cleanly\.

Gravity \(fluid\)\.Trained atg=1\.0g=1\.0, tested at never\-seeng=1\.5g=1\.5: held\-out error6\.8×10−116\.8\\times 10^\{\-11\}, indistinguishable from in\-distribution \(7\.7×10−117\.7\\times 10^\{\-11\}\), showing agreement to the reported numerical precision; the non\-separable control fits in\-distribution \(3\.2×10−53\.2\\times 10^\{\-5\}\) and fails extrapolation at1\.061\.06,∼1010×\\sim 10^\{10\}\\timesworse\.decisive\.ggmultiplies the potential term12​h2\\tfrac\{1\}\{2\}h^\{2\}in the learned energy, providing the required parameter map in this configuration\. The very small reported errors do not establish exact transfer for general fluid flows or identify smoothness or absence of chaos as their cause\.

### C\.5Implementation\-level counterfactual stress tests

We also evaluate interventions whose correct parameter dependence is not supplied to the tested model\. The reported pre\-registered predictions are that the model’s counterfactual error increases and an oracle control succeeds\. The oracle uses the same class and matched budget but includes or trains on the intervened physics\. Success of this control shows that the configuration can be learned with the additional information; it does not prove that the intervention is unidentifiable for every model or isolate a unique failure cause\.

Cavity: the width enters every field\-gather and current\-deposit inside a Gaussian; theσs=0\.10\\sigma\_\{s\}\\\!=\\\!0\.10\-trained model missesσs=0\.16\\sigma\_\{s\}\\\!=\\\!0\.16physics at111×111\\timesits in\-distribution error while the oracle\-kernel control reaches1\.4×1\.4\\times\(a first oracle run at half the training budget failed its bar; the confound was found and the control re\-run size\-matched, on record\)\. Fluid: trained atγ=2\\gamma=2, refused atγ=1\.7/2\.3\\gamma=1\.7/2\.3with errors0\.75/0\.270\.75/0\.27\(∼109×\\sim\\\!10^\{9\}\\timesin\-distribution\) while theγ\\gamma\-oracle fits each refused law at1\.3×10−6/3\.7×10−71\.3\\times 10^\{\-6\}/3\.7\\times 10^\{\-7\}\. Grid: refused at7\.0×7\.0\\times, real but below the pre\-registered10×10\\timesbar, reporteddirectional, with the milder effect plausibly reflecting the gentler generator perturbation \(a smoothing\-order step rather than a kernel\-width shift\)\.

These interventions change parameter dependence that the tested model does not correctly supply\. The resulting failures delimit this implementation\. They do not establish that nonlinear parameter dependence is intrinsically unidentifiable: known or restricted nonlinear maps can support transfer\.

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/falsifier_suite.png)Figure 4:Refusal tests with oracle controls\. Bars: the factored model’s error at the refused intervention, as a multiple of its in\-distribution error \(log scale\), in the three regimes; the dashed red line is the10×10\\timesrefusal threshold fixed before the run\. The numbers and the matched oracle controls are discussed in §[C\.5](https://arxiv.org/html/2610.00280#A3.SS5)\.An empirical fit\-gap alarm\.A one\-degree\-of\-freedom oscillator test compares a trained quadratic\-plus\-bounded energy parameterization on a quartic truth with a quadratic control\. The one\-step fit error is2\.7×10−32\.7\\times 10^\{\-3\}versus9\.0×10−89\.0\\times 10^\{\-8\}, and the fitted curvature atx=2x=2is−4\.5\-4\.5rather than13\.013\.0\. A bounded correction can reproduce quartic behaviour on compact support\. Consequently, these results establish fitting difficulty for the tested implementation, not global representational impossibility or an effect caused by coercivity\.

Across the reported quartic\-strength sweep, the fit gap grows, and the structured model changes from42×42\\timesbetter than an equal\-capacity generic model atβ=0\\beta=0to213×213\\timesworse atβ=4\\beta=4\. This is an empirical alarm\. Optimization error and differences between nonnested function classes can also produce such a gap\. The single quadratic control does not establish a general false\-positive rate\.

A complementary ambiguity test comparesGGwith\|G\|\|G\|on positive training support\. These maps are observationally equivalent there and can differ under a negative intervention\. The reported procedure returns ambiguity in 40 of 40 such development worlds\. This outcome does not establish universal refusal: the frozen confirmation selector also accepts some out\-of\-family worlds\.

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/g36_bsweep.png)Figure 5:Fit\-gap diagnostics on the quartic test, for a confining\-quadratic energy class trained on a quartic world of strengthβ\\beta\. \(a\) One\-step fit floor relative toβ=0\\beta=0with the10×10\\timesthreshold \(dashed\)\. \(b\) CurvatureV′′​\(2\)V^\{\\prime\\prime\}\(2\)of the reference \(blue\) and of the learned energy \(dashed magenta\)\. \(c\) One\-step fit floor of the structured class \(magenta\) against a capacity\-matched unstructured network \(orange\), with the crossover marked \(dashed vertical line\)\. The three panels are read in the text below\.

## Appendix DWhat long\-horizon stability structurally requires

Long\-horizon stability decomposed, experimentally, into two separately necessary conditions, one about the learned energy’s*shape*and one about the coupling’s*bookkeeping*, plus one residual boundary we characterized to its correlate\.

### D\.1Condition 1: a confining learned energy

Result \(cavity\)\.With an unconstrained learned field energy the model passes fit and counterfactuals but fails the stability test: every seed’s 50,000\-step rollout diverges \(deviations to2\.9×1072\.9\\times 10^\{7\}\)\. Imposing a confining form \(per\-mode quadratics with learned positive stiffness plus bounded learned corrections, on per\-mode\-scaled inputs\) takes the same suite from0/50/5to5/55/5seeds bounded with a flat drift slope, an eight\-order improvement, with no regression in fit or counterfactuals\.decisive\.

A symplectic integrator conserves the*learned*energy to backward\-error order, and that constrains the state only if the energy’s level sets are compact \(and the total energy, including the coupling, is coercive; Prop\.[4](https://arxiv.org/html/2610.00280#Thmproposition4)\)\. A tanh\-MLP energy flattens far from data: out there, level sets open up, and the flow, faithfully conserving a flat function, can drift to infinity at constant learned “energy\.” Probing the failed models confirmed exactly this geometry \(the learned potential saturates where the divergences escape\)\. Confinement restores compact level sets, so excursions must return\. The learned correction only reshapes the landscape*inside*the confined region, where data constrain it\. The scope is stated in Proposition[4](https://arxiv.org/html/2610.00280#Thmproposition4): this is*vacuum\-field*stability, since energies unbounded below \(gravitational or bare\-Coulomb attraction\) are the recorded exception, inherited from physics itself rather than from learning\. One parameterization lesson travelled program\-wide: learned physical coefficients must live in units where their target isO⁡\(1\)O\(1\)\(the raw\-input stiffness spanned0\.50\.5–128128and could not be traversed by training; per\-mode scaling fixed it\), the same lesson later reappearing in the dissipation port’sσ=raw2\\sigma=\\mathrm\{raw\}^\{2\}form \(§[F](https://arxiv.org/html/2610.00280#A6)\)\.

### D\.2Condition 2: energy\-consistent coupling, the mechanism, measured

Result \(cavity, grid, fluid\)\.Confinement alone is*not*sufficient: the non\-factored control, given the identical confining energy, still diverges where its*coupling*is black\-boxed \(4/5 seeds unbounded on the cavity; diverges on the grid\)\. The fluid is the informative exception that makes the rule precise: its non\-separable control black\-boxes only the gravity*entry*while the flux\-form coupling remains energy\-consistent, and it stays bounded \(5/55/5seeds, max\|Δ​E\|/E0=9\.2×10−5\|\\Delta E\|/E\_\{0\}=9\.2\\times 10^\{\-5\}\) yet fails extrapolation at1010×10^\{10\}\\times, exactly the dissociation Propositions[4](https://arxiv.org/html/2610.00280#Thmproposition4)and[5](https://arxiv.org/html/2610.00280#Thmproposition5)jointly predict: stability tracks coupling consistency, counterfactuals track separable entry \(the0/50/5\-bounded fluid model is the unstructured baseline, whose coupling*is*black\-boxed\)\. The mechanism was measured directly: instrumenting the coupling’s two ends \(the force on particles and the current into fields\) shows the non\-factored model injects∼9×\\sim 9\\timesthe net energy per step of the factored model under identical conditions, a phantom power source at the particle–field interface\.decisive, and the measurement corrected our own earlier account \(the original failure had*two*components, not one; on record\)\.

In the true physics, the force on the particle and the particle’s back\-reaction current are two derivatives of*one*interaction energy; their energy exchange cancels identically\. Factoring enforces this by making both ends share one coefficient, so the model*cannot*represent a force without its energetically matched back\-reaction\. A black\-box coupling learns the two ends as independent functions; any mismatch between them is a non\-conservative loop that does net work every cycle, and a long rollout is an integrator of exactly such work\. This is why the failure grows with horizon while fit metrics stay excellent, and why no amount of confinement \(which bounds the*learned*energy, not the injected physical energy\) can rescue it\. For the fluid the same role is played by the flux form: mass conservation to4\.5×10−164\.5\\times 10^\{\-16\}is structural, and functions as the validity meter that charge conservation provides in the electromagnetic systems\.

### D\.3The residual worst\-case energy tail

Result \(cavity\)\.The structured configuration reported throughout is bounded and trend\-free at100×100\\timeshorizon, but rare episodes spike to55–12%12\\%energy deviation \(typical episodes hold11–3%3\\%\), missing a pre\-registered5%5\\%worst\-case bound by∼2×\\sim 2\\times\. We report this as aboundaryand characterized it\. The long\-carried “close\-encounter” attribution was*refuted*by direct test: across 400 episodes the correlation of closest approach with worst\-case deviation isr=\+0\.13r=\+0\.13\(the wrong sign\), and on attracting pairs that genuinely reach zero separation there is still no correlation \(−0\.06\-0\.06\); the strongest actual correlate is*initial kinetic energy*\(r=\+0\.24r=\+0\.24\)\. Both pre\-registered levers fail: multi\-step \(pushforward\) training widens seed variance and costs fit;67×67\\timesclose\-encounter oversampling changes nothing \(0\.1240\.124vs0\.0910\.091natural\)\. The grid closes the same bound outright \(0\.4%0\.4\\%worst case\), so the tail is cavity\-specific content strain, not a defect of the form\.

What the refutation teaches is as useful as a fix: the tail is not a rare\-*event*problem \(which coverage would cure\) but a high\-*energy*problem, since fast episodes traverse the learned content’s least\-constrained region at velocity, so small content errors integrate into transient energy excursions that then relax \(the excursions return to the11–3%3\\%band; the flow remains confined\)\. Multi\-step training fails because its window \(≪\\llthe excursion timescale\) never sees the error it is meant to shrink; oversampling fails because the strained region is reached by*energy*, not by proximity\. The mechanism statement is: the boundary is set by where data constrain the learned content, and its currency is kinetic energy\.

### D\.4Metric discipline for stability claims

Energy\-drift alone can be satisfied trivially \(a model predicting trivial motion has perfect drift\), so every stability claim co\-reports rollout error; boundedness alone hides excursions, so worst\-case per\-seed deviations are reported alongside means; and reported means never silently drop diverged seeds \(survivor\-conditioning is disclosed wherever aggregation could hide a divergence\)\. Intervals retain their originally reported construction; they are not reinterpreted as min–max ranges\. Small\-seed summaries are descriptive and are not used as calibrated evidence without a justified uncertainty analysis\. The counterfactual sign\-flip tests report their exact binomial tests separately; non\-overlap of error bars is not itself a significance test\. These rules exist because each was, at some point in this program, the difference between a true and a flattering number \(Appendix[K](https://arxiv.org/html/2610.00280#A11)\)\.

## Appendix EDissipation, driving, and stochastic consistency

### E\.1Nonnegative damping and its same\-channel representational limit

Result \(cavity\)\.Two ports, identical everywhere except the sign constraint\. The*guarantee*port \(rate≥0\\geq 0by construction\) transfers a never\-seen dissipation magnitude with the reported rate error \(§[C\.4](https://arxiv.org/html/2610.00280#A3.SS4)\) and*refuses*anti\-dissipation: asked to runσ=−0\.5\\sigma=\-0\.5, it clamps at zero \(rate error1\.031\.03\), failing*by construction*, as intended\. The*sign\-free*port represents the anti\-dissipative dynamics faithfully: it tracks the genuinely divergent trajectory \(true energy growth∼22,000×\\sim 22\{,\}000\\timesover 2,000 steps\) to∼1%\\sim 1\\%energy error across 3 seeds\. This is representation fidelity of a divergent process, not a stability claim, and not a second\-law violation by the model of a stable world\.

The sign constraint restricts the same scalar damping channel:σ≥0\\sigma\\geq 0contracts its specified positive quadratic diagnostic, whereasσ<0\\sigma<0expands it away from zero\. This is not an entropy\-production theorem for everything the model can express\. The sign\-free control measures the representational consequence of relaxing that constraint\. External work and thermal noise remain separate sources of subsystem energy\.

### E\.2Decay, terminal state, and explicit driving

Result \(cavity\)\.Trained on dissipative trajectories, the port recovers the medium’s resistivity to0\.4%0\.4\\%\(σ=0\.502\\sigma=0\.502, truth0\.50\.5\), matches the field\-energy decay rate to0\.00380\.0038relative error and the driven steady state \(tail ratio0\.9980\.998; injection tracked at6\.046\.04against a true6\.256\.25, a discrepancy of about3\.4%3\.4\\%\); the entropy\-production diagnostic flags an engineered violator5/55/5and the structured model0/50/5, and itself flags the unstructured baseline\.decisive\. Metric scope, disclosed: the decay\-rate is computed on*field*energy, since kinetic energy is undamped in this medium and would swamp a total\-energy terminal ratio\.

Identification is easy*because*the structure is right: the port is a single scalar multiplying the exact resistive structure, so fittingσ\\sigmais a one\-parameter regression on an exponential, and the architecture converts a behaviour \(“obey the second law at the right rate”\) into an estimation problem the data can actually pin\. The driven control matters because a model could “pass” dissipation tests simply by damping everything; tracking a known antenna’s injection at6\.04/6\.256\.04/6\.25while never injecting through the port shows injection and dissipation live in the right components \(drive vs port\), which is the design rule \(represent the known external input separately from passive damping\) tested here\.

### E\.3The stochastic second law: recovering the correct temperature

Result \(cavity\)\.Under a fluctuating thermal bath \(fluctuation–dissipation\-matched noise on a charge\-0 linear reservoir\), the trained model relaxes from a cold start to the correct stationary temperature:T=0\.502±0\.007T=0\.502\\pm 0\.007against a truth of0\.50\.5\(0\.3%0\.3\\%\),*both*field sectors within0\.4%0\.4\\%, energy\-histogram Wasserstein distance0\.0160\.016\(gate0\.100\.10\), with independently learnedσ=0\.497\\sigma=0\.497\. Controls: the same model in a noise\-doubled \(FDT\-mismatched\) bath equilibrates near2​T2Tand is flagged5/55/5; the correct model is flagged0/50/5; the unstructured baseline reaches a temperature about106×10^\{6\}\\timestoo high \(finite, a gross temperature failure, not a numerical divergence\)\. The reference Langevin system was calibrated first \(6/66/6tests; Gibbs temperature to0\.6%0\.6\\%\)\.decisive\. Scope: one\(T,σ\)\(T,\\sigma\)pair, a charge\-0 linear bath; the reservoir’sTTand its FDT relation are supplied as known physics, and the claim is that the*learned*σ\\sigmaand reversible core are quantitatively consistent with them\.

The magnetic sector is undamped and undriven in this test, so its equilibration probes energy exchange through the learned coupling\. Agreement of sector temperatures and the reported energy histogram is consistent with the supplied bath in this charge\-zero linear setting\. It does not uniquely identify the coupling, prove detailed balance, or establish the invariant law of the full nonlinear simulator\. The noise\-doubled control tests sensitivity to a specified FDT mismatch\.

### E\.4The fluid’s second law: same structure, different physics

Result \(fluid\)\.Viscous decay rate matched to1\.8×10−81\.8\\times 10^\{\-8\}; the no\-port control cannot dissipate at all \(rate error1\.01\.0, so the port is necessary content, not decoration\); wind injection tracked exactly \(factor17\.417\.4, equal to ground truth\); the violator flagged5/55/5; anti\-friction refused by the guarantee \(rate clamps to zero exactly\)\.decisive\. A measurement lesson that generalises: the diagnostic must read the port’s*fluctuation*energy, not total energy, since the constant12​g​h02​L\\tfrac\{1\}\{2\}gh\_\{0\}^\{2\}Lbackground would otherwise swamp the signal, the fluid analog of the kinetic\-vs\-field scoping in §[E\.2](https://arxiv.org/html/2610.00280#A5.SS2)\.

## Appendix FOne model that estimates the physics and selects its form

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/g37_hardened.png)Figure 6:Automated structure identification on the one\-degree\-of\-freedom stress ladder \(development instrument\)\. Bars: fraction of runs on which the selection loop picks the true general form, across a conservative world, dissipative worlds of increasing rate, and driven worlds of increasing amplitude\. Line \(right axis\): one\-step error of the next\-simpler form relative to the true form, with equal error as the dashed line\. The ladder result and the flagship\-cavity margins are given in the text\.The reversible/irreversible split has a consequence that turns architecture into epistemology: “conservative” is not a different model class but theσ=0\\sigma=0member of the dissipative family\. A single model can therefore*decide from data*whether each world it watches dissipates, the smallest possible instance of discovering*which laws apply*\(one bit per system: does the second law bind? plus one number: how fast\)\. This section earns that claim in three escalating forms \(supervised by a condition index, tested by rate\-clamping controls, and label\-free\) and then extends discovery from constants to*functions*\.

Section[3\.2](https://arxiv.org/html/2610.00280#S3.SS2)of the main text states the division of labour component by component: what the architecture supplies as general form, what the data must estimate as specific content, and how each estimated quantity is tested\. The value of the approach rests on that division, since a wrong fit of the estimated content gives wrong numbers under the same imposed form\.

### F\.1Supervised discovery: one checkpoint, four worlds

Result \(cavity; 10 seeds; one checkpoint per seed, scored as\-is\)\.One model \(shared learned energy, coupling, and scales; per\-condition ratesσc=rawc2\\sigma\_\{c\}=\\mathrm\{raw\}\_\{c\}^\{2\}; a 10×\\timesport learning rate; and an*energy\-trajectory auxiliary loss*, using 50\-step windows matching the model’s episode energy series to the data’s; its energy*readout*uses the reference functional, disclosed known\-physics supervision, given to the informed baseline too, and removed entirely in the label\-free variant of §[F](https://arxiv.org/html/2610.00280#A6)\.4, which is the answer to this soft spot\) trained jointly on conservative, dissipative \(σ=0\.5\\sigma=0\.5\), driven, and external\-field episodes, told only*which condition*each episode belongs to, never what anyσ\\sigmais\. It drives the rates apart by over four orders of magnitude, per system, correctly:σcons=1\.2×10−5\\sigma\_\{\\rm cons\}=1\.2\\times 10^\{\-5\}\(implied spurious energy loss over the full 50,000\-step horizon:0\.6%0\.6\\%\);σdiss=0\.481\\sigma\_\{\\rm diss\}=0\.481; the external\-field condition reaches*exact*zero\. On that single set of weights, the four behaviours hold: composition match1\.01\.0; the charge\-sign flip10/1010/10seeds \(p=0\.001p=0\.001\); stability bounded10/1010/10with flat slope \(the5%5\\%worst\-case bound fails on the same characterized tail as the specialist, §[D](https://arxiv.org/html/2610.00280#A4); disclosed, not averaged away\); second\-law decay error3\.6%3\.6\\%\(median\), driven injection6\.336\.33vs true6\.256\.25, violator flagged10/1010/10, model clean\. Controls: without the port the same model cannot fit the dissipative world \(decay error0\.990\.99\); an*informed*equal\-capacity baseline \(given the condition index, drive and field features,*and*the auxiliary loss\) still fails everything \(rollout∼106\\sim 10^\{6\}, second\-law rate error∼7\\sim 7\); and the sign\-free port learns a small*negative*conservative rate on 3/5 seeds \(−1\.6\-1\.6to−3\.2×10−4\-3\.2\\times 10^\{\-4\}\), positing a spurious energy source on ambiguous data, exactly the failure the one\-way port forbids by construction\.decisiveat the behaviour level\. The fluid reproduces the entire result:σinviscid=1\.4×10−4\\sigma\_\{\\rm inviscid\}=1\.4\\times 10^\{\-4\}vsσviscous=0\.500\\sigma\_\{\\rm viscous\}=0\.500, with the sign\-mode control showing no spurious anti\-dissipation\.

Two design decisions carry the identification, and both are general lessons rather than one\-off tricks\.*Signal:*a single step constrainsσ\\sigmaonly through the factor

e−σ​Δ​t≈1−σ​Δ​t,e^\{\-\\sigma\\,\\Delta t\}\\approx 1\-\\sigma\\,\\Delta t,\(8\)a small sensitivity for a short step, while the*episode\-level energy series*carriesσ\\sigmawith signal proportional to horizon; and unlike full\-state multi\-step training \(which fails here, §[D](https://arxiv.org/html/2610.00280#A4)\), the energy is smooth and non\-chaotic, so rollout\-training the*invariant*extracts the long\-horizon signal without the long\-horizon chaos\.*Parameterization:*σ=raw2\\sigma=\\mathrm\{raw\}^\{2\}can reach zero*exactly*\(atraw=0\\mathrm\{raw\}=0\), where a softplus approaches it only asymptotically; the gradientd​σ/d​raw=2​rawd\\sigma/d\\mathrm\{raw\}=2\\,\\mathrm\{raw\}itself vanishes at zero, so the parameterization buys exact representability ofσ=0\\sigma=0, not healthy gradients there, and the training signal for smallσ\\sigmacomes from the episode\-level energy series \(whose sensitivity∂e−σ​t/∂σ=−t​e−σ​t\\partial e^\{\-\\sigma t\}/\\partial\\sigma=\-t\\,e^\{\-\\sigma t\}grows with horizon\), not from a single step, where∂e−σ​Δ​t/∂σ=−Δ​t\\partial e^\{\-\\sigma\\Delta t\}/\\partial\\sigma=\-\\Delta tis small because the step is short\. The sign\-free control then explains*why the guarantee matters precisely here*: on nearly\-conservative data the likelihood surface aroundσ=0\\sigma=0is almost flat, so an unconstrained port wanders to small negative rates, harmless at short horizon but compounding at10510^\{5\}steps\. The one\-way port converts an ill\-conditioned sign inference the data cannot settle into a constraint physics already settles\.

### F\.2Identifiability: the energy cannot absorb the dissipation

Result \(cavity\)\.Clampingσdiss\\sigma\_\{\\rm diss\}at wrong values and letting every shared component retrain to compensate leaves a clean fit\-loss minimum at the truth \(→0\.0120\.25\\\!\\to\\\!0\.012,→0\.0040\.4\\\!\\to\\\!0\.004,→0\.0004\\mathbf\{0\.5\\\!\\to\\\!0\.0004\},→0\.0080\.6\\\!\\to\\\!0\.008,→0\.0150\.75\\\!\\to\\\!0\.015\), with the conservative\-condition energy uncorrupted\. Aλ\\lambda\-sweep exonerates the shrinkage penalty for the unified run’s residual∼4%\\sim 4\\%low bias inσdiss\\sigma\_\{\\rm diss\}\(L1 contributes−0\.08\-0\.08percentage points; atλ=0\\lambda=0,σ=0\.50005\\sigma=0\.50005\): the bias is*protocol\-specific*\(the four\-condition/auxiliary\-loss interaction; mechanism recorded as open\), and a prior attribution in our own record was corrected accordingly\. Fresh\-data replication corroborates: the cheap two\-condition protocol reproducesσdiss=0\.4988\\sigma\_\{\\rm diss\}=0\.4988\(0\.2%0\.2\\%\)\.

The sharp referee question \(“couldn’t the flexible energy quietly absorb the dissipation, making the learnedσ\\sigmadecorative?”\) has a structural answer made empirical: a Hamiltonian flow conserves its own generating energy, but can change a different reference diagnostic\. The clamp experiment shows that the tested shared components did not compensate for the specified wrong rates under this protocol\. It does not establish global uniqueness of the conservative/dissipative decomposition\.

### F\.3Fresh\-data robustness and the precision boundary

Result \(cavity\)\.Regenerating all training data with three fresh generation seeds: the sign\-flip dissociation survives outright \(15/1515/15seeds, pooled479/480479/480\)\. Theσ\\sigma\-identification survives*at the order\-of\-magnitude level*:σcons\\sigma\_\{\\rm cons\}andσdiss\\sigma\_\{\\rm diss\}stay∼3\.7\\sim 3\.7orders apart on every draw, but the strict pre\-registered per\-value gate \(σcons<10−4\\sigma\_\{\\rm cons\}<10^\{\-4\}on every fit\) marginally fails:7/97/9values below, maximum1\.07×10−41\.07\\times 10^\{\-4\}; fresh\-draw inference is∼7×\\sim 7\\timeslooser than the shared\-data run, and the worst seed’s implied 50,000\-step spurious loss is5\.2%5\.2\\%, marginally over the5%5\\%bound\. We keep the gate failure as aboundary\(an early draft that relabeled it was itself corrected; the recorded verdict was right, Appendix[K](https://arxiv.org/html/2610.00280#A11)\)\.

Near\-zero rates show greater relative estimation sensitivity in these runs\. Boundary constraints, parameterization, finite\-window signal, and optimization can all contribute, and the experiment does not isolate them\. Absolute errors and their long\-horizon consequences are therefore reported alongside relative rate accuracy\.

### F\.4Label\-free inference of the medium from observation

Result \(cavity\)\.An encoder reads the first 100 steps and estimatesσ^\\hat\{\\sigma\}using a frozen core’s short\-rollout prediction loss, without rate labels\. In the reported five\-seed experiment, every held\-out conservative estimate is below every dissipative estimate\. A sliding window tracks a medium switch from00to0\.50\.5with estimates0\.0780\.078and0\.4570\.457\. Intermediate\-rate correlation is0\.900\.90against the declared0\.950\.95gate, with slope0\.7750\.775\. The conservative estimate is approximately0\.070\.07, and the corresponding rollout error is0\.600\.60, versus0\.0050\.005in the reference\-rate comparison\. Short and shuffled windows weaken the separation, although a permutation\-invariant cue remains in one seed\.

These results demonstrate regime discrimination with residual estimator bias, not an information\-theoretic limit\. Direct fitting on the same windows achieves smaller errors, as reported below\. Freezing the core separates context estimation from content adaptation\. Accuracy at two training rates does not establish accuracy on every intermediate trajectory or state region\.

### F\.5Discovering functions, not just constants

Result \(grid; fluid\)\.On the grid, within the declared confining functional parameterization and supplied operators, the learned constitutive derivatives come out*linear*\(r2=1\.0000r^\{2\}=1\.0000\) with slopesfE′=1\.043f\_\{E\}^\{\\prime\}=1\.043,fB′=1\.001f\_\{B\}^\{\\prime\}=1\.001: the model discovers that vacuum electromagnetism’s energy is quadratic, and reports their slope ratio,rslope=aB/aE=0\.959±0\.017r\_\{\\rm slope\}=a\_\{B\}/a\_\{E\}=0\.959\\pm 0\.017\(reference ratio11\)\. Its interpretation as squared wave speed depends on the field\-variable and bracket normalization, which is not derived in this report\. We therefore retain it as a constitutive\-slope diagnostic rather than an independent wave\-speed measurement\. The approximately4%4\\%E\-side discrepancy is reported with its direction\. Smoothing and discretization are plausible contributors, but their causal contribution is not established here\. The slope\-regression estimator was pre\-registered \(a pointwise\-ratio probe bug, which diverges near zero field, is on record with its correction\)\. In the fluid, the learned mass\-metric and potential\-derivative are linear with slopes1\.0001\.000\(R2=1\.000R^\{2\}=1\.000\); since gravity is supplied, this yields an effective coefficient1\.000±5×10−61\.000\\pm 5\\times 10^\{\-6\}per unit supplied gravity, not an independent gravity measurement\.decisive, with the scope stated where it belongs: the reported estimates agree with these tested truths within the supplied class\. Class membership alone does not ensure identifiability or successful optimization\.

Why can an energy*functional*be recovered from one\-step data at all, and modulo what? With the update rule pinned to the curl\-form stencils, observed field increments constrain the operator imageL∇EL\\nabla E, not∇E\\nabla Eitself: any admissibleCCwithL∇C=0L\\nabla C=0\(Casimirs of the fixed bracket, additive constants, and any direction the curl operator annihilates\) leaves the dynamics unchanged, soEEis identifiable only up to that equivalence class, and only on the states the data visit\. We distinguish operator\-null directions from observable quantities: a product diagnostic removes one scaling ambiguity, and spatial derivatives remove additive force constants\. These operations do not establish a unique underlying gauge\. The gauge test below exhibits distinct admissible functions with identical trajectories, which is why the nonlinear\-recovery errors are reported gauge\-aligned \(mean difference removed\) alongside the raw values\. The amplitude sweep broadens the tested operating support, but does not establish full rank of the observation\-sensitivity operator\. Recovery concerns the operator\-visible content on the sampled states\. A highR2R^\{2\}against known functions is empirical agreement, not an identifiability proof\.

Gauge test \(distinct admissible functions, identical trajectories\)\.We checked the equivalence class directly on the oracle\-content fluid model\. Shifting the potential force by a constant,V′​\(h\)→V′​\(h\)\+cV^\{\\prime\}\(h\)\\to V^\{\\prime\}\(h\)\+c\(equivalentlyE→E\+c⋅massE\\to E\+c\\cdot\\mathrm\{mass\}, a Casimir direction of the fixed bracket\), leaves every trajectory unchanged to machine precision: over1616episodes and200200steps the rollout nMSE between the shifted and unshifted models is10−2710^\{\-27\}forc∈\{0\.5,−0\.3,2\.0\}c\\in\\\{0\.5,\-0\.3,2\.0\\\}at both the standard \(p=2p=2\) and polytropic \(p=3p=3\) truths\. Shifting the mass density instead,M⁡\(h\)→M⁡\(h\)\+cM\(h\)\\to M\(h\)\+c, changes the flux and the trajectories \(nMSE0\.390\.39to5\.85\.8over the same shifts\)\. TheV′V^\{\\prime\}intercept is therefore a genuine non\-identifiable direction, which is why only thehh\-dependence \(slope and shape\) ofV′V^\{\\prime\}is the intended recovery target\. The tested constant shifts ofMMare distinguishable; this does not establish uniqueness over all admissible functions or exclude compensating changes in other learned components\.

Nonlinear recovery, downstream prediction, and the linear comparator\.The fluid experiment uses a polytropic truthp=3p=3, withV′​\(h\)=h2V^\{\\prime\}\(h\)=h^\{2\}andM⁡\(h\)=hM\(h\)=h, at excitation amplitudesa∈\{0\.03,0\.05,0\.10\}a\\in\\\{0\.03,0\.05,0\.10\\\}, step sizes0\.010\.01and0\.0050\.005, and three seeds per cell, giving1818cells in all\. A learned\-content arm is compared with an oracle\-content arm \(same class and integrator, exact reference functions\) and, atΔ​t=0\.01\\Delta t=0\.01and every amplitude, with an explicitly linear constitutive comparator

M⁡\(h\)=softplus⁡\(α\)​h,V′​\(h\)=softplus⁡\(c\)​h,M\(h\)=\\mathrm\{softplus\}\(\\alpha\)\\,h,\\qquad V^\{\\prime\}\(h\)=\\mathrm\{softplus\}\(c\)\\,h,\(9\)trained on identical data \(the tanh corrections zeroed and frozen; its two parameters need an Adam step of10−210^\{\-2\}to converge within the same3030epochs, after which its loss is flat from epoch2020\)\. Function\-level comparisons use the uniform grid over\[1−3​a,1\+3​a\]\[1\-3a,\\,1\+3a\]\. Because the additive constant ofV′V^\{\\prime\}is a Casimir direction \(V′\+cV^\{\\prime\}\+cleaves every trajectory unchanged to10−2710^\{\-27\}\), the gauge\-aligned error removes the mean difference before comparing, and the operator image is the exact derivatived​V′/d​hdV^\{\\prime\}/dhthat the momentum equation sees\.

Table 4:Nonlinear fluid truthV′=h2V^\{\\prime\}=h^\{2\}atΔ​t=0\.01\\Delta t=0\.01, means over three seeds \(seed ranges within±0\.01\\pm 0\.01for function errors and±15%\\pm 15\\%for prediction errors\)\. Gauge\-aligned and operator\-image errors are relative RMSE on the uniform operating range; “best linear” is the best linear approximant of the truth on that range\. Prediction errors are nMSE on held\-out episodes: the rollout at the training gravity and theg=1\.5g=1\.5intervention\.Across the1818cells the raw relative RMSE of the learnedV′V^\{\\prime\}against the truth is0\.0410\.041/0\.0530\.053/0\.0640\.064atΔ​t=0\.01\\Delta t=0\.01and0\.0440\.044/0\.0580\.058/0\.0740\.074atΔ​t=0\.005\\Delta t=0\.005for amplitudes0\.030\.03/0\.050\.05/0\.100\.10\(seed ranges within±0\.005\\pm 0\.005\); most of it is the unidentifiable constant \(gauge\-aligned0\.0040\.004/0\.0080\.008/0\.0250\.025; the linear comparator learnsV′=2\.00​hV^\{\\prime\}=2\.00hwith zero intercept, so its raw error of0\.90\.9–1\.01\.0is entirely that constant\)\. The fitted quadratic coefficient is0\.780\.78–1\.081\.08\(truth11\); learnedMMis within0\.0160\.016–0\.0420\.042, whereas the linear comparator, whose class containsM=hM=h, recovers it to10−410^\{\-4\}; the oracle\-content arm reproduces the reference to10−1510^\{\-15\}\. Every cell is bounded, and the learned arm’s long\-horizon energy deviation equals the oracle’s and the linear comparator’s to two digits \(0\.00080\.0008/0\.00210\.0021/0\.00800\.0080\), so stability is not where constitutive content shows\. It shows in prediction: the converged linear comparator’s one\-step loss is6×6\\times/8×8\\times/5×5\\timesthe nonlinear model’s ata=0\.03a=0\.03/0\.050\.05/0\.100\.10\(8\.1×10−48\.1\\times 10^\{\-4\}against1\.0×10−41\.0\\times 10^\{\-4\}ata=0\.05a=0\.05\), and its rollout and intervention errors are1414–17×17\\timeslarger ata=0\.03a=0\.03,13×13\\timesata=0\.05a=0\.05, and2\.42\.4–2\.6×2\.6\\timesata=0\.10a=0\.10\. Function\-level error on the uniform range does not separate the classes: a linear law approximatesh2h^\{2\}to within0\.24%0\.24\\%/0\.7%0\.7\\%/2\.5%2\.5\\%there, the nonlinear model’s gauge\-aligned error is no lower than that floor, and its derivative error is larger \(0\.130\.13–0\.240\.24against0\.050\.05–0\.170\.17; the linear slope estimator likewise returnsR2≥0\.993R^\{2\}\\geq 0\.993on the quadratic truth, so slope agreement alone is not a nonlinearity test\)\. The nonlinear content that the data constrain is therefore established at the level of curvature and of prediction on the sampled distribution, not at the level of uniform\-range function error\. The step\-size check separates numerics from content: atΔ​t=0\.005\\Delta t=0\.005the intervention error is3\.6×10−53\.6\\times 10^\{\-5\}/1\.9×10−41\.9\\times 10^\{\-4\}/3\.3×10−33\.3\\times 10^\{\-3\}against1\.1×10−41\.1\\times 10^\{\-4\}/4\.5×10−44\.5\\times 10^\{\-4\}/1\.8×10−21\.8\\times 10^\{\-2\}atΔ​t=0\.01\\Delta t=0\.01, three\- to five\-fold lower at essentially unchanged constitutive error, a numerical contribution that must be held fixed in any constitutive\-error\-to\-prediction comparison\.

Observation\-window inference: encoder versus a direct per\-window fit\.On the identical test windows and the identical frozen core, a direct per\-window fit ofσ\\sigma\(grid then local refinement over the window prediction error\) reaches a conservative\-rate floor of0\.0090\.009\(windowK=25K=25\),0\.0110\.011\(K=50K=50\), and0\.0070\.007\(K=100K=100\) on clean windows, rising to0\.0290\.029–0\.0390\.039at2%2\\%observation noise, with rate correlation0\.900\.90–0\.980\.98across the trained grid; the amortized encoder on the same inputs floors at0\.0510\.051/0\.0580\.058/0\.0700\.070with correlation0\.540\.54/0\.760\.76/0\.900\.90, and is brittle off its training window length \(absolute error atσ=0\.5\\sigma=0\.5of0\.400\.40atK=25K=25against0\.020\.02for the direct fit\)\. The0\.070\.07floor is therefore predominantly amortized\-encoder error in this comparison\. Direct fitting achieves errors near0\.010\.01clean and0\.030\.03–0\.040\.04at2%2\\%noise; these are estimator outcomes, not information\-theoretic lower bounds\. The grid’s approximately4%4\\%slope\-ratio discrepancy is a separate systematic recovery error\. Smoothing and discretization are plausible contributors; the reported agreement does not uniquely establish their causal contribution\.

### F\.6Discovery rung 2: selecting the general form

The discoveries above identify content*within*a fixed general form; selecting*among*forms begins to answer who designs the structure\. In the lumped one\-degree\-of\-freedom harness \(a Hamiltonian core, a non\-negative dissipation port, and an optional explicit drive; learned inertia, stiffness, andσ\\sigma\) we score the candidates\{H,H\+port,H\+port\+drive\}\\\{H,\\ H\{\+\}\\text\{port\},\\ H\{\+\}\\text\{port\}\{\+\}\\text\{drive\}\\\}by a pre\-specified selection statistic, the Bayesian information criterion on the one\-step fit,BIC=N​ln⁡MSE\+p​ln⁡N\\mathrm\{BIC\}=N\\ln\\mathrm\{MSE\}\+p\\ln N\. Under a hardened protocol \(eight worlds spanning aσ→0\\sigma\\\!\\to\\\!0dissipation ladder and adrive→0\\text\{drive\}\\\!\\to\\\!0ladder, five seeds each, with1%1\\%observation noise\) the minimum\-BIC candidate is the true form on all4040runs \(conservative→H\\,\\to H; dissipative→H\+\\,\\to H\{\+\}port; driven→H\+\\,\\to H\{\+\}port\+\{\+\}drive\), a*non\-nested*wrong form \(dissipation on the wrong variable\) is never selected, and the correct form’s one\-step error separates from the next\-simpler candidate by1\.11\.1to3\.1×3\.1\\times, the separation shrinking toward the conservative boundary, where atσ≲0\.05\\sigma\\\!\\lesssim\\\!0\.05selection leans on the complexity penalty rather than the fit, a characterized near\-degenerate boundary\. The general model’s learned diagnostics agree \(σ^=0\.305\\hat\{\\sigma\}=0\.305against a true0\.300\.30; the drive term switches on only when driven\)\. Scored this way, the port\-necessity and injection ablations the program already runs*are*a model\-selection statistic\. The same harness recovers the lumped constants to11–3%3\\%and performs label\-free dissipation discovery \(σ^=0\.002\\hat\{\\sigma\}=0\.002conservative versus0\.2520\.252dissipative\), the capability of this section carried into the one\-DoF form\. Scope, in the same spirit as the discovery results above: the worlds are synthetic with the truth inside the candidate set, so the claim is that the selection statistic recovers the correct form*when it is present*, the selection analog of the “selects the correct member” claim of §[F\.1](https://arxiv.org/html/2610.00280#A6.SS1);σ\\sigmais weakly identified under a strong drive and near the conservative boundary, disclosed \(App\.[M](https://arxiv.org/html/2610.00280#A13)\)\. See also the lift to the flagship cavity below \(Fig\.[7](https://arxiv.org/html/2610.00280#A6.F7)\)\.

Lifted to the flagship\.The same pre\-specified statistic, applied to the paper’s central system, the1616\-mode, two\-particle EM cavity of §[B](https://arxiv.org/html/2610.00280#A2), with the8\.98\.9k\-parameter energy network*shared*across all three candidate forms so only the structural hypothesis \(does it dissipate? is it driven?\) is penalized, and, per this program’s standing rule, escalated in seeds before any headline: at three seeds it selected the true form in all three worlds;*the five\-seed escalation softened that*\(Fig\.[7](https://arxiv.org/html/2610.00280#A6.F7)\)\. Wherever there is dissipative or driven structure to find, selection is decisive and seed\-stable: the dissipative world picksH\+H\{\+\}port byΔ\\DeltaBIC22572257and the driven worldH\+H\{\+\}port\+\{\+\}drive by10541054, with the learned resistivity recovering physics it was not told \(σ^=0\.502\\hat\{\\sigma\}=0\.502dissipative,0\.5020\.502driven; true0\.50\.5\)\. On the*conservative*world the five\-seed verdict flips toH\+H\{\+\}port by2929BIC atσ^=0\.0126\\hat\{\\sigma\}=0\.0126, a fit difference of1%1\\%of wMSE, swung by a single seed\. Stating the degeneracy’s scope precisely matters, because it is a*fit\-horizon*artifact, not an equivalence of physics: over the500500\-step evaluation windowσ^=0\.0126\\hat\{\\sigma\}=0\.0126sheds only6%6\\%of the field energy \(the two candidates are indistinguishable*there*\), but the cavity port is palindromic \(field energy decays ase−2​σ^​te^\{\-2\\hat\{\\sigma\}t\}\), so over the5050k\-step stability horizon this paper’s stability claims live at, that sameσ^\\hat\{\\sigma\}dissipates99\.8%99\.8\\%of it\. A rolled\-outH\+H\{\+\}port selection would be drastically wrong physics at long horizon\. The five\-seed verdict is2/32/3, superseding the three\-seed3/33/3\(Appendix[K](https://arxiv.org/html/2610.00280#A11)\); the surviving claim is that the flagship selector finds*dissipation and drive*decisively and recovers their rates, while on the conservative world the residual learned rateσ^cons\\hat\{\\sigma\}\_\{\\rm cons\}is an artefact of the one\-step objective whose long\-horizon consequences make removing it necessary rather than cosmetic\. Adding the energy\-trajectory auxiliary loss alone did not remove it \(σ^cons\\hat\{\\sigma\}\_\{\\rm cons\}moved only1\.2×1\.2\\timesagainst a10×10\\timesbar, and the incorrect selection widened\), so the residual is an identifiability boundary of the objective\. AnL1L\_\{1\}parsimony prior in the unified model restores the correct selection \(3/33/3, with the conservative formHHwinning that world by280280BIC and an independent profile\-likelihood tie\-break agreeing\) and leaves the identifiable rate unbiased \(σ^diss=0\.502\\hat\{\\sigma\}\_\{\\rm diss\}=0\.502\), but the residual conservative rate is unchanged at the protocol training budget \(σ^cons=0\.0126\\hat\{\\sigma\}\_\{\\rm cons\}=0\.0126\), still failing the stricter eval\-horizon gate\. Longer training then removes it: the residual is a finite\-training transient that shrinks by about11\.4×11\.4\\timesper decade of epochs, and a pre\-registered long\-training run crossed the10−410^\{\-4\}eval\-horizon bound at8\.08\.0–8\.58\.5k epochs and reachedσ^cons=5×10−6\\hat\{\\sigma\}\_\{\\rm cons\}=5\\times 10^\{\-6\}by27,50027\{,\}500epochs, below even the unified model’s1\.27×10−51\.27\\times 10^\{\-5\}\. Plain training at sufficient budget therefore closes the question with no special mechanism, and the decay accelerates \(≈t−2\.5\\approx t^\{\-2\.5\}\) while a static\-regression estimate of any true residual stays at zero throughout, so the transient falls to meet a floor that never rises\. At the selection protocol’s budget the correct selection is carried by theL1L\_\{1\}prior and the tie\-break; nothing about the conservative world remains open \(App\.[M](https://arxiv.org/html/2610.00280#A13)\)\.

### F\.7The selection loop and its fit\-gap alarm

The selector compares a finite menu of supplied forms\. The fit\-gap alarm \(§[C\.5](https://arxiv.org/html/2610.00280#A3.SS5)\) compares the best structured fit against a capacity\-matched unstructured reference under the stated fitting protocol\. In the reported perturbation sweep the ordering reverses atβ≥0\.25\\beta\\geq 0\.25, while smaller perturbations lie in the alarm’s observed blind region\. This is evidence for an empirical screening signal, not a test that certifies whether the true dynamics belong to the menu\.

The proposed workflow is to fit the candidates and reference, inspect the fit gap, and then select or investigate further\. An alarm can reflect optimization failure, measurement error, an inadequate reference comparison, or candidate misspecification\. Its absence does not establish class correctness\. Candidate selection can therefore be wrong, including accepting an out\-of\-family world\. Expanding or repairing the menu is a subsequent scientific decision, not an automatic consequence of a correct diagnosis\.

The reported ingredients include40/4040/40selections on the11\-DoF stress ladder and strong BIC separation for dissipative and driven cavity cases \(Δ\\DeltaBIC≥1054\\geq 1054, five seeds\)\. They also include a conservative near\-degeneracy: the five\-seedHH\-versus\-H\+H\{\+\}port tie\-break changes even though the small port affects evaluation\-horizon behaviour\. These experiments use synthetic worlds and small menus\. Automatic proposal of new forms is not implemented; neither BIC ranking nor the separate marginal conformal retention guarantee supplies a universal correct\-form\-or\-refusal outcome\.

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/g37_cavity.png)Figure 7:Discovery on the flagship electromagnetic cavity: the same selection statistic lifted from the one\-degree\-of\-freedom surrogate to the1616\-mode two\-particle cavity\. \(a\)Δ\\DeltaBIC of each candidate form \(HH,H\+H\{\+\}port,H\+H\{\+\}port\+\{\+\}drive\) relative to the selected form, which hasΔ\\DeltaBIC=0=0and so no bar\. \(b\) True dissipation rate \(blue\) against the rate learned by the selected form \(magenta\)\. The margins, and the near\-degenerate conservative tie\-break, are discussed in the text\.

## Appendix GGenerality, noise, and the strongest competitor

### G\.1The same program in a different physics

The fluid block \(§[B](https://arxiv.org/html/2610.00280#A2)\) is not an extra benchmark but the generality experiment: mass replaces charge as the structural invariant, a scalar height field replaces vector fields, and gravity and viscosity replace coupling and resistivity, while every pillar reproduces at the same∼\\sim8\.7k scale: fit7\.8×10−117\.8\\times 10^\{\-11\}with the baseline failing in\-distribution at long horizon \(0/50/5bounded at 10k steps, median\-vs\-mean disclosure on record\); gravity counterfactual transfer \(at the supplied value\) with the non\-separable control failing at1010×10^\{10\}\\times; machine\-precision mass conservation; the complete second\-law suite; and the certified refusal test\.decisive\. The transfer of the*whole program*, including its refusal test, is what licenses the word “general” in our title: the behaviours attach to the imposed form \(conservative flow \+ one\-way port \+ known drive \+ factored constants\), not to electromagnetism, and the criterion’s predictions \(which parameters transfer, which are refused\) were made in advance from the fluid’s generator alone\.

## Appendix HSynthesis: which structure buys which behaviour, and why

The findings admit a compact mechanistic summary because the program was built to dissociate: each behaviour was tested against controls differing in exactly one structural element\. Table[5](https://arxiv.org/html/2610.00280#A8.T5)states the resulting map; the paragraphs below give the*why*in five sentences each, with the evidence rows that pin it\.

Table 5:The structure\-to\-behaviour map, with the dissociating evidence\. Entries summarize the evaluated controls and their scope; they do not establish universal necessity of each architectural component\.### H\.1Interpretation and limits of the map

Locality and shared functions support composition in the evaluated models\. Supplied parameter maps prescribe responses outside the parameter training support, provided the learned content remains accurate on the visited states\. Neither observation establishes that a generic learned model can never acquire a useful extrapolation prior\.

Architectural constraints restrict representability, but their guarantees must be stated for the quantities they actually control\. The scalar port dissipates its quadratic diagnostic; the full learned energy and stochastic stationary behaviour are separate empirical outcomes\. The limits also have different causes\. Gauge ambiguity is a property of the observation operator, while an encoder’s residual rate error may reflect estimation or optimization\. A finite\-horizon energy tail is not explained solely by a correlation with initial energy\. The resulting practical lesson is to evaluate the supplied representation, recovered quantities, numerical update, and downstream task separately\.

## Appendix IThe stated exceptions, tested, and a suite of everyday physics

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/zoo_x1_ladder.png)Figure 8:The bare\-gravity softening ladder\. \(a\) Relative energy drift \(blue circles\) and angular\-momentum drift \(light\-blue squares\) of the reference integrator against the softeningε\\varepsilon, with the energy and angular\-momentum gates \(dashed and dotted\) and red crosses where the reference is non\-finite\. \(b\) Fraction of episodes in which a body escapes, at each rung\. The reading, that the ladder measures the reference’s own numerical reliability, is in the text\.![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/zoo_x2_port_profile.png)Figure 9:Shock dissipation as a learned, state\-dependent, localised port on the dam\-break fluid at three times \(a\)–\(c\)\. Blue \(left axis\): the coarse height profilehh\. Dashed magenta \(right axis\): the learned port dissipation, normalised per panel\. Yellow: the reference entropy mask, which the model never sees\. The text describes how the learned dissipation tracks the moving shock front\.Every scope limit stated earlier in this paper is tested here: bare gravity was*excluded*by the confinement hypothesis of Proposition[4](https://arxiv.org/html/2610.00280#Thmproposition4), contact by smoothness, shocks by the smooth\-regime fluid scope, and interacting baths by the earlier linear\-bath disclaimer\. This section pays those notes\. We test all four carried exceptions plus an eleven\-system suite of everyday classical physics, chosen to*tile*a declared taxonomy \(bracket×\\timesport×\\timessmoothness×\\timesstate×\\timesdrive\), not sampled, so the aggregate claim is coverage of the tiling, and the taxonomy was committed before any system was built\. Each system gets the full pass card of its type \(conservative, dissipative, or chaotic, the three pre\-registered variants with the pre\-registered constants\), a certified baseline \(LR sweep, rollout\-stabilizing noise\), a discovery gate, and where declared, a refusal\-test world\. Table[7](https://arxiv.org/html/2610.00280#A9.T7)is the ledger; verdicts are printed by the harness, and the pre\-registration \(with its numbered amendments, all committed*before*the affected runs\) is in the record appendix\.

### I\.1The pass\-card ledger: eleven systems, twelve cards, three card types

Result\.All eleven common\-life systems earn their pass card \(twelve cards with the pendulum’s double\-pendulum arm\), eight clean and four with exactly one carried boundary \(the pre\-registered pass rule\), every boundary with a measured mechanism\. Denominators used below: the eleven suite systems plus the four exceptions are the fifteen registry worlds of the pre\-registered≥14/15\\geq 14/15selection gate; the selection matrix has sixteen system rows because the double\-pendulum card is scored as its own row, plus two out\-of\-menu controls\. The conservative cards hold their energy drift below the 5% worst\-case gate at100×100\\timeshorizons \(2\.16%2\.16\\%pendulum;1\.4×10−71\.4\\times 10^\{\-7\}spinning top, whose Casimir is machine\-exact*by construction*\); the dissipative cards recover their rates and long\-run states \(RLC port rate0\.29990\.2999vs0\.3000\.300; frictionμ^​N\\hat\{\\mu\}Nto0\.012%0\.012\\%; motor power balance3\.1×10−173\.1\\times 10^\{\-17\}, an architectural identity, not a fit\); the chaotic cards are graded distributionally and pass climate 3/3 and counterfactual\-climate 10/10 seeds while carrying the same boundary twice \(the double pendulum, Lorenz\-63\): the structured model outperforms the certified baseline at the sub\-Lyapunov horizon by only13×13\\times\(the double pendulum\) or loses to it \(0\.38×0\.38\\times, Lorenz\-63\) while reproducing the climate correctly, so the split buys long\-horizon accuracy, not short\-horizon accuracy\.

The suite exists to answer whether the program’s approach holds up on ordinary physics, not only on the cavity it was developed on\. The answer has a clear shape: structure helps precisely on the axes the theory says it must \(invariants, long\-run states, rates, transfer\) and is weaker where it promises nothing \(short\-horizon trajectory error on chaotic systems\)\.

### I\.2Refusal tests at scale: the refusal quota, and what a refusal instrument must be

Result\.Eleven refusal/guard entries pass across six non\-separability types \(the harness tally in Table[7](https://arxiv.org/html/2610.00280#A9.T7); the pre\-registered quota was six across four\): geometry \(a pivot\-geometry world hidden in the*kinetic*term refuses the fixed\-kinetic class at1\.5×106×1\.5\\times 10^\{6\}\\timesits own base floor\), exponent \(drag law,1\.1×104×1\.1\\times 10^\{4\}\\times\), state\-dependence \(nonlinear resistance; conductivityκ⁡\(u\)\\kappa\(u\)at1\.6×107×1\.6\\times 10^\{7\}\\times; diffusivityD⁡\(u\)D\(u\)\), material law \(static\-vs\-kinetic friction\), separability itself \(an FPU quartic trained*across*two stiffness settings, absorbable at any single setting, refused by the factorization\), and locality \(a fourth\-order stiffness operator vs a nearest\-neighbor class\)\. The instrument was refined through pre\-registered amendments: refusal is*within\-class*\(the class’s floor on the refusal test world over its floor on its own base world\), and the learnability certificate is either a learned right\-structured oracle separating by≥10×\\geq 10\\timeson the same world, or a hand\-set truth member of the oracle class at reference\-competitive floor, because cross\-class common\-base ratios conflate integrator truncation with class membership \(measured: a learned Tao oracle read2×105×2\\times 10^\{5\}\\timesagainst a common base while its hand\-set truth floor sat at0\.0031×0\.0031\\times\)\.Disclosure:the refusal instrument was redesigned mid\-block for this measured reason; under the redesign*two verdicts flipped*FAIL→\\toPASS \(the pendulum’s geometry and the spring lattice’s separability refusal tests, both via truth anchors\), and the same redesign initially*failed*the heat rod’s refusal test, forcing the bond\-form oracle fix, so the redesign moved verdicts in both directions\. Both pre\- and post\-redesign results are in the record; the flips changed the quota tally, not any pass\-card count\.

Three pitfalls we encountered and worked around, kept in the record because any replication will meet them: \(i\)*oracle classes must provably contain their refusal test world*\(a pointwise mobilityM\(u\)∇EM\(u\)\\nabla Edoes not contain a flux\-form∂x\(κ⁡\(u\)​∂xu\)\\partial\_\{x\}\(\\kappa\(u\)\\partial\_\{x\}u\)world, since they differ atO⁡\(∂κ​∂u\)O\(\\partial\\kappa\\,\\partial u\), and a two\-cell pair stencil does not contain the biharmonic’s boundary rows, while curvature bonds contain it exactly\); \(ii\)*refusal\-test worlds can be degenerate*\(our first static\-friction world hadμs\\mu\_\{s\}exactly at the drive amplitude, so every trajectory eventually stuck forever and class and oracle floors collapsed together\); and \(iii\)*optimizer adequacy is part of a floor instrument*\(refusal test floors compare classes, so they run under a certified fitting procedure,2×2\\timesepochs, cosine anneal, motivated by a measured three\-decade gap between a hand\-set truth floor and a fixed\-LR fit\)\.

### I\.3The four exceptions

Contact \(closed\)\.The guard\-first protocol is the point: the smooth class must*detectably refuse*the bouncing world against its own soft\-wall base before the augmented class is credited, and it does, after an instructive false start \(free fall is a non\-stationary base world that saturates any bounded\-input network; the amendment is in the record\)\. The complementarity class then recovers contact to machine precision:g^=1−3×10−14\\hat\{g\}=1\-3\\times 10^\{\-14\},e^=0\.8−3×10−12\\hat\{e\}=0\.8\-3\\times 10^\{\-12\}, per\-bounce energy ratioe^2\\hat\{e\}^\{2\}exact, settling time exact, and the augmented class’s own restitution is clampede^<1\\hat\{e\}<1*architecturally*, theσ≥0\\sigma\\geq 0lesson at contact, so the energy\-gaining violator lives in the world and is flagged by the meter, not representable by the model\.

The interacting thermal bath\.The earlier linear\-bath scope lifts: with soft\-Coulomb interactions*on*, the drift is fitted and the noise is*measured*\(the measure\-the\-noise discipline\), givingγ^\\hat\{\\gamma\},T^=s^2/2​γ^\\hat\{T\}=\\hat\{s\}^\{2\}/2\\hat\{\\gamma\}, and a two\-sector equipartition meter that stays quiet on the FDT bath and flags a sector\-asymmetric violator at66%66\\%mismatch\. The magnetic\-sector mechanism check survives interactions: a workless force must be implemented worklessly \(a discrete force\-kickB⁡\(p×z^\)B\(p\\times\\hat\{z\}\)heats at\(h​B\)2​\|p\|2\(hB\)^\{2\}\|p\|^\{2\}per kick, so reference and model both carry Boris rotations\), andB^\\hat\{B\}is recovered with equipartition preserved\. \(Two thermostat\-grade integrator upgrades, BAOAB and Boris, were forced by the system’s*own*calibration gates, which is what calibration\-first is for\.\)

Bare gravity: a numerical\-reliability ladder\.The reference is evaluated at softening valuesε∈\{0\.3,0\.1,0\.03,0\.01,0\}\\varepsilon\\in\\\{0\.3,0\.1,0\.03,0\.01,0\\\}with an adaptive reference integrator\. Onlyε=0\.3\\varepsilon=0\.3meets the stated energy gate, with drift9\.0×10−59\.0\\times 10^\{\-5\}against10−410^\{\-4\}\. At0\.10\.1the drift is4\.2×10−44\.2\\times 10^\{\-4\}, at0\.030\.03it is6\.7%6\.7\\%, and at0\.010\.01the numerical integration becomes non\-finite\. The original escape diagnostic reports100%100\\%there; this cannot be interpreted as verified physical escape\. Reported angular\-momentum residuals near10−1410^\{\-14\}do not restore energy accuracy or validate a non\-finite trajectory\.

At the accepted reference setting, the fitted pair function hasr2=0\.9999999r^\{2\}=0\.9999999, the reported baseline contrast is1\.3×1061\.3\\times 10^\{6\}, and theG→2​GG\\to 2Gerror is0\.0070\.007of the reported oracle error\. The worst\-case energy deviation is nevertheless813%813\\%\. These summaries describe a limited numerical test, not a transition in physical confinement\. Softening removes the collision singularity but does not make attractive gravity globally coercive\. Symplecticity of an adaptive update requires a specified construction; it is not established by the integrator label in this record\.

Shocks and the localized port\.The hardest case is shock dissipation, which is not a constant rate but a*state\-dependent, spatially localized*one\. Against a certified Rusanov reference \(mass4×10−164\\times 10^\{\-16\}; fine\-grid energy strictly decreasing; Stoker front to0\.8%0\.8\\%\), coarse\-grained dam breaks are fit by one flux family whose three variants differ only in the port: none, constant, orλθ​\(local features\)≥0\\lambda\_\{\\theta\}\(\\text\{local features\}\)\\geq 0\. The pre\-registered localization score \(dissipation mass inside the true entropy mask over the mask’s width fraction\) measures whether the learned port*finds the front*; the pre\-registration commits to either outcome, score≥5\\geq 5or the representational limit of coarse local flux stencils on shocks, characterized\. Table[7](https://arxiv.org/html/2610.00280#A9.T7)carries the verdict\.

### I\.4One menu, fifteen worlds, two controls

Result\.The selection matrix \(Table[8](https://arxiv.org/html/2610.00280#A9.T8)\) runs the*same*menu \(\{H,H\+port,H\+port\+drive,port​only,\(±drive\)\}\\\{H,\\ H\{\+\}\\mathrm\{port\},\\ H\{\+\}\\mathrm\{port\}\{\+\}\\mathrm\{drive\},\\ \\mathrm\{port\\ only\},\\ \(\\pm\\mathrm\{drive\}\)\\\}plus each system’s declared bracket variant\) across every world with the frozen selector \(min\-BIC over*structural*channels, L1 on port/drive amplitudes, the clamp\-CI tie\-break, and the budget rule that refuses to trust a conservative call whose port sibling has not drained\)\. Both out\-of\-menu controls fire: the anti\-dissipative world \(architecturally outside every member, sinceσ=raw2\\sigma=\\mathrm\{raw\}^\{2\}cannot anti\-damp\) and the delay oscillator \(non\-Markovian\) both return*none\-of\-these*via the reference\-free alarm\. The pre\-registered≥14/15\\geq 14/15gate ismissed: 13/16 rows are correct as reported \(12/16 under the strict “accepted only if” reading of the budget rule, which drops bare gravity’s self\-flagged row, both numbers printed by the harness\), and the miss is recorded in the integrity ledger\.

The matrix’s errors are as informative as its hits, and all three are structural failures of the*alarm’s*scope, not the selector’s: on the contact world the smooth member outright*outperforms*the flexible 4\-step\-history reference \(0\.0300\.030vs0\.4040\.404, misfit concentrating in rare event steps, where flexibility cannot collect\); on friction the reference is better but only by2\.2×2\.2\\times, far under the10×10\\timesinversion the alarm needs; and on the coarse shock world the reference is itself incompetent at this budget\. A floor\-ratio alarm requires dense misfit*and*a competent reference; impulsive and shocked worlds offer neither\. Rare\-event misspecification belongs to the guard\-first protocol, which fired for all three systems at card level \(3\.5×104×3\.5\\times 10^\{4\}\\timeson friction\)\. Writing that division of labor down \(alarms for dense misfit, matched\-structure guards for events, budget flags for fake\-conservative calls\) is the section’s methodological yield\.

The report\-only zoo matrix is superseded as a headline by one canonical selector evaluation, which the sealed development\-then\-confirmation protocol \(§[3](https://arxiv.org/html/2610.00280#S3)\) freezes before any confirmation world is drawn\. The frozen selector is then scored across four separately implemented generator families, with the per\-family four\-way accuracy \(0\.840\.84–0\.890\.89\), true\-candidate coverage \(0\.7270\.727–0\.9000\.900\), selective error among one\-candidate calls \(6\.8%6\.8\\%–15\.6%15\.6\\%\), and out\-of\-family non\-rejection \(2%2\\%–24%24\\%\) reported in Table[6](https://arxiv.org/html/2610.00280#A9.T6)\. This is cross\-implementation empirical validation with visible selective risk\. Table[6](https://arxiv.org/html/2610.00280#A9.T6)reports the per\-family measures\. Only the true\-candidate coverage is the quantity Proposition[1](https://arxiv.org/html/2610.00280#Thmproposition1)bounds \(marginally, at1−α1\-\\alpha\); the error among one\-candidate decisions and out\-of\-family non\-rejection are measured endpoints that the proposition does not predict, and an empty prediction set is not a certificate that the truth lies outside the menu, since a true candidate is excluded with probability up toα\\alpha\.

Table 6:Frozen selector confirmation by generator family\.Table 7:The pass\-card ledger\.bmarks the single boundary the pre\-registered pass rule allows a system to carry, each with a measured mechanism \(Appendix[M](https://arxiv.org/html/2610.00280#A13)\)\. Competence is the rollout error relative to the certified history\-based baseline; the refusal column is the within\-class refusal ratio\.Table 8:The selection matrix: one menu, fifteen worlds, two controls \(the frozen selector = min\-BIC over structural channels \+ L1 \+ clamp\-CI tie\-break; alarm = reference\-free misspecification inversion vs a 4\-step\-history MLP\)\. Rows print the selector’s output; errors are kept and mechanised, never relabeled\.

## Appendix JDiscussion: what structure buys, what it costs, where it ends

### J\.1What the experiments support

Within the supplied representations, the models recover useful constitutive content and fitted rates on the tested support\. Quadratic\-channel contraction is a mathematical property; 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\.

### J\.2What remains conditional

A wrong prior can fit observed data and fail after intervention\. An empirical fit\-gap alarm can also fail or react to optimization error; the confirmation selector has measured false acceptance\. The nonnegative port excludes the opposite sign of that channel, but does not exclude driven or stochastic energy increases\. The achieved precision of rate estimators is protocol\-dependent, not an information lower bound\. Functional recovery is limited by operator nullspaces, excitation, and state support\. These restrictions motivate the supplied\-versus\-learned accounting rather than universal discovery claims\.

### J\.3The boundary map

Stated once, together, as results: \(1\) the cavity’s Stability, worst\-case tail, bounded, trend\-free, characterized to its correlate \(initial kinetic energy\), both candidate levers refuted,5%5\\%bound carried; the grid closes it\. \(2\) State\-support envelopes for extrapolation fidelity: grid magnitude/flip rollouts degrade out\-of\-band \(grid\-specific; the cavity holds\|q\|=2\|q\|=2to horizon 1000 and added coverage changes nothing\)\. \(3\) Perception: single\-frame momentum invisibility, located, with the minimal multi\-frame fix quantified; from\-pixels binding and closed\-loop pixel rollouts open\. \(4\) Stochastic thermodynamics beyond one\(T,σ\)\(T,\\sigma\)and a charge\-0 linear bath: open\. \(5\) The universal scaling question: our scaling\-claim evidence is an existence result over four families, five rungs, and a certified protocol; whether some training procedure manufactures stable black\-box competence at scale, and whether its implicit bias then locks in or flips, is open, pre\-registered, and twice answered “inconclusive” by our own escalation rules\. \(6\) Regimes: 2D/shock fluids, contact, and singular attraction are future physics for this program\.

### J\.4Threats to validity

We audit ourselves the way we expect referees to\.*Shared\-data seeds*: all headline claims were replicated on fresh independent data draws; remaining intervals measure optimization variance and say so\.*Metric power*: every counterfactual verdict is conditioned on trained\-setting reliability \(the competence\-first lesson, applied as a gate\)\.*Survivor bias in aggregates*: diverged seeds are counted, not dropped, and worst cases are reported beside means\.*Undertrained giants*: the scaling study’s competence rungs carry learning\-rate sweeps, plateau evidence with published curves, unlimited\-data and longer\-horizon retries, and disclosed compute caps, and where competence still failed we labeled the verdict inconclusive rather than victorious\.*Experimenter degrees of freedom*: thresholds pre\-registered, verdict text generated directly from the pre\-registered constants, and the full ledger of corrections \(including two overturned verdicts, one retired metric, one superseded statistic, and one refuted mechanism attribution\) published in Appendix[K](https://arxiv.org/html/2610.00280#A11)\.

### J\.5Outlook

Three rungs order the future work\.*Up the discovery ladder*: from one learned bit \(does this world dissipate?\) toward selecting which symmetries and which bracket structure apply, since the fluid result suggests the program’s form survives regime changes, so the selection problem is the bottleneck\.*Down the perception stack*: the momentum boundary is located here and its minimal fix quantified, since a single frame is velocity\-blind and a short window recovers momentum by differencing \(§[M](https://arxiv.org/html/2610.00280#A13.SSx32)\); grounding physical state from raw pixels, including joint learned detection and binding, remains the open direction\.*Across the scaling question*: the experiment our own verdicts demand, a black\-box family engineered for*stable*competence at multiple scales, then the same flip, would either finally earn the universal leg or discover a generic bias that substitutes for one clause of the criterion; both outcomes are progress, and the measurement program published here is built to decide between them\.

### J\.6Three distinct boundaries

First, in\-support observations do not identify off\-support truth over an unrestricted function class\. Supplied or learned restrictions can prescribe a continuation; the tested scaling results do not rule out all such learning\. Second, failure to ground the tested frozen\-token pipelines is an empirical limitation, not an impossibility for token\-based representations\. Third, a static renderer depending only on position maps different velocities at that position to the same image\. It cannot identify arbitrary velocities without additional information\. Statistical image predictors may still exploit restricted support or correlations, while multiple frames supply motion information directly\. These are different mathematical and empirical limitations and should not be combined into three universal impossibility claims\.

## Appendix KVerdict discipline and the integrity ledger

### K\.1The rules

\(1\) Metrics and thresholds are pre\-registered before results exist; printed verdict text is generated directly from the pre\-registered constants rather than transcribed by hand, so every “PASS” in a results table is the output of a function whose thresholds are the registered ones\. \(2\) Every diagnostic ships with a positive control that must trip; a diagnostic that has never flagged anything is treated as untested, not as good news\. \(3\) Near\-binary outcomes use seed\-level exact binomial statistics; episodes within a seed are correlated and are never pooled as independent\. \(4\) Counterfactual verdicts require metric power: the model under test must get the*trained*setting right on held\-out data, else its counterfactual score is uninformative and excluded\. \(5\) Headline verdicts require an escalation \(ten seeds, fresh data draws\) before any thesis\-level guard moves\. \(6\) A FAIL is reported as a FAIL; a refuted attribution is corrected at its source with a dated marker, and every claim that a correction was applied program\-wide is accompanied by the search that verifies it\.

### K\.2The ledger

Chronological; each entry names what changed, why, and where the full record lives\.

Two integrator defects \(cavity build\)\.First\-order operator splitting \(secularO⁡\(Δ​t\)O\(\\Delta t\)energy error\) and, introduced by its fix, a tensor\-view aliasing bug \(30×30\\timesworse\)\. Both caught by pre\-registered conservation tests before any training; fixed \(palindromic composition; snapshotting\)\. Lesson exported: the halve\-Δ​t\\Delta tscaling test distinguishes integrator orders instantly\.

A self\-force in the PIC build \(grid\)\.A stationary charge felt its own field; root\-caused in two layers \(B\-spline order pairing with charge\-conserving deposition; lattice self\-energy gradient, smoothed40×40\\times\)\. Caught before training; without this, every second\-law number on the grid would have carried a phantom dissipation channel\.

A statistic superseded \(grid flip\)\.Per\-episode circulation means withttintervals were retired when a broken control scored a perfect 32/32 episode block, because episodes within a seed are one correlated draw\. All headline flips were re\-scored at the seed level with exact binomials and got*stronger*\(cavity 10/10 with controls below chance\)\.

A metric retired by its own control \(magnitude counterfactual\)\.The pre\-registered gyration\-frequency criterion failed on the oracle itself \(4/5 seeds; chaotic phase makes the estimator ill\-posed\)\. Retired as a failed pre\-registration*of a metric*; the verdict rests on the surviving nMSE arm\.

An estimator bug corrected \(functional discovery\)\.A pointwise relative\-error probe diverges near zero field and mis\-scored the discovery as a failure; the pre\-registered slope\-regression estimator is the instrument of record \(rslope=0\.959±0\.017r\_\{\\rm slope\}=0\.959\\pm 0\.017, the±\\pma95%95\\%Student\-ttinterval over three grid seed estimates0\.9510\.951,0\.9630\.963,0\.9640\.964; a computed interval, not the observed min–max range\)\. Lesson: measure slopes with regressions, never pointwise ratios near zeros\.

A control upgraded \(unified model\)\.The mixed\-data baseline was re\-run*informed*\(given the condition index, drive/field features, and the auxiliary loss\) so that its failure isolates structure rather than information asymmetry\. It still fails\.

An attribution refuted and propagated \(stability tail\)\.The “close\-encounter” mechanism carried since the early cavity runs was disproved by direct correlation test \(wrong sign; the correlate is initial kinetic energy\) and the coverage lever refuted \(67×67\\timesoversampling, no effect\)\. Corrected at every occurrence with dated markers; the grep is in the record\.

A gate failure kept against temptation \(fresh\-dataσ\\sigma\)\.The strict per\-valueσcons<10−4\\sigma\_\{\\rm cons\}<10^\{\-4\}gate fails on 2/9 fresh fits \(max1\.07×10−41\.07\\times 10^\{\-4\}\)\. An early draft relabeled the recorded “NOT confirmed” outcome as a bug; the correction was reversed \(the boolean was right, the prose was wrong\) and the runner’s prose is now generated from the pre\-registered constant\. The order\-of\-magnitude identification stands; the strict gate is carried as a boundary\.

Scaling verdict \#1 overturned \(FNO ladder\)\.“Extrapolation at chance across100×100\\timesscale” was retracted by our own review: no rung was competent, the fit\-improves prediction had failed under the fixed\-epoch protocol, and a diverged rollout’s flip is a coin toss\. Relabeleddirectional; superseded by the competence\-first design\.

Scaling verdict \#2 overturned \(competence\-first scaling study\)\.A 3\-seed EARNED \(two competent rungs below chance\) did not survive the pre\-registered 10\-seed fresh\-data escalation: one rung’s trained\-sign reliability proved seed\-fragile \(mean0\.867<0\.900\.867<0\.90\) and dropped out; the recorded verdict isinconclusive\-single\-anchor\. The thesis guard stayed up; the main text’s scaling claim \(§[4\.5](https://arxiv.org/html/2610.00280#S4.SS5)\) is the one the surviving evidence supports\. Every stale table was marked superseded; the pipeline that produced each number is labeled\.

Flagship selection verdict softened by its own escalation \(flagship cavity,3/3→2/33/3\\to 2/3\)\.The three\-seed flagship lift read DISCOVERED \(3/33/3, every runner\-up≥278\\geq 278BIC\), its smallest margin being exactly the conservative cell whose near\-degeneracy we had already characterized\. The five\-seed repeat flipped that cell toH\+H\{\+\}port by2929BIC withσ^=0\.0126\\hat\{\\sigma\}=0\.0126\(a1%1\\%wMSE difference swung by one seed\); the recorded verdict is2/32/3\(BOUNDARY\)\. The surviving claim is stated in §[F](https://arxiv.org/html/2610.00280#A6): dissipative/driven selection is decisive and seed\-stable \(Δ\\DeltaBIC≥1054\\geq 1054\) and the rates are recovered; the conservative tie\-break is fit\-degenerate but*not*horizon\-degenerate, sinceσ^=0\.0126\\hat\{\\sigma\}=0\.0126sheds6%6\\%of the field energy over the500500\-step window but99\.8%99\.8\\%over the5050k\-step stability horizon \(E∝e−2​σ^​tE\\propto e^\{\-2\\hat\{\\sigma\}t\}, palindromic port\), an arithmetic that corrected our first “same physics” framing; the corrected reading \(§[F](https://arxiv.org/html/2610.00280#A6)\) makes removing this residual necessary\. Every stale3/33/3/≥278\\geq 278site was corrected at source, including the figure, whose annotations are now generated from the verdict JSON\.

The response\-levelω\\omegastatistic is a POST HOC promotion, disclosed\.The odd/even decomposition was pre\-registered as a*mechanism probe*to explain the high\-flip anchor seeds, not as a scaling\-claim verdict statistic\. Its outcome, a clean, seed\-robust separation between the structured model and every trained black box, led us to promoteω\\omegato the mechanism\-level half of the scaling\-claim evidence\. We flag the promotion as post hoc: the*claim*it supports \(black boxes do not acquire the charge\-flip symmetry\) is exactly the pre\-registered scaling hypothesis, andω\\omegais an additional measurement of it, not a moved goalpost; the accompanying random\-init control \(also post hoc\) is what fixes its interpretation \(0\.5 is the parity floor, not “half the symmetry”\)\.

### K\.3Why we publish this

The ledger is calibration\. Each entry is an instance where the program’s rules produced a less flattering number than the first analysis offered, which is precisely the property a reader needs a measurement program to have before trusting the numbers that survived\.

## Appendix LPropositions

The two propositions below carry the boundedness and transfer claims; their scope clauses are stated inline, and the port and selector propositions are in the main text with proofs in App\.[C\.2](https://arxiv.org/html/2610.00280#A3.SS2)\.

### L\.1Energy\-consistent coupling and boundedness

###### Proposition 4\(Energy\-consistent coupling and boundedness\)\.

*Setting\.*The imposed flow class: state==particles⊕\\oplusfield; dynamics generated by

H=T⁡\(p\)\+Uϕ​\(field\)\+q​Cψ​\(particle,field\)H=T\(p\)\+U\_\{\\phi\}\(\\text\{field\}\)\+q\\,C\_\{\\psi\}\(\\text\{particle\},\\text\{field\}\)\(10\)through a fixed symplectic \(or symplectic\-plus\-port\) integrator, withUϕU\_\{\\phi\}confining \(bounded below, radially unbounded on the field sector\) and the coupling*factored*: one shared learned coefficient serves both the force on particles and the current into the field\.

*Claim\.*\(i\) The exact*conservative*flow \(no port, no drive\) conservesHH\. If in addition the totalHHis coercive, its level sets are compact and rollout energy is bounded for all time\. Coercivity ofHHis an assumption here, not a consequence ofUϕU\_\{\\phi\}confining andTTbounded below: with an unbounded indefinite coupling those do not suffice, since

T=p2,U=f2,C=−2pf⟹H=\(p−f\)2,T=p^\{2\},\\qquad U=f^\{2\},\\qquad C=\-2pf\\qquad\\Longrightarrow\\qquad H=\(p\-f\)^\{2\},whose sublevel sets are unbounded alongp=fp=f\. We therefore require a coercivity\-preserving bound on the interactionq​Cψq\\,C\_\{\\psi\}, or coercivity ofHHdirectly\. For exact flows, local well\-posedness and regularity on the relevant compact sublevels are also required\. With a drive or a port not aligned toHH, finite\-horizon state bounds require control of all positive energy contributions, including port work, and the discrete symplectic integrator bounds oscillation of a shadow energy only under the usual step\-size, regularity, and domain conditions \(constants in the formal version\)\. \(ii\) If the coupling is*not*factored \(force and back\-reaction learned independently\) there exist arbitrarily small function\-approximation mismatches whose closed\-loop work is nonzero per cycle, that is, a phantom power source; no confinement ofUϕU\_\{\\phi\}bounds the resulting physical energy injection\.

###### Proof sketch\.

\(i\) is conservation ofHHby the exact flow together with compactness of the sublevel sets of a coercive continuous function\. \(ii\) Let the force beF=−∂x\(qC\)\+δ1F=\-\\partial\_\{x\}\(qC\)\+\\delta\_\{1\}and the currentJ=∂ϕ\(q​C\)\+δ2J=\\partial\_\{\\phi\}\(qC\)\+\\delta\_\{2\}with independent learned errorsδ1,δ2\\delta\_\{1\},\\delta\_\{2\}; the power exchanged at the interface isx˙⋅F\+ϕ˙⋅J\\dot\{x\}\\cdot F\+\\dot\{\\phi\}\\cdot J, whose non\-conservative partx˙⋅δ1\+ϕ˙⋅δ2\\dot\{x\}\\cdot\\delta\_\{1\}\+\\dot\{\\phi\}\\cdot\\delta\_\{2\}has nonzero circulation along a periodic orbit wheneverδ1\\delta\_\{1\}andδ2\\delta\_\{2\}are not derivatives of one function, which holds for generic independent approximations of arbitrarily small norm; the integrated work then grows linearly in the number of cycles regardless of the confinement ofUϕU\_\{\\phi\}\. ∎

*Measured instantiations\.*\(i\): the confinement dissociation \(cavity and fluid\)\. \(ii\): the9×9\\timescoupling power imbalance measured at the interface and the non\-factored control’s divergence under identical confinement on cavity and grid\. The fluid is the joint instantiation of both propositions rather than a third divergence: its non\-separable control keeps the energy\-consistent flux coupling \(so Proposition[4](https://arxiv.org/html/2610.00280#Thmproposition4)predicts boundedness; measured:5/55/5,9\.2×10−59\.2\\times 10^\{\-5\}\) while black\-boxing only thegg\-entry \(the reported implementation fails the counterfactual test by1010×10^\{10\}\\times\)\.

*Scope\.*Excludes energies unbounded below \(gravitational/bare\-Coulomb attraction\), physics’ own exception, inherited; vacuum\-field stability is the claim\.

### L\.2Supplied\-map transfer and unrestricted extrapolation

###### Proposition 5\(Supplied\-map transfer and unrestricted extrapolation\)\.

Let the true and learned generators be

G∗​\(x,θ\)=θ​g∗​\(x\)\+h∗​\(x\),Gϑ​\(x,θ\)=θ​gϑ​\(x\)\+hϑ​\(x\)\.G\_\{\*\}\(x;\\theta\)=\\theta\\,g\_\{\*\}\(x\)\+h\_\{\*\}\(x\),\\qquad G\_\{\\vartheta\}\(x;\\theta\)=\\theta\\,g\_\{\\vartheta\}\(x\)\+h\_\{\\vartheta\}\(x\)\.\(11\)If both learned components equal their true counterparts onℛ\\mathcal\{R\}, thenGϑ​\(x,θ′\)=G∗​\(x,θ′\)G\_\{\\vartheta\}\(x;\\theta^\{\\prime\}\)=G\_\{\*\}\(x;\\theta^\{\\prime\}\)there for every admissibleθ′\\theta^\{\\prime\}\. This follows by substitution\. With the same initial condition and uniqueness of the differential equations, the continuous trajectories agree while they remain inℛ\\mathcal\{R\}\. Numerical predictions retain their separate integration error\. Separately, an unrestricted class of functions of\(x,θ\)\(x,\\theta\)contains functions agreeing on the observed support and differing outside it, so in\-support data alone do not determine off\-support dynamics over that class\.

This is a sufficient supplied\-map construction\. More generally, a known nonlinear map can transport accurate content without multiplicative separability\. The nonidentifiability statement is not a converse to the sufficient construction, and does not exclude transfer under additional known or identifiable restrictions\. Parameter\-support shift defines our extrapolation tests; it is not a necessary condition for counterfactual answerability or a causal positivity assumption\.

### L\.3Restriction of the passive channel

For the same isolated scalar coordinate\-damping channel and positive quadratic diagnostic, imposingσ≥0\\sigma\\geq 0excludes the anti\-damped channelσ<0\\sigma<0\. Representing that sign reversal requires relaxing the constraint or changing the model\. This does not forbid external work, thermal energy injection, or other driven dynamics\.

## Appendix MThe complete per\-experiment record

This appendix reports one block per finding, in program order, and each block follows the same structure: the setup, the gates fixed in advance, the result with its per\-seed statistics, the controls, and the verdict class with any boundary the finding carries\. Negative results, refutations, and retired metrics are reported on equal footing with successes\. The per\-seed measurements, model checkpoints, and run logs behind these numbers are provided as supplementary material; the values here are final and should not be re\-rounded\.

### Cavity reference calibration

Ring cavity, palindromic symplectic integrator; 16 episodes/config, 50,000 steps\. Gates: relative energy drift<10−4<10^\{\-4\}; Gauss residual<10−10<10^\{\-10\}and constant\. Result: drift9\.3×10−69\.3\\times 10^\{\-6\}\(same\-sign\) /1\.38×10−51\.38\\times 10^\{\-5\}\(mixed; binding case,7\.25×7\.25\\timesmargin\); Gauss≤2\.0×10−14\\leq 2\.0\\times 10^\{\-14\}, exactly conserved\. Build defects caught pre\-training: first\-order splitting \(secular; fixed palindromic\), tensor\-view aliasing \(fixed by snapshotting\)\.decisivegate; recalibration required on any configuration change\.

### Cavity behaviour suite: the first pass and its informative failure

4 models×\\times5 seeds\. one\-step competence: structured recovers content to0\.2%0\.2\\%\. Counterfactual, charge\-sign flip: structured 0\.95 vs non\-factored 0\.33, baseline 0\.24 \(circulation match\)\. reset\-kick ratio0\.995±0\.140\.995\\pm 0\.14; the intervention test∼1\\sim 1\.Stability FAIL: all learned models exceed the 5% energy gate, since unconstrained learned field energy flattens away from the training data \(probed directly: tanh\-MLP saturation where divergences escape\), deviations to2\.9×1072\.9\\times 10^\{7\}\. Pre\-registered overall verdict FAIL, reported as such; mechanism identified in the confinement result below\. Also seeded the counterfactual informative\-fail that became the criterion\.

### Composition and extrapolation dissociate

Two counterfactual types on identical models\. Unseen mixed\-charge configuration \(marginals in\-support\): passed by both field\-mediated models, failed by the baseline\. Charge\-sign flip \(support\-moving\): passed only by the factored model\. Assigns composition to locality and extrapolation to factoring; formalized as the structural criterion \(§[C](https://arxiv.org/html/2610.00280#A3)\)\.decisive\(upgraded statistics in the seed\-level flip result\)\.

### Confinement restores stability; the coupling condition appears

Confining per\-mode energy \(quadratic, learned stiffness on per\-mode\-scaled inputs\+\+bounded correction\)\. Gates: 4/5 pass \(one\-step competence; counterfactual primary\+composition; reset\-kick@1\.00\.85±0\.050\.85\\pm 0\.05; the intervention test0\.88±0\.430\.88\\pm 0\.43\)\. Stability transformed:0/5→5/50/5\\to 5/5bounded, slope1\.1×10−71\.1\\times 10^\{\-7\}, but worst\-case episodes55–12%12\\%vs the 5% bound, carried \(characterized in the stability\-tail result\)\. Same\-magnitude field\-bump channel misses the≤2\\leq 2gate \(2\.72±0\.952\.72\\pm 0\.95\), disclosed\.Non\-factored control NOT rescuedby confinement \(4/5 unbounded\): stability needs the coupling condition too\. Parameterization lesson: learned coefficients in units where targets areO⁡\(1\)O\(1\)\.

### The coupling power imbalance, measured

Instrumented force/current interface under identical confining energy\. Non\-factored model injects∼9×\\sim 9\\timesthe net energy per step of the factored model\. Also self\-corrected the first\-pass account \(the original failure had two components: in\-band pumping*and*the flat tail\)\.decisive; the pool\-mean is a mechanism localizer, not a rollout number\.

### Magnitude counterfactual \(cavity\)

Trainq=\+1q=\+1, test\|q\|=2\|q\|=2; 5 seeds\. nMSE arm: structured0\.096±0\.0350\.096\\pm 0\.035vs oracle0\.088±0\.0360\.088\\pm 0\.036\(ratio≈1\.09\\approx 1\.09\); baseline7\.7×1037\.7\\times 10^\{3\}; non\-factored4\.24\.2with gyration\-frequency ratio−0\.46\-0\.46\(rotation reversed\)\. Pre\-registered frequency\-within\-5% criterion RETIRED: the oracle itself fails it 4/5 seeds \(chaotic phase; ill\-posed estimator\), a failed pre\-registration of a metric\.decisiveon the surviving arm\.

### Rate counterfactuals; guarantee vs representation

Ports trained atσ=0\.5\\sigma=0\.5\(label given\), tested atσ=0\.25\\sigma=0\.25and−0\.5\-0\.5; 3 seeds\. Learned scale1\.0045±0\.00071\.0045\\pm 0\.0007\(\+0\.45%\+0\.45\\%systematic, truth outside interval\)\. Rate transfer: error0\.0017±0\.00340\.0017\\pm 0\.0034\. Anti\-dissipation: guarantee port clamps \(rate error1\.031\.03, a by\-construction refusal\); sign port tracks the genuinely divergent truth \(energy growth∼22,000×\\sim 22\{,\}000\\times/2,000 steps\) to∼1%\\sim 1\\%; amplification ratio2\.15±0\.832\.15\\pm 0\.83\(the pre\-registered\>10×\>10\\timesinstability prediction NOT confirmed, reported\)\. No stability claim for anti\-dissipation; representation fidelity only\.decisive\.

### Multi\-step training does not close the stability tail

Pushforward training \(K\-step windows\): widens seed variance \(worst cases to0\.280\.28–0\.320\.32\), mildly costs fit; the structured model remains the reported configuration\. Mechanism: training window≪\\llexcursion timescale\. Negative, kept\.

### Deterministic second law PASS

Dissipative \+ driven cavity; 5 seeds; port model\.σ=0\.502±0\.0003\\sigma=0\.502\\pm 0\.0003\(truth 0\.5\); field\-energy decay\-rate error0\.0038±0\.00210\.0038\\pm 0\.0021\(gate≤0\.10\\leq 0\.10\); terminal ratio0\.9950\.995; driven tail ratio0\.998±0\.0030\.998\\pm 0\.003; injection6\.046\.04vs true6\.256\.25\. Production diagnostic: engineered violator flagged 5/5; structured 0/5; baseline itself flagged\. Metric scope: rates on field energy \(kinetic undamped\)\. Identification note: port under\-fit at shared learning rate;10×10\\timesLR group fixed it \(41%→2\.7%41\\%\\to 2\.7\\%override diagnostic\)\.decisive; deterministic scope \(stochastic half below\)\.

### Symbolic regression positioning

SINDy \(STLSQ\), generous disclosed dictionary\. Clean data: most accurate as positioned \(worst\-family coefficient error0\.07%0\.07\\%, zero false positives\)\. Observation noise: collapses between0\.5%0\.5\\%and2%2\\%\(particle\-force family first; derivative amplification\)\. Dictionary/dimensionality boundary stated for the grid\. A weak\-form \(integral, test\-function\) variant with the identical library is far more robust \(worst\-family error0\.007/0\.028/0\.1020\.007/0\.028/0\.102at0\.5/2/5%0\.5/2/5\\%noise against0\.02/0\.23/0\.940\.02/0\.23/0\.94for the central\-difference form\)\. Both sides are reported\.

### Structural advantages survive observation noise

Noise sweep over0\.5/2/5%0\.5/2/5\\%\(training\-time, inputs and targets\)\. At5%5\\%: sign\-flip match1\.01\.0; one\-step fit∼\\simunchanged; stability envelope loosens gracefully \(0\.045→0\.2850\.045\\to 0\.285at 10k steps\); baseline at chance throughout\. Scope: eval\-time estimation is the perception test\.decisive\.

### The criterion’s refusal test \(cavity\)

Non\-separable parameter: particle\-shape widthσs\\sigma\_\{s\}\(0\.10→0\.160\.10\\to 0\.16\), inside a Gaussian in every gather/deposit\. Pre\-registered: \(a\) structured fails≥10×\\geq 10\\times; \(b\) oracle\-kernel control succeeds≤3×\\leq 3\\times\. Result: structured4\.78±0\.024\.78\\pm 0\.02=111×111\\timesin\-distribution \(CONFIRMED\); size\-matched oracle0\.0607±0\.0130\.0607\\pm 0\.013=1\.4×1\.4\\times\(CONFIRMED; a first half\-budget control missed its bar, confound found, re\-run size\-matched, on record\)\.decisive\.

### Grid reference calibration

Yee/Boris/Esirkepov PIC,64264^\{2\}, staggered\-consistent gather, binomial smoothing×3\\times 3\. Gates at 50,000 steps: drift≤4\.6×10−5\\leq 4\.6\\times 10^\{\-5\}\(gate10−210^\{\-2\}\); Gauss residual3\.1×10−133\.1\\times 10^\{\-13\}\(gate10−1010^\{\-10\}\)\. Numerical heating did not materialize\. Build defects caught pre\-training: plane\-wave*test*direction \(simulator right\); static self\-force \(two\-layer root cause; smoothing40×40\\times\)\.decisivegate; per\-configuration scope\.

### The criterion reproduces on the grid

3 seeds plus the external\-field arm; equal capacity \(8,711/8,591/11,306\)\. One\-step competence: structured one\-step1\.6×10−91\.6\\times 10^\{\-9\}, rollout nMSE0\.0440\.044; baseline broken \(LR\-sweep certified: best3\.3×10−43\.3\\times 10^\{\-4\}, still∼2×105×\\sim 2\\times 10^\{5\}\\timesworse\)\. Composition:0\.979/0\.896/0\.4690\.979/0\.896/0\.469\(non\-factored barely meets the 0\.90 gate, QUALIFIED\)\. Stability: structured 3/3 bounded, max0\.00390\.0039, flat; both controls diverge\. reset\-kick1\.141\.14; the intervention test1\.031\.03\.\|q\|=2\|q\|=2: out\-of\-band divergence beyond∼T=75\\sim T\{=\}75\(boundary; resolved grid\-specific below\)\. Flip: superseded statistics below\.decisivepattern\.

### Functional discovery: vacuum Maxwell

Learned constitutive derivatives, slope\-regressed over±1\.5\\pm 1\.5field\-std \(pre\-registered estimator; the pointwise probe bug is on record\):fE′f^\{\\prime\}\_\{E\}slope1\.0431\.043,fB′f^\{\\prime\}\_\{B\}1\.0011\.001, bothr2=1\.0000r^\{2\}=1\.0000;rslope=0\.959±0\.017r\_\{\\rm slope\}=0\.959\\pm 0\.017\(truth 1\), a reported∼4%\\sim 4\\%E\-side slope discrepancy \(discretization\), disclosed with direction\. In\-band scope; linear \(vacuum\) constitutive law\.decisive\.

### Grid sign\-flip transfer at the seed level

10 seeds, exact binomial, majority\-per\-seed\. Structured 10/10 \(pooled 315/320,p=0\.001p=0\.001\); baseline 4/10; non\-factored 5/10\. Transfer is orbit\-sense at trajectory level \(nMSE∼3\\sim 3on non\-divergent seeds\), resolved further below\.decisive\(regime\)\.

### One model, four worlds \(ten seeds\)

Unified cavity model: per\-conditionσc=rawc2\\sigma\_\{c\}=\\mathrm\{raw\}\_\{c\}^\{2\}, shared content; L110−510^\{\-5\};10×10\\timesport LR; energy\-trajectory auxiliary loss \(50\-step windows; targets from data; readout uses the reference energy functional, disclosed known\-physics supervision\)\.σcons=1\.2×10−5\\sigma\_\{\\rm cons\}=1\.2\\times 10^\{\-5\}\(range3×10−63\\times 10^\{\-6\}–2\.5×10−52\.5\\times 10^\{\-5\}; implied 50k loss0\.6%0\.6\\%\);σdiss=0\.481\\sigma\_\{\\rm diss\}=0\.481\(∼4%\\sim 4\\%low, protocol\-specific per the identifiability result\);σB0=\\sigma\_\{B\_\{0\}\}=exact 0\. One checkpoint per seed, scored as\-is: composition1\.01\.0; flip 10/10 \(p=0\.001p=0\.001\); stability bounded 10/10, slope3\.6×10−73\.6\\times 10^\{\-7\}, but the pre\-registered5%5\\%energy\-deviation threshold is not met on 6 of 10 seeds, so boundedness and fidelity are reported separately \(per\-seed max\[0\.333,0\.663,0\.052,0\.036,0\.031,0\.044,0\.107,0\.286,0\.057,0\.037\]\[0\.333,0\.663,0\.052,0\.036,0\.031,0\.044,0\.107,0\.286,0\.057,0\.037\]; worst5\.5×5\.5\\timesthe specialist’s, same characterized tail\); second\-law rate error3\.7%3\.7\\%, injection6\.32/6\.256\.32/6\.25, violator 10/10, clean\. Controls: no\-port fails dissipation \(0\.99\); informed baseline \(condition\+\+drive\+\+B0B\_\{0\}features\+\+aux loss\) still fails \(rollout∼106\\sim 10^\{6\}; second\-law error∼7\\sim 7\); sign\-free port drifts negative on 3/5 seeds \(−1\.6\-1\.6to−3\.2×10−4\-3\.2\\times 10^\{\-4\}\)\.decisiveat behaviour level; bound carried\.

### Cavity sign\-flip transfer at the seed level, with controls below chance

10 seeds\. Structured 10/10 \(pooled 318/320,p=0\.001p=0\.001\); baseline 2/10, non\-factored 1/10,*below*chance: trained\-rotation lock\-in \(configuration\-scoped\)\.\|q\|=2\|q\|=2arm: the circulation metric is non\-discriminating for sign\-preserving moves \(controls pass it 7–9/10 without the intended mechanism\), scope correction applied program\-wide; magnitude claims rest on the fidelity arms\.decisive\.

### What the grid flip transfers, horizon\-resolved

Phase\-insensitive observables on existing checkpoints \(10 seeds, medians\): total energy0\.8%0\.8\\%at all horizons; speed error4\.6%​@​25→12%​@​50→33%​@​100→36%​@​2004\.6\\%@25\\to 12\\%@50\\to 33\\%@100\\to 36\\%@200; controls fail at every horizon \(8×8\\times–67×67\\times\)\. Mechanism: out\-of\-band drift \(the\|q\|\|q\|envelope\), not phase decorrelation\. Upgrades “orbit\-sense only” to “energy exactly; kinematics in\-envelope\.”

### The stochastic second law in full

Thermal cavity \(charge\-0 linear bath, FDT\-matched noise;T=0\.5T=0\.5,σ=0\.5\\sigma=0\.5\); 5 seeds; temperature from the last 3,000 of 6,000 steps, cold start\. Reference calibration first: 6/6, GibbsTTto0\.6%0\.6\\%\. Model:T=0\.502±0\.007T=0\.502\\pm 0\.007\(0\.3%0\.3\\%; gate10%10\\%\); sectors0\.502/0\.5020\.502/0\.502\(magnetic sector thermalizes only via the learned reversible coupling\); Wasserstein0\.016±0\.0140\.016\\pm 0\.014\(gate0\.100\.10\); learnedσ=0\.497\\sigma=0\.497\. Controls: FDT violator \(noise variance doubled, same model\) settles≈2​T\\approx 2T, flagged 5/5; clean 0/5; baseline runs hot∼106×\\sim 10^\{6\}\\times\(finite\)\. Convergence note: an early4040\-epoch run showed a spurious28%28\\%magnetic over\-heat that was pure under\-training and disappeared by120120epochs; we record it so that a training\-horizon artefact is not mistaken for physics\.decisive; one\(T,σ\)\(T,\\sigma\), known\-reservoir scope\.

### Fluid reference calibration

1D shallow\-water, spectral implicit\-midpoint\. 50,000 steps \(250×250\\timestraining horizon\): mass3×10−153\\times 10^\{\-15\}; inviscid drift1\.2×10−71\.2\\times 10^\{\-7\}\(linear\) /7\.5×10−57\.5\\times 10^\{\-5\}\(amplitude 0\.1\); wave speed≤0\.6%\\leq 0\.6\\%worst mode \(gate 1%\); viscous decay matches2​ν2\\nuto10−1010^\{\-10\}; 18 tests\. Smooth\-regime scope\.

### Four behaviours in the fluid \+ the gravity counterfactual

5 seeds,∼\\sim8\.7k params\. One\-step competence: structured7\.8×10−117\.8\\times 10^\{\-11\}; baseline in\-distribution median0\.280\.28\(per\-seed\[0\.28,428,0\.17,6\.1,0\.22\]\[0\.28,428,0\.17,6\.1,0\.22\], mean dominated by one seed; disclosed\) and 0/5 bounded at 10k steps \(h<0h<0,\|u\|→2400\|u\|\\to 2400\)\. counterfactual gravity: traing=1\.0g=1\.0, test1\.51\.5:6\.8×10−116\.8\\times 10^\{\-11\}vs in\-dist7\.7×10−117\.7\\times 10^\{\-11\}\(exact\); non\-separable control3\.2×10−53\.2\\times 10^\{\-5\}in\-dist→1\.06\\to 1\.06extrapolated; baseline2\.5×1032\.5\\times 10^\{3\}\. Stability: 5/5 vs 0/5; max7\.1×10−57\.1\\times 10^\{\-5\}\. Mass:4\.5×10−164\.5\\times 10^\{\-16\}vs baseline1\.091\.09\. Metric note: absolute nMSE primary \(smooth, non\-chaotic; oracle ratios inflate at10−1010^\{\-10\}base\)\.decisive\.

### Fluid second law and the viscosity counterfactual

Second law: decay\-rate error1\.8×10−81\.8\\times 10^\{\-8\}; no\-port control error1\.01\.0\(port necessary\); violator flagged 5/5; wind injection factor17\.4=17\.4=truth\. Viscosity transferν:0\.5→0\.25\\nu:0\.5\\to 0\.25, error2\.9×10−82\.9\\times 10^\{\-8\}; anti\-friction clamps to exactly 0 \(guarantee\)\. Diagnostic on fluctuation energy \(constant12​g​h02​L\\tfrac\{1\}\{2\}gh\_\{0\}^\{2\}Lbackground excluded, lesson\)\.decisive\.

### Fluid discovery: functional content \+ unified port

M⁡\(h\)M\(h\)slope1\.0001\.000\(R2=1\.000R^\{2\}=1\.000\);V′​\(h\)V^\{\\prime\}\(h\)slope1\.0001\.000; effective coefficient1\.000±5×10−61\.000\\pm 5\\times 10^\{\-6\}per unit suppliedgg\(gravity supplied, not discovered\)\. Unified \(condition\-index only\):σinviscid=1\.4×10−4\\sigma\_\{\\rm inviscid\}=1\.4\\times 10^\{\-4\},σviscous=0\.50004\\sigma\_\{\\rm viscous\}=0\.50004; sign\-mode control shows no spurious anti\-dissipation \(≈−6×10−6\\approx\-6\\times 10^\{\-6\}\)\. The reported agreement concerns the tested class and support, not a general identifiability or optimization guarantee\.decisive\.

### The fluid refusal test, certified

Pressure exponentγ\\gammainsidehγh^\{\\gamma\}; trainedγ=2\\gamma=2\(in\-dist1\.0×10−101\.0\\times 10^\{\-10\}\): refused atγ=1\.7\\gamma=1\.7\(0\.750\.75;7\.4×109×7\.4\\times 10^\{9\}\\times\) andγ=2\.3\\gamma=2\.3\(0\.270\.27;2\.7×109×2\.7\\times 10^\{9\}\\times\)\. Learnability certified by theγ\\gamma\-oracle \(same class trained on each refused law\):1\.3×10−6/3\.7×10−71\.3\\times 10^\{\-6\}/3\.7\\times 10^\{\-7\},\>5\>5orders below the refusal\. Registration scope: direction\-level pre\-registration, with the oracle control added later \(dated\)\.decisive\.

### The\|q\|\|q\|boundary is grid\-specific

Cavity,B0B\_\{0\}, 5 seeds, horizon 1000 \(5×5\\timesthe grid’s\):\|q\|=2\|q\|=2speed error0\.027%​@​25→1\.9%​@​10000\.027\\%@25\\to 1\.9\\%@1000; energy\-vs\-truth0\.12%0\.12\\%; model drift0\.57%0\.57\\%; trajectory decorrelates to nMSE0\.450\.45\(bounded, phase\-only; comparable to in\-envelope\|q\|=1\.5\|q\|=1\.5\)\. Widening training to\|q\|∈\{1,1\.5\}\|q\|\\in\\\{1,1\.5\\\}at equal volume: no change \(1\.00×1\.00\\times\), nothing to fix\. Grid\-side mechanism \(grid\-scale coupling at speed\) remains a hypothesis, stated\.

### The grid refusal test \(directional\)

Smoothing widthnsmooth:3→5n\_\{\\rm smooth\}:3\\to 5; 3 seeds\. Structured refusal7\.0×7\.0\\timesin\-distribution \(pre\-registered bar10×10\\times: NOT met,directional\); oracle\-kernel control0\.34×0\.34\\times\(learnable, certified\); baseline broken \(2×108×2\\times 10^\{8\}\\times\)\. Milder than the cavity’s111×111\\times\(gentler generator perturbation\); farther width untested\.

### Identifiability, and a corrected attribution

Clamp/profile: retrain everything else atσdiss∈\{0\.25,0\.4,0\.5,0\.6,0\.75\}\\sigma\_\{\\rm diss\}\\in\\\{0\.25,0\.4,0\.5,0\.6,0\.75\\\}, fit loss0\.012/0\.004/0\.0004/0\.008/0\.0150\.012/0\.004/\\mathbf\{0\.0004\}/0\.008/0\.015: clean minimum at truth; conservative energy uncorrupted\.λ\\lambda\-sweep: atλ=0\\lambda=0,σ=0\.50005\\sigma=0\.50005; at the default,0\.49970\.4997, so L1 contributes−0\.08\-0\.08pp; the unified model’s∼4%\\sim 4\\%bias is protocol\-specific \(mechanism open\); its caveat corrected \(dated\)\.decisive\.

### The stability tail characterized, and an attribution refuted

Diagnostic across 400 episodes: closest\-approach vs worst\-case deviationr=\+0\.13r=\+0\.13\(predicted≤−0\.5\\leq\-0\.5: wrong sign\); attracting pairs reaching separation≈0\\approx 0:r=−0\.06r=\-0\.06; strongest correlate: initial kinetic energy \(\+0\.24\+0\.24\)\. Lever:67×67\\timesclose\-encounter oversampling \(fraction0\.0870\.087vs natural0\.00130\.0013\): worst\-case0\.124±0\.0790\.124\\pm 0\.079vs0\.091±0\.0300\.091\\pm 0\.030, no closure; no fit/counterfactual regression\. Both pre\-registered predictions resolved against the hypothesis; the attribution corrected program\-wide with dated markers\. The tail is a high\-initial\-energy content boundary, carried\.

### Scaling probe 1

100×100\\timesFNO ladder \(10k/100k/1M\), certified 3\-LR/20\-epoch protocol\. No rung competent \(rollouts diverge everywhere; one\-step MSE worsens up the ladder under fixed epochs, so the pre\-registered fit\-improves prediction failed\); flip 6/10–2/10–4/10 \(uninformative: powerless metric on broken models\)\. Self\-retracted headline; superseded by the competence\-first design\. Lesson: competence is the power condition\.

### Label\-free regime identification

Frozen unified core; self\-supervisedσ^\\hat\{\\sigma\}encoder \(100\-step window; loss = short\-rollout match; no rate labels\)\. Separation: all held\-out conservativeσ^\\hat\{\\sigma\}\(0\.070±0\.0080\.070\\pm 0\.008\) below all dissipative \(0\.449±0\.0080\.449\\pm 0\.008\), 5/5 seeds,z=7\.2z=7\.2\. Mid\-rollout switch re\-identified \(0\.078→0\.4570\.078\\to 0\.457\)\. the second\-law test withσ^\\hat\{\\sigma\}:9\.8%9\.8\\%mean \(per\-seed\[9\.7,11\.3,8\.3,9\.5,10\.0\]%\[9\.7,11\.3,8\.3,9\.5,10\.0\]\\%, disclosed straddle\)\. Boundaries: intermediate trackingr=0\.90r=0\.90\(gate0\.950\.95\),directional;σ^cons\\hat\{\\sigma\}\_\{\\rm cons\}floor≈0\.07\\approx 0\.07\(finite\-window limit\), so conservative rollouts degrade \(0\.600\.60vs0\.0050\.005\); controls reduce but don’t zero the signal \(K=5:z=1\.3z=1\.3; shuffled: sign\-cancelling, one seed retainsz=3\.6z=3\.6\)\. Gate text generated from the pre\-registered constants \(an earlier misprint corrected, on record\)\.

### The perception boundary mapped

Lossy 1\-D rendering \(position blob \+ field traces; velocity never rendered\); corruption sweep; best\-case estimator per level; frozen core\. Single frame: position0\.0340\.034rad;vxv\_\{x\}0\.1650\.165\(floor0\.2900\.290\),vyv\_\{y\}0\.2420\.242\(floor0\.2610\.261, at floor\); energy\-shell miss8\.8%8\.8\\%, kinetic \(0\.0820\.082\) not field \(0\.0250\.025\)\. Three strided frames:vxv\_\{x\}0\.0520\.052,vyv\_\{y\}0\.0430\.043; shell3\.5%3\.5\\%; continuation22\.8→4\.522\.8\\to 4\.5\(oracle0\.0130\.013\)\. Field\-trace ablation:vxv\_\{x\}0\.165→0\.2970\.165\\to 0\.297\(to floor\), so the single\-frame signal is the field imprint\. Per\-component floors are reported here; an earlier combined floor of 0\.391 overstated the margins and is corrected\.boundary, measured; from\-pixels binding open\.

### Fresh\-data replication, with one gate that fails

Three fresh generation seeds, disjoint\. Flip: 15/15 seeds \(pooled 479/480; controls 5/15, 1/15\)\.σ\\sigma: orders preserved on every draw \(σdiss=0\.4988\\sigma\_\{\\rm diss\}=0\.4988,0\.2%0\.2\\%; corroborates the identifiability result\); strict per\-value gateσcons<10−4\\sigma\_\{\\rm cons\}<10^\{\-4\}FAILS marginally \(7/9; max1\.07×10−41\.07\\times 10^\{\-4\}\); fresh inference∼7×\\sim 7\\timeslooser than shared\-data \(worst\-seed implied 50k loss5\.2%5\.2\\%, over the 5% bound\)\. An intermediate draft relabeled the recorded FAIL as a bug; reversed, the boolean was right \(ledger\)\. Boundary kept\.

### Scaling probe 2: does model capacity buy the counterfactual?

![Refer to caption](https://arxiv.org/html/2610.00280v1/figures/s3_scissors.png)Figure 10:Scale does not replace structure \(supporting study\)\. \(a\) Composition score against parameter count for the local convolutional family \(filled green: competent; open orange: not\) and a global family \(purple diamond\), with the competence gate \(dashed\)\. \(b\) Fraction of correct sign\-flip counterfactuals at the two competent rungs, with chance \(dashed\) and the small structured model \(magenta star\)\. \(c\) Oddnessω\\omegaof the charge\-flip response for trained black\-box seeds \(green\) and the structured model \(star\), against the random\-initialisation band \(yellow\)\. The reading is given in this appendix\.Four families×\\timesfive rungs \(10410^\{4\}–10810^\{8\}\), plateau training, a pre\-registered three\-branch verdict, MIN\_COMPETENT=2=2\. 100M\-first gate: FNO/100M composition0\.5420\.542\(decorrelated; worse than freezing the state at0\.5620\.562\); CNN/100M intractable \(∼\\sim1\.7h/epoch\), unstable; 10k local anchor composition1\.0001\.000→\\tocompetence is bought by locality; the study then pivoted to the tractable local ladder \(dated amendment\)\. First test, composition within the CNN family:1\.000→0\.938→0\.688→1\.000\\to 0\.938\\to 0\.688\\todivergence; an unlimited\-data probe at 1M improves the fit \(0\.014→0\.0040\.014\\to 0\.004\) but not composition \(0\.6460\.646\), a scale or optimisation effect rather than data starvation \(compute cap disclosed\)\. Second test \(3\-seed\): both competent rungs below chance \(46/14446/144,0/1440/144\), giving a provisional pass reported as “pending escalation\.”Escalation \(10 seeds, fresh data\): OVERTURNED, since 100k trained\-sign reliability0\.600\.60–1\.001\.00\(mean0\.867<0\.900\.867<0\.90; drops out\); 10k flip pooled196/480=0\.41196/480=0\.41\(“intermediate”\); verdictinconclusive\-single\-anchor; guard stays up\. Structured reference, same harness:141/144=0\.979141/144=0\.979\. Incompetent\-rung artifacts handled by the power gate \(a10810^\{8\}FNO at0\.760\.76pooled with reliability0\.530\.53is a powerless metric, excluded by design\)\. Directional signal kept: competent\-seed flips0\.000\.00–0\.170\.17\(lock\-in\); competence\-by\-locality; erosion with scale\. The main text’s scaling claim is exactly what survives\.

### Odd/even response decomposition

Evaluation\-only on the twenty trained CNN flip seeds\. Feeding identical states at±q\\pm q\(charge feature flipped, fields and kinematics held\) and splitting the particle velocity update into odd and even parts inqq: the oddness fraction isω=0\.33\\omega=0\.33–0\.510\.51across black\-box seeds against0\.9940\.994for the 9k structured model \(metric validated\)\. Per\-seedω\\omegadoes not predict per\-seed flip rate, Spearmanρ=−0\.25\\rho=\-0\.25\(p=0\.29p=0\.29\) over twenty seeds; anchor\-onlyρ=0\.57\\rho=0\.57\(p=0\.08p=0\.08, not significant\)\. Pre\-registered outcome \(a\): the three high\-flip anchor seeds are sampling variation, not an implicit odd\-in\-qqbias\. A random\-init control showsω≈0\.5\\omega\\approx 0\.5is the no\-preferred\-parity floor \(an untrained CNN gives meanω=0\.45\\omega=0\.45, seeds0\.030\.03–0\.940\.94\) and trained CNNs sit at it \(10410^\{4\}0\.450\.45,10510^\{5\}0\.500\.50\), not near0\.9940\.994: training acquires*none*of the symmetry, it only concentratesω\\omegaat the floor\.ω\\omegais thus a decisive response\-level separator \(structured0\.9940\.994vs every*trained*black box0\.330\.33–0\.510\.51, no overlap\) where the rollout\-based scaling study is inconclusive\. This closes the open question flagged in §[M](https://arxiv.org/html/2610.00280#A13.SSx34)and supplies the mechanism\-level half of the evidence that scale does not substitute for structure\.

### Quartic fit\-gap experiment

The one\-degree\-of\-freedom quadratic\-plus\-bounded implementation is fitted toV⁡\(x\)=12​x2\+14​x4V\(x\)=\\tfrac\{1\}\{2\}x^\{2\}\+\\tfrac\{1\}\{4\}x^\{4\}and compared with a quadratic control\. Its reported fit error is2\.7×10−32\.7\\times 10^\{\-3\}versus9\.0×10−89\.0\\times 10^\{\-8\}, and fitted curvature atx=2x=2is−4\.5\-4\.5versus the reference13\.013\.0\. The independentβ\\betasweep reports fit\-floor ratios1,16\.8,150,2040,4\.6×104,1\.8×105,9\.1×1051,16\.8,150,2040,4\.6\\times 10^\{4\},1\.8\\times 10^\{5\},9\.1\\times 10^\{5\}forβ=0,0\.1,0\.25,0\.5,1,2,4\\beta=0,0\.1,0\.25,0\.5,1,2,4\. These are empirical fitting failures\. Bounded corrections can match quartic behaviour on compact domains, so the result does not prove that the admissible class cannot represent the observed truth\. Coercivity also does not forbid negative local curvature\. The relative contributions of capacity, correction bounds, and optimization remain unresolved\.

### Discovery rung 2: selecting the general form

Candidate forms\{H,H\+port,H\+port\+drive\}\\\{H,\\ H\{\+\}\\text\{port\},\\ H\{\+\}\\text\{port\}\{\+\}\\text\{drive\}\\\}in the lumped11\-DoF harness \(Hamiltonian core, non\-negative dissipation port, optional explicit drive; learned inertia, stiffness,σ\\sigma\), scored by a pre\-specified selection statistic,BIC=N​ln⁡MSE\+p​ln⁡N\\mathrm\{BIC\}=N\\ln\\mathrm\{MSE\}\+p\\ln N\. On each generating world the minimum\-BIC candidate is the true form\. Hardened protocol \(amendment of 2026\-07\-31\): eight worlds on aσ→0\\sigma\\\!\\to\\\!0and drive→0\\to\\\!0ladder, five seeds,1%1\\%noise,40/4040/40correct; a non\-nested wrong form \(dissipation on the wrong variable\) never selected; correct\-form one\-step error separates from the next\-simpler candidate1\.11\.1–3\.1×3\.1\\times\(penalty\-assisted only atσ≲0\.05\\sigma\\\!\\lesssim\\\!0\.05, a characterized near\-degenerate boundary\)\. The general model’s learned diagnostics agree \(σ^=0\.305\\hat\{\\sigma\}=0\.305vs true0\.300\.30; the drive switches on only when driven\)\. The same harness recovers the lumped constants to11–3%3\\%and does label\-free dissipation discovery \(σ^=0\.002\\hat\{\\sigma\}=0\.002conservative vs0\.2520\.252dissipative\), the §[F](https://arxiv.org/html/2610.00280#A6)capability in the one\-degree\-of\-freedom form\. Scope: synthetic worlds with the truth inside the candidate set, so the claim is that the statistic recovers the correct form when it is present;σ\\sigmais weakly identified under a strong drive and near the conservative boundary, disclosed\.

*Flagship lift, with its five\-seed escalation\.*The identical BIC statistic was applied to the paper’s central system, the1616\-mode, two\-particle EM cavity \(§[B](https://arxiv.org/html/2610.00280#A2), f64 symplectic core\), with the8\.98\.9k\-parameter energy network*shared*across all three candidate forms, so BIC penalizes only the structural hypothesis, not capacity\. Three worlds \(conservative/dissipative/driven\),400400epochs, per\-dimension standardized one\-step fit\. At three seeds:3/33/3, every runner\-up≥278\\geq 278BIC worse\. The pre\-committed five\-seed escalation*softened*the conservative cell: dissipative and driven remain decisive and seed\-stable \(H\+H\{\+\}port byΔ\\DeltaBIC22572257;H\+H\{\+\}port\+\{\+\}drive by10541054\), but the conservative world flips toH\+H\{\+\}port by2929BIC, withσ^=0\.0126\\hat\{\\sigma\}=0\.0126; the underlying fit difference is1%1\\%of wMSE, swung by one seed\. The pre\-registered five\-seed verdict is2/32/3, a boundary case, superseding the three\-seed3/33/3; ledgered in Appendix[K](https://arxiv.org/html/2610.00280#A11)\. The port form recovers the resistivity it was not told on every world:σ^=0\.013\\hat\{\\sigma\}=0\.013conservative \(≈0\\approx\\\!0\),0\.5020\.502dissipative,0\.5020\.502driven \(true0\.50\.5\)\. Two reproducibility notes: the fit metric must be the per\-dimension standardized \(1/std\-residual\) weighted MSE, since the port’s per\-step signal is∼0\.1%\\sim\\\!0\.1\\%of a state’s residual, so the raw full\-state MSE is diluted and selects the simplest form regardless; and the conservativeHH\-vs\-H\+H\{\+\}port tie\-break is degenerate at the fit but*not*at evaluation horizons:σ^=0\.0126\\hat\{\\sigma\}=0\.0126sheds6%6\\%of the field energy over the500500\-step window yet99\.8%99\.8\\%over the5050k\-step stability horizon \(E∝e−2​σ^​tE\\propto e^\{\-2\\hat\{\\sigma\}t\}\), so the flipped selection carries drastically wrong long\-horizon physics, the near\-degeneracy characterized on the11\-DoF ladder, now demonstrated to flip a selection on the distributed system, and the reason removing this residual is registered as a follow\-up study\.

### The selection loop and its guard

No new runs: the min\-BIC form selector presupposes the true form is in its candidate set; the misspecification alarm’s reference\-free detector \(structured\-vs\-capacity\-matched\-unstructured floor, inverting atβ≥0\.25\\beta\\geq 0\.25with no ground truth\) is the fallible diagnostic for that presupposition\. Composed \(§[F](https://arxiv.org/html/2610.00280#A6), Fig\.[6](https://arxiv.org/html/2610.00280#A6.F6)\): propose forms→\\tofit under one protocol→\\toguard \(alarm⇒\\Rightarrowinvestigate fit failure and candidate adequacy\)→\\toselect→\\torecover the form and rates\. Every edge is backed by a recorded run: the selector at40/4040/40on the11\-DoF stress ladder, and on the flagship cavity decisive wherever dissipative/driven structure exists \(Δ\\DeltaBIC≥1054\\geq 1054at five seeds; the conservative tie\-break’s five\-seed flip, at2/32/3, is the disclosed boundary, superseding the earlier three\-seed3/33/3\)\. Scope: this is a first version on synthetic worlds with small candidate sets, and loop closure that auto\-proposes new forms is not yet built\.

### Energy\-auxiliary unification: a pre\-registered negative result

The misfit\-residual hypothesis, that the residual conservativeσ^\\hat\{\\sigma\}absorbs one\-step objective error, curable by the unified model’s energy\-trajectory auxiliary \(which reachedσ^cons=1\.27×10−5\\hat\{\\sigma\}\_\{\\rm cons\}=1\.27\\times 10^\{\-5\}on this same cavity\), was pre\-registered and*failed under its pre\-registered thresholds*: with the exact energy meter and the unified model’s window protocol,σ^cons\\hat\{\\sigma\}\_\{\\rm cons\}moved0\.0126→0\.01070\.0126\\to 0\.0107\(1\.2×1\.2\\timesagainst a10×10\\timesbar\), every fit degraded, and the conservative flip widened to750750BIC\. The unified model’s L1\-on\-σ\\sigmawas deliberately omitted \(it would confound the test\); how its10−510^\{\-5\}decomposes among its levers \(L1, energy aux, joint multi\-world training\) is now an open attribution question, stated as such\. The one\-stepσ\\sigmafloor is an objective\-level identifiability boundary\. It suggests a*trusted\-meter principle*: an energy supervisor must be exact or computed from grounded data only, since a learned functional supervising its own trajectories opens a self\-consistency escape instead of constraining the physics\.

### Decomposition of the training procedure

The L1 arm \(λ=10−5\\lambda=10^\{\-5\}, the unified model’s value\): selection*restored*to3/33/3\(conservative→H\\to Hby280280BIC; the paired profile\-likelihood clamp tie\-break, per\-seedHH\-vs\-H\+H\{\+\}port wMSE differences,−2\.0×10−3±4\.4×10−3\-2\.0\\times 10^\{\-3\}\\pm 4\.4\\times 10^\{\-3\}, CI includes00, independently selectsHH\), with the identifiable rate unbiased \(σ^diss=0\.502\\hat\{\\sigma\}\_\{\\rm diss\}=0\.502; identifiability reconfirmed in\-harness\)\. Butσ^cons=0\.0126\\hat\{\\sigma\}\_\{\\rm cons\}=0\.0126is unmoved, failing the pre\-registered eval\-horizon gate \(≤10−4\\leq 10^\{\-4\}\): the L1 stops the port from converting this residual into a*fit advantage*without removing the residual itself\. The pre\-registered verdict for this arm is FAIL on that gate, since selection integrity is repaired while long\-horizon negligibility\-when\-selected is still open\. The second arm \(L1\+\+aux at the loweredW=0\.01W=0\.01\) is strictly worse, since every fit degrades33–4×4\\timesand even the dissipative selection misses \(2/32/3\) withσ^cons\\hat\{\\sigma\}\_\{\\rm cons\}still unmoved, confirming the energy\-aux negative at a tenth of the weight\.

### The residual near\-zero rate

\(a\) The pre\-registeredσ∗\\sigma^\{\*\}prediction, the static regression of the trainedHHcore’s one\-step error onto the port decay direction, gives4\.1×10−44\.1\\times 10^\{\-4\},30×30\\timesbelow the400400\-epoch residual rate: this was not predicted, because the residual rate at protocol budget is not a property of the converged core’s residual field\. \(b\) The misfit ladder*refutes the infeasibility framing we expected*:σ^cons\\hat\{\\sigma\}\_\{\\rm cons\}drains as∼1/epochs\{\\sim\}1/\\text\{epochs\}\(0\.069/0\.0126/0\.0035/0\.000260\.069/0\.0126/0\.0035/0\.00026at40/400/1600/640040/400/1600/6400; log\-log slope−1\.055\-1\.055\), reaching the10−410^\{\-4\}eval\-horizon bound at an extrapolated∼2\.7×104\{\\sim\}2\.7\\times 10^\{4\}epochs, feasible, a transient rather than a wall\. Post hoc synthesis, flagged: the asymptote matchesσ∗\\sigma^\{\*\}’s magnitude, consistent with the residual being a*co\-training transient*that relaxes toward the converged core’s static\-regression floor, explaining at once why the energy aux could not drain it \(wrong lever\), why L1 restores selection without draining \(it suppresses the transient’s fit advantage, not the transient\), and why training alone eventually does\. This also dissolves the unified model’s attribution question: no special lever is needed, only budget\.

### Removing the residual rate with longer training

A pre\-registered continuous run to the budgeted horizon:σ^cons​\(27,500\)=5\.1/4\.1×10−6\\hat\{\\sigma\}\_\{\\rm cons\}\(27\{,\}500\)=5\.1/4\.1\\times 10^\{\-6\}\(2 seeds; bar10−410^\{\-4\}, P\-cap\-1 CONFIRMED, the crossing at8\.08\.0–8\.58\.5k epochs, so the residual\-rate price was∼3×\\sim 3\\timesconservative\), and the measured decay steepens to≈t−2\.5\\approx t^\{\-2\.5\}\(slopes−2\.49/−2\.55\-2\.49/\-2\.55, outside the pre\-registered1/t1/tband, P\-cap\-2 NOT CONFIRMED as pre\-specified; the ladder’s early points were not asymptotic and its exponent is retired\)\. The mechanism’s floor signature held cleanly:σ∗​\(t\)\\sigma^\{\*\}\(t\)stays at\|σ∗\|≤6×10−5\|\\sigma^\{\*\}\|\\leq 6\\times 10^\{\-5\}, essentially zero, from epoch5,0005\{,\}000onward \(one seed’s2,5002\{,\}500checkpoint reads1\.64×10−41\.64\\times 10^\{\-4\}, still12×12\\timesunder the pre\-registered gate\) whileσ^​\(t\)\\hat\{\\sigma\}\(t\)relaxes monotonically down to it, terminating*below*the unified model’s1\.27×10−51\.27\\times 10^\{\-5\}: the question is closed empirically\. Nothing about the conservative cell remains open\.

### Wave 1, the ODE zoo \(8 systems, full pass cards\)

Pre\-registered card variants with the pre\-registered constants; every verdict printed by the harness\. The pendulum PASS \(one\-step competence2\.2×105×2\.2\\times 10^\{5\}\\times; worst energy bound0\.93%0\.93\\%;g^\\hat\{g\}to0\.3%0\.3\\%; geometry refusal test refuses at1\.5×106×1\.5\\times 10^\{6\}\\timeswith a hand\-set truth anchor at0\.03×0\.03\\timesbase\)\. The double pendulum PASSb\(climate 3/3, counterfactual climate 10/10 seeds; one\-step\-competence sprint13\.4×13\.4\\timesunder the10310^\{3\}gate, carried\)\. The RLC circuit PASSb\(σ^=0\.2999\\hat\{\\sigma\}=0\.2999vs0\.3000\.300;ω^2=1\.0004\\hat\{\\omega\}^\{2\}=1\.0004; the counterfactual test6\.5×6\.5\\timescarried\)\. The drag projectile PASS \(clean twice; violator prints DIVERGED and is flagged a fortiori\)\. The spinning top PASS under the gauge\-free gates \(one\-step competence5×1011×5\\times 10^\{11\}\\times; Casimir machine\-exact;κ\\kappa\-differences in\-tol; uniform\-scale counterfactual\)\. Lorenz\-63 PASSb\(climate 3/3, the counterfactual test 10/10; one\-step competence0\.38×0\.38\\timescarried, the split buys the long game\)\. Dry friction PASSb\(guard3\.5×104×3\.5\\times 10^\{4\}\\times;μ^​N\\hat\{\\mu\}Nto0\.012%0\.012\\%; material refusal test passes by truth anchor; counterfactual carried on a stick\-degenerate target world, mechanism recorded\)\. The DC motor PASS \(power balance5\.6×10−175\.6\\times 10^\{\-17\}architectural; saturation refusal test\)\.

### Wave 2, 1D fields \(4 systems; exact\-propagator references\)

All references calibrate at machine precision against analytic mode solutions \(10−1610^\{\-16\}\-level; mass drift4×10−144\\times 10^\{\-14\}\), so gate failures are the model’s, never the integrator’s\. The spring lattice, the heat rod, advection–diffusion, and the plucked string all PASS\. Highlights: the heat rod’s flux\-form refusal test refuses at1\.6×107×1\.6\\times 10^\{7\}\\timeswith the bond\-form mobility oracle separating at7600×7600\\times; advection–diffusion’s mass conservation is architectural \(1\.8×10−141\.8\\times 10^\{\-14\}on model rollouts\) and its two instrument bugs \(a sign convention the model exposed by fitting−w\-wexactly; a projection instrument that conflates transport with decay, replaced by the FFT magnitude\) are on record; the plucked string’s locality refusal test is certified by the curvature\-bond class that represents the biharmonic exactly\.

### The instrument findings

\(1\) the drive\-time bug \(batch\-mean drive times poisoned every driven system while sparing the undriven ones\); \(2\) the Lie–Poissonκ\\kappagauge \(Casimir multiples are flow\-null; measured as5×10115\\times 10^\{11\}one\-step competence beside a6×1086\\times 10^\{8\}axis\-wise counterfactual failure\); \(3\) certified floor\-fitting procedure \(a hand\-set truth floor at2\.8×10−102\.8\\times 10^\{\-10\}vs a fixed\-LR stall at1\.4×10−21\.4\\times 10^\{\-2\}\); \(4\) the amendment\-5 refusal instrument \(within\-class refusal \+ oracle separation or truth anchor; common\-base ratios retired as truncation\-confounded\); \(5\) oracle classes must provably contain their refusal\-test worlds \(pointwise vs flux\-form mobility; pad\-2 stencils vs curvature bonds\); \(6\) refusal\-test worlds can be degenerate \(μs\\mu\_\{s\}at the drive amplitude sticks forever\); \(7\) violator meters need a divergence branch; \(8\) event\-time quantization bounds every instrument near events \(calibration, model substeps, and the one\-step floors of*all*classes including alarm references\); \(9\) reference\-grade thermostats are part of the instrument \(BAOAB; Boris rotations for workless magnetic forces\); \(10\) architectural guarantees relocate violators from the class to the world \(e^<1\\hat\{e\}<1,σ≥0\\sigma\\geq 0\)\.

### The four exceptions

bouncing contact CLOSED \(guard\-first fires vs the soft\-wall base; contact recovered to machine precision:g^=1−3×10−14\\hat\{g\}=1\-3\\times 10^\{\-14\},e^=0\.8−3×10−12\\hat\{e\}=0\.8\-3\\times 10^\{\-12\}, oracle floor10−2410^\{\-24\}\)\. The interacting thermal bath characterized boundary \(all discoveries and both FDT meters pass, incl\. the sector\-asymmetric violator at66%66\\%and the magnetic sector via Boris; stability and equipartition\-B fail through the single quantifiedT^\\hat\{T\}\-bias chain\)\. The shallow\-water dam break closed on the characterized\-limit arm \(guards refuse3636–40×40\\times; the localized port finds the fronts, since localization passes with rich features at the measured cost of theg^\\hat\{g\}discovery gauge; coarse\-stencil limit measured\)\. Bare gravity: the softening ladder is the deliverable, using an adaptive reference with per\-rung numerical calibration\.

### The selection matrix

One menu across every world; the frozen selector in its final form \(structural\-channel BIC \+ L1 \+ drained\-channel collapse \+ CI\-window clamp tie\-break \+ the budget flag\)\. Every declared bracket variant is selected correctly \(Lie–Poisson, metriplectic, skew\-transport, port\-Hamiltonian\); both out\-of\-menu controls fire; bare gravity’s correct\-but\-undrained row self\-flags\. The pre\-registered≥14/15\\geq 14/15gate is*missed*at12/1512/15registry systems: the three misses \(dry friction, bouncing contact, the shallow\-water dam break\) share one mechanism, since floor\-ratio alarms require dense misfit and a competent reference, and impulsive/shocked worlds offer neither; their misspecification was caught by the guard\-first protocol at card level instead\. The division of labor \(alarms for dense misfit, matched\-structure guards for events, budget flags for fake\-conservative calls\) is the matrix’s methodological yield, and the three selector passes with their per\-pass results are all in the record\.

## Appendix NExtended related work and the novelty audit

Table 9:Closest structured methods: what each is supplied, what it learns, what changes at test time, and what is actually guaranteed\. The present work’s advance is the explicit supplied\-versus\-learned ledger with estimation error carried into downstream predictions, not any ingredient individually\.This appendix records the extended related\-work comparison, so the main text’s differentiator claims are checkable line by line\.

### N\.1The nearest neighbors, adversarially read

Dissipative HNNs \(D\-HNN\)\([Sosanya and Greydanus, 2022](https://arxiv.org/html/2610.00280#bib.bib6)\)learn an additive Hamiltonian\-plus\-Rayleigh decomposition and can, on the system they are fit to, exhibit a small dissipative component on conservative data\. This is the phenomenon\-level precedent closest to ourσ→0\\sigma\\to 0result, which we discuss openly here\. The differences that matter are as follows\. D\-HNN fits one system at a time, with no cross\-condition discovery on shared content, no condition\-index protocol, and no label\-free variant\. Its dissipative head is unconstrained in sign, which means it has no second\-law guarantee\. This is precisely the degree of freedom our sign\-free control shows drifting negative on ambiguous data\. D\-HNN does evaluate prediction at unseen friction coefficients, so it is not correct to say it lacks counterfactual evaluation\. What it does not include is the long\-horizon, violator\-control, and label\-free context\-inference program, together with the explicit supplied\-versus\-learned accounting and downstream error propagation that accompany the decomposition here \(Table[9](https://arxiv.org/html/2610.00280#A14.T9)\)\.

Dissipative SymODEN\([Zhong et al\., 2020a](https://arxiv.org/html/2610.00280#bib.bib4)\)andport\-Hamiltonian NNs\([Desai et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib7)\)learn dissipation and control terms with physically\-motivated structure \(PSD damping and port interconnection, respectively\)\. Both share the same single\-system scope as D\-HNN: they perform identification rather than discovery, include no counterfactual criterion, and run no measurement program\.

GENERIC/metriplectic learning\([Hernández et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib8);[Lee et al\., 2021](https://arxiv.org/html/2610.00280#bib.bib9);[Gruber et al\., 2023](https://arxiv.org/html/2610.00280#bib.bib10)\)imposes the same reversible/irreversible skeleton we use, with degeneracy conditions enforced architecturally, the closest structural ancestry for our port\. These works establish relevant structure\-preserving learning methods\. Our comparison concerns the combination of shared\-content rate estimation, supplied\-versus\-learned accounting, and recovery\-linked prediction tests, not absence of individual capabilities from all earlier work\.

### N\.2Adjacent lines

Learned simulators \(GNS, MeshGraphNets, FNO\) supply our scale frontier and the locality lesson; system identification \(SINDy and variants\) supply the clean\-data competitor we position against; DeLaN and pH system identification supply the robotics\-side friction\-learning precedent \(single\-system, torque\-supervised\); and multi\-environment/context adaptation lines \(e\.g\. CoDA\-style\) share the “shared content, per\-condition parameter” protocol shape without physical structure or behavioural targets\.

### N\.3The central claim, restated

The paper has one central claim, the one its main text is organized around:*within explicitly supplied physical representations, data recover useful constitutive functions and dissipation parameters, and the recovered content, its identifiability limits, and its estimation error are carried into the downstream predictions that use it*\. The cross\-condition dissipation decision, the quadratic\-energy dissipation guarantee, and the behavioural program are supporting evidence for that claim, not a second contribution of record; the guarantee is scoped to the quadratic diagnostic energy \(Prop\.[3](https://arxiv.org/html/2610.00280#Thmproposition3)\), and the label\-free inference carries its measured rate floor into every prediction it feeds\.