When Do Conservation Laws Survive Learned Representations? Certified Horizons for Latent World Models

# When Do Conservation Laws Survive Learned Representations? Certified Horizons for Latent World Models
Source: [https://arxiv.org/html/2606.24945](https://arxiv.org/html/2606.24945)
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###### Abstract

We ask a representation\-learning question about physical world models: when does a conservation law remain*certifiable*after a model learns a latent representation? A certified horizon bounds — in advance, from measurable model defects — how many steps a rollout provably stays on a physical invariant’s level set\. The key design choice is*what*is certified: not a learned latent Hamiltonian or a learned scalar witness \(a model can conserve either while drifting in true energy\), but thedecoded physical invariantobtained by decoding the latent state and evaluating the known invariant\. Around this object we derive shell\-horizon certificates whose budget decomposes into representation, readout, and latent\-dynamics defects, with a monotone*alignment bridge*through which a soft learned witness yields a certified horizon for the decoded invariant, and test them across state, learned\-lift, and pixel observations on conservative systems\. Conservation certificates*can*survive learned representation, but not all geometric priors survive equally: hard canonical symplectic structure yields the longest horizons in known phase coordinates yet does not cross a learned chart, whereas a controlled\-Lipschitz\-aligned soft invariant survives in the learned\-representation settings we test; pixel certification is recovered on a readout\-stable sub\-tube; and the Kepler problem exposes a geometric boundary\. The central object is therefore not a latent Hamiltonian, but a decoded physical invariant whose robustness to representation learning can be measured, certified, and falsified\.

### 1\. Introduction

A world model is only deployable if it can say*when to stop trusting itself*, and modern world models earn their leverage by not predicting in observation space at all: they learn a latent representation and roll forward there\. That is exactly what makes the trust question subtle\. For physical systems there is a natural currency for trust — conserved quantities: if a rollout drifts off the energy shell afternnsteps, thennnis a concrete, checkable horizon\. But conservation is a statement about*physical*coordinates, and a latent world model has, by construction, replaced those coordinates with learned ones\. The question of this paper is therefore a representation\-learning question:*when does a conservation law remain*certifiable*after a model learns a latent representation*— bounded ahead of time from measurable model defects, rather than merely observed after the fact?

Two facts make the question non\-trivial\. First, the dominant guarantees for learned dynamics are*pointwise and spectral*— they bound one\-step error growth through a Jacobian or Koopman spectrum — so they decay geometrically and certify how fast trajectories*separate*, not whether a conserved*structure*is maintained over many steps\. Conservation offers a different, slower currency: a spectral horizon bounds how fast trajectories diverge \(decaying likelog⁡\(1/ϵ\)/λ\\log\(1/\\epsilon\)/\\lambda\), whereas a conservation horizon bounds how long a conserved scalar stays put — often far longer\. Second, and this is the crux, the moment the model is latent one must decide*what*is being conserved\. The learned latent HamiltonianHθH\_\{\\theta\}and a learned scalar witnessCωC\_\{\\omega\}are convenient and architecturally natural, but neither is the physical invariant: a model can conserve its own internal scalar perfectly while the true energy drifts\. Representation learning does not merely complicate certification — it creates an*object problem*about what the certificate is attached to in the first place\. Latent world models may preserve*something*, but unless that something aligns with a decoded physical invariant, it is not a certificate\.

Our answer fixes the certified object to thedecoded physical invariantH⋆​\(Π​Dψ​z\)H^\{\\star\}\(\\Pi D\_\{\\psi\}z\): decode the latent state \(DψD\_\{\\psi\}\), read out physical coordinates \(Π\\Pi\), and evaluate the*known*physical invariantH⋆H^\{\\star\}\. Concretely, for a pendulum observed as pixels,H⋆H^\{\\star\}is the true mechanical energy: the model never receives it during training or rollout, and we evaluate it only after decoding the latent state and reading out\(q,p\)\(q,p\)— so the certificate measures whether*physical*energy, not a model\-internal surrogate, stays on its shell\. Figure 1 makes this the spine of the paper —HθH\_\{\\theta\}andCωC\_\{\\omega\}are witnesses/proxies on a side branch with no path to the certified object; certificates are evaluated only where the state pipeline and the latent\-decoded pipeline converge onH⋆​\(Π​Dψ​z\)H^\{\\star\}\(\\Pi D\_\{\\psi\}z\)\. Two principles then govern every claim below: the certified object is*always*the decoded physical invariant, and every success or failure is attributed honestly along a state→\\tolift→\\topixel ladder of increasing representational distance from physical coordinates\.

![Certificates are evaluated on the decoded physical invariant Hˆ\star(\Pi D_\psi z), where the state pipeline and the latent-decoded pipeline converge --- not on the learned latent Hamiltonian H_\theta or the witness C_\omega, drawn as a demoted side branch with no path to the certified object.](https://arxiv.org/html/2606.24945v1/x1.png)Figure 1:Certificates are evaluated on the decoded physical invariantH⋆​\(Π​Dψ​z\)H^\{\\star\}\(\\Pi D\_\{\\psi\}z\), where the state pipeline and the latent\-decoded pipeline converge — not on the learned latent HamiltonianHθH\_\{\\theta\}or the witnessCωC\_\{\\omega\}, drawn as a demoted side branch with no path to the certified object\.With the object fixed, the scientific content becomes a question about*representation robustness*: which structural priors let a conservation certificate survive the move from physical state to a learned chart? We state the thesis once:*conservation\-aligned certificates can survive learned representation, but not all geometric priors survive equally\.*Hard canonical symplectic structure — the default inductive bias of Hamiltonian world models — works in known phase coordinates but does not automatically cross a learned chart, because symplecticity constrains the*form*of the flow \(it preserves a symplectic form\), not*which*scalar equals physical energy\. A*soft*learned invariant, trained to be conserved in the latent space and then*aligned*to physical energy through a controlled\-Lipschitz bridge, is more robust to exactly the representation change that defeats the hard prior\. Pixel observation adds an orthogonal*perceptual*bottleneck — a decoder/readout sits between latent and physical coordinates — and a singular system \(the Kepler two\-body problem\) exposes a purely geometric limit to joint\-invariant certification\. The practical reading is conditional, not dismissive: certify in coordinates where the latent symplectic form is the physical one, or pair the hard structure with an explicit alignment of the conserved scalar — otherwise the prior protects a*shadow*quantity that has drifted fromH⋆H^\{\\star\}once the chart is learned\.

We test this across three representations on conservative systems \(pendulum, harmonic oscillator, Kepler two\-body\), holding the certified object fixed and varying only what the model sees\. Four findings, attributed in full in §5: \(i\) in known canonical state, hard symplectic structure is the clear winner — on Kepler the joint\-invariant shell is non\-vacuous in3/33/3seeds for symplectic versus0/30/3for plain, a∼7\.6×\\sim 7\.6\\timessmaller per\-step invariant drift; \(ii\) under a learned lift the hard prior collapses to the unconstrained baseline while the soft invariant survives, with a structure\-gain ratio≈1\.30\\approx 1\.30andP​\(soft\>plain\)=0\.88P\(\\text\{soft\}\>\\text\{plain\}\)=0\.88over eight seeds; \(iii\) under pixels the stabilized soft witness recovers a non\-vacuous decoded\-energy certificate on a*readout\-stable sub\-tube*— beating plain in8/108/10seeds atϵ=2\.0\\epsilon=2\.0, non\-vacuous in9/109/10\(plain4/104/10\), alignment\-positive in10/1010/10, excluded fraction≤2\.3%\\leq 2\.3\\%; and \(iv\) the Kepler*lift*hits an intrinsic near\-periapsis stiffness floor that survives capacity scaling and every legal metric re\-charting we tried — a clean negative\-with\-mechanism, not a structure failure\.

Contributions\.1\.A representation\-robustness framing for conservation certificates\.We fix the certified object to the decoded physical invariantH⋆​\(Π​Dψ​z\)H^\{\\star\}\(\\Pi D\_\{\\psi\}z\)— notHθH\_\{\\theta\}orCωC\_\{\\omega\}— and study when a certificate built on it survives a learned chart \(Figure 1; §3–4\)\. 2\.Certificate theory\(§3\): a unified family of decoded\-invariant horizon bounds — a state shell, a latent decoded\-energy shell with an explicit three\-way error budget, a controlled\-Lipschitz*alignment bridge*that bridges a soft witness to the certified decoded invariant, and a vector joint\-invariant shell\. 3\.A hard\-vs\-soft survival result under learned representation\(§5\): hard symplectic wins in canonical coordinates \(3/33/3vs0/30/3\) but collapses across a learned lift, while a controlled\-Lipschitz\-aligned soft invariant survives \(≈1\.30\\approx 1\.30,P=0\.88P=0\.88\);*monotonicity alone is insufficient*\. 4\.Two honest representation\-limit results: a pixel certificate recovered only on a readout\-stable sub\-tube \(excluded≤2\.3%\\leq 2\.3\\%\), and a Kepler\-lift geometric boundary robust to capacity and legal re\-charting\.

### 2\. Related work

These four lines each supply useful structure for*learning*dynamics; our contribution is orthogonal — we ask which physical invariant is actually*certified*after encoding and decoding, and how hard versus soft conservation priors behave under a learned representation\. \(We do not claim prior lines ignore conservation, that symplectic models are useless, or that we are first to use invariants\.\)

Structure\-preserving and invariant learning\.Two related lines build conservation into learned dynamics\. Hamiltonian/symplectic models impose phase\-space structure: Hamiltonian Neural Networks\[Greydanus et al\.,[2019](https://arxiv.org/html/2606.24945#bib.bib4)\], Hamiltonian Generative Networks\[Toth et al\.,[2020](https://arxiv.org/html/2606.24945#bib.bib11)\]\(which infer a latent phase space from images\), SympNets\[Jin et al\.,[2020](https://arxiv.org/html/2606.24945#bib.bib6)\], and the geometric\-numerical\-integration / backward\-error tradition\[Hairer et al\.,[2006](https://arxiv.org/html/2606.24945#bib.bib5), Reich,[1999](https://arxiv.org/html/2606.24945#bib.bib9)\]that explains symplectic integrators’ long\-time energy behavior\. A second line learns invariants directly: meta\-learned conserved quantities\[Alet et al\.,[2021](https://arxiv.org/html/2606.24945#bib.bib1)\], data\-driven conservation\-law discovery\[Kaiser et al\.,[2018](https://arxiv.org/html/2606.24945#bib.bib7)\], and Koopman\-operator methods whose unit\-eigenvalue eigenfunctions express conserved observables\[Lusch et al\.,[2018](https://arxiv.org/html/2606.24945#bib.bib8)\]— our soft “B\-self” witnessCωC\_\{\\omega\}belongs here\.*Distinction\.*These methods learn or impose structure to improve the*model*; they do not ask what is*certified*once the model operates on a learned representation\. Our certified object is the decoded physical invariantH⋆​\(Π​Dψ​z\)H^\{\\star\}\(\\Pi D\_\{\\psi\}z\), notHθH\_\{\\theta\}or a learned witness; and a learned invariant is awitness, not a certificateunless aligned to the decoded invariant through a bi\-Lipschitz bridge \(§3\.3\), where monotonicity alone is insufficient\. \(HGN is the closest structural template; we treat “learning a latent phase space from pixels” as established prior art\.\)

Equivariant / geometric deep learning\.A second line imposes symmetry through equivariant networks\[Satorras et al\.,[2021](https://arxiv.org/html/2606.24945#bib.bib10), Cohen and Welling,[2016](https://arxiv.org/html/2606.24945#bib.bib3)\]\.*Distinction\.*Our focus is not generic equivariance but conservation\-aligned*certificate objects*under learned latent representations; equivariance constrains how a representation transforms, whereas we ask which decoded physical invariant admits a horizon certificate after encoding/decoding\. \(A separate spacetime\-equivariant world\-model direction is related but not central here; this line is positioning only\.\)

Certified / long\-horizon world models\.A third line bounds how far a learned model can be trusted: spectral and Lyapunov\-style certificates bound one\-step error growth; conformal and reachability methods calibrate prediction sets or safe tubes\[Angelopoulos and Bates,[2021](https://arxiv.org/html/2606.24945#bib.bib2)\]; and a recent line establishes spectral certified horizonsTj​\(ϵ\)∼log⁡\(1/ϵ\)/λjT\_\{j\}\(\\epsilon\)\\sim\\log\(1/\\epsilon\)/\\lambda\_\{j\}for equivariant world models\[Wang,[2026](https://arxiv.org/html/2606.24945#bib.bib12)\]\.*Distinction\.*Existing certificates are largely*pointwise/spectral*and decay geometrically; this paper buildsconservation\-alignedhorizons tied to a decoded physical invariant, whose budget is a per\-step invariant drift, not a Jacobian spectrum \(the spectral line is the complementaryTstatespecT\_\{\\rm state\}^\{\\rm spec\}baseline\)\.

> Positioning\.The novelty is not “using Hamiltonian priors”; it iscertifying decoded physical invariants through learned representations, and showing that hard canonical symplectic structure is coordinate\-sensitive \(it wins in known phase coordinates but does not automatically survive a learned chart\) while a controlled\-Lipschitz\-aligned soft invariant does\.

### 3\. Conservation\-aligned certificates

#### 3\.1 Setup

LetH⋆H^\{\\star\}be a known physical invariant of a continuous system,Σc=\{x:H⋆​\(x\)=c\}\\Sigma\_\{c\}=\\\{x:H^\{\\star\}\(x\)=c\\\}an energy shell, andKc,ρK\_\{c,\\rho\}a tube around it\. The certificate is the largest number of model steps for which a rollout provably stays within toleranceϵ\\epsilonof the shell\. Every bound below has the same shape — a tolerance budgetmc,ρ​ϵm\_\{c,\\rho\}\\,\\epsilonminus an initialization defect, divided by a per\-step drift — and differs only in*which*defects enter as the representation becomes more abstract\. We writemc,ρ=infx∈Kc,ρ∥∇H⋆​\(x\)∥\>0m\_\{c,\\rho\}=\\inf\_\{x\\in K\_\{c,\\rho\}\}\\lVert\\nabla H^\{\\star\}\(x\)\\rVert\>0for the shell’s sensitivity and use⌊⋅⌋\\lfloor\\cdot\\rfloorsince horizons are integer step counts\.

#### 3\.2 State and latent decoded\-energy shells \(Propositions A, B\)

Proposition A \(state shell\)\.With one\-step state defectδstate=supK\|H⋆​\(fθ​x\)−H⋆​\(x\)\|\\delta\_\{\\rm state\}=\\sup\_\{K\}\\lvert H^\{\\star\}\(f\_\{\\theta\}x\)\-H^\{\\star\}\(x\)\\rvertand initialization defectϵ0state\\epsilon\_\{0\}^\{\\rm state\}, telescoping and the mean\-value theorem give\|H⋆​\(xn\)−H⋆​\(x0\)\|≤n​δstate\\lvert H^\{\\star\}\(x\_\{n\}\)\-H^\{\\star\}\(x\_\{0\}\)\\rvert\\leq n\\,\\delta\_\{\\rm state\}and

Tshellstate=⌊mc,ρ​ϵ−ϵ0stateδstate⌋\.T\_\{\\rm shell\}^\{\\rm state\}=\\left\\lfloor\\frac\{m\_\{c,\\rho\}\\,\\epsilon\-\\epsilon\_\{0\}^\{\\rm state\}\}\{\\delta\_\{\\rm state\}\}\\right\\rfloor\.
Proposition B \(latent decoded\-energy shell\)\.Forx^n=Π​Dψ​Fθn​Eϕ​\(o0\)\\hat\{x\}\_\{n\}=\\Pi D\_\{\\psi\}F\_\{\\theta\}^\{\\,n\}E\_\{\\phi\}\(o\_\{0\}\), with tube decoded\-energy driftδphystube=supz∈Kz\|H⋆​\(Π​Dψ​Fθ​z\)−H⋆​\(Π​Dψ​z\)\|\\delta\_\{\\rm phys\}^\{\\rm tube\}=\\sup\_\{z\\in K\_\{z\}\}\\lvert H^\{\\star\}\(\\Pi D\_\{\\psi\}F\_\{\\theta\}z\)\-H^\{\\star\}\(\\Pi D\_\{\\psi\}z\)\\rvertand initial decoded defectϵ0=\|H⋆​\(Π​Dψ​Eϕ​o0\)−H⋆​\(x0\)\|\\epsilon\_\{0\}=\\lvert H^\{\\star\}\(\\Pi D\_\{\\psi\}E\_\{\\phi\}o\_\{0\}\)\-H^\{\\star\}\(x\_\{0\}\)\\rvert,

\|H⋆​\(x^n\)−H⋆​\(x0\)\|≤ϵ0\+ϵΠ,num\+n​δphystube,Tshelllatent=⌊mc,ρ​ϵ−ϵ0−ϵΠ,numδphystube⌋\.\\lvert H^\{\\star\}\(\\hat\{x\}\_\{n\}\)\-H^\{\\star\}\(x\_\{0\}\)\\rvert\\leq\\epsilon\_\{0\}\+\\epsilon\_\{\\Pi,\\rm num\}\+n\\,\\delta\_\{\\rm phys\}^\{\\rm tube\},\\qquad T\_\{\\rm shell\}^\{\\rm latent\}=\\left\\lfloor\\frac\{m\_\{c,\\rho\}\\,\\epsilon\-\\epsilon\_\{0\}\-\\epsilon\_\{\\Pi,\\rm num\}\}\{\\delta\_\{\\rm phys\}^\{\\rm tube\}\}\\right\\rfloor\.The budget splits three ways: aninitial embedding errorϵ0\\epsilon\_\{0\}\(representation\), adecoder/readout errorϵΠ,num\\epsilon\_\{\\Pi,\\rm num\}\(readout\), and alatent\-dynamics driftδphystube\\delta\_\{\\rm phys\}^\{\\rm tube\}\(dynamics\)\. In the lift settingϵΠ,num≈0\\epsilon\_\{\\Pi,\\rm num\}\\approx 0and the certificate is dynamics\-limited; in pixels the decoder/readout term can dominate \(§5\.3\); in the Kepler lift the embedding termϵ0\\epsilon\_\{0\}dominates \(§5\.4\)\. Full proofs are in Appendix A\.

#### 3\.3 The alignment bridge \(Proposition C\)

A soft witnessCωC\_\{\\omega\}is not a priori the physical energy; it must be*aligned*\. LetδC=sup\|Cω​\(F​z\)−Cω​\(z\)\|\\delta\_\{C\}=\\sup\\lvert C\_\{\\omega\}\(Fz\)\-C\_\{\\omega\}\(z\)\\rvert, letgCg\_\{C\}be anLgL\_\{g\}\-Lipschitz calibration, and define the decoder\-side defectηD=supz\|Ψ​\(z\)−gC​\(Cω​z\)\|\\eta\_\{D\}=\\sup\_\{z\}\\lvert\\Psi\(z\)\-g\_\{C\}\(C\_\{\\omega\}z\)\\rvertwhereΨ​\(z\)=H⋆​\(Π​Dψ​z\)\\Psi\(z\)=H^\{\\star\}\(\\Pi D\_\{\\psi\}z\)\. The decoder\-side bound \(C1\) reads

\|Ψ​\(Fn​z\)−Ψ​\(z\)\|≤2​ηD\+Lg​n​δC,Talign=⌊mc,ρ​ϵ−2​ηDLg​δC⌋,\\lvert\\Psi\(F^\{n\}z\)\-\\Psi\(z\)\\rvert\\leq 2\\eta\_\{D\}\+L\_\{g\}\\,n\\,\\delta\_\{C\},\\qquad T\_\{\\rm align\}=\\left\\lfloor\\frac\{m\_\{c,\\rho\}\\,\\epsilon\-2\\eta\_\{D\}\}\{L\_\{g\}\\,\\delta\_\{C\}\}\\right\\rfloor,with a physical\-initial variant \(C2\) absorbing an encoder\-side defect\. The load\-bearing condition isbi\-Lipschitzcalibration,0<κ≤gC′≤Lg0<\\kappa\\leq g\_\{C\}^\{\\prime\}\\leq L\_\{g\}: only then isCωC\_\{\\omega\}quantitatively comparable to the decoded\-energy level sets\. This upgrades the soft witness from a regularizer to a theorem\-shaped*bridge*to the decoded invariant, and it is why affine is too rigid and*uncontrolled*monotonicity is too loose\. We stress:gC∘Cωg\_\{C\}\\circ C\_\{\\omega\}is the*bridge*\(a witness alignment\); the certified object remainsΨ\\Psi\.

#### 3\.4 The joint\-invariant shell

For systems with more than one conserved quantity we certify the*vector*invariant\. For Kepler we use the normalizedI~=\(H~,L~\)\\tilde\{I\}=\(\\tilde\{H\},\\tilde\{L\}\)\(energy and angular momentum, each standardized\)\. With per\-step driftδI=sup∥I~​\(F​z\)−I~​\(z\)∥\\delta\_\{I\}=\\sup\\lVert\\tilde\{I\}\(Fz\)\-\\tilde\{I\}\(z\)\\rVertand worst\-case sensitivityσ∗=infx∈Kσmin​\(D​I~​\(x\)\)\\sigma\_\{\*\}=\\inf\_\{x\\in K\}\\sigma\_\{\\min\}\(D\\tilde\{I\}\(x\)\)— guarded by a rank conditionrank⁡D​I~=2\\operatorname\{rank\}D\\tilde\{I\}=2on the tube —

Tjoint​\(ϵ\)=⌊σ∗​ϵ−ϵ0δI⌋\.T\_\{\\rm joint\}\(\\epsilon\)=\\left\\lfloor\\frac\{\\sigma\_\{\*\}\\,\\epsilon\-\\epsilon\_\{0\}\}\{\\delta\_\{I\}\}\\right\\rfloor\.The rank/σ∗\\sigma\_\{\*\}guard later makes a metric re\-charting*legal*: a chart cannot buy horizon by collapsingσ∗\\sigma\_\{\*\}\(§5\.4\)\.

#### 3\.5 Conditional outlook \(D–F\)

Backward\-error / shadow\-Hamiltonian alignment \(D\), near\-integrable action drift \(E\), and shadowing \(F\) are stated as program\-level / conditional theorems \(Appendix A\.5\), not load\-bearing for the empirical claims here\. Proposition D supplies the discussion’s intuition:*symplecticity constrains the geometry of the flow, not which scalar equals physical energy*, so a symplectic learned map need not preserveH⋆H^\{\\star\}under an arbitrary chart\.

### 4\. Methods

This section says, concretely, what is trained and how a certificate is computed from it\. Nothing here uses the ground\-truth invariant as a training signal for the soft witness;H⋆H^\{\\star\}enters only at*evaluation*\.

#### 4\.1 Systems and data

We instantiate the framework on three conservative systems: the simplependulum, theharmonic oscillator, and theKepler two\-bodyproblem — the last being the only*singular*system \(an inverse\-square potential whose invariant stiffens near periapsis\), which is what later exposes a geometric boundary \(§5\.4\)\. For each system the physical invariantH⋆H^\{\\star\}\(and, for Kepler, the angular momentumLL\) is known in closed form and is used only for evaluation\. Trajectories are integrated from initial conditions sampled across a band of energy shells\. To keep the certifier honest we use disjoint seed triples per run — a training seed, an evaluation seed, and a separate calibration seed for the alignment audit — so the data a certificate is read on is never the data it was fit on\. Runs use88seeds \(alignment bridge\),1010seeds \(pixel recovery\), and33seeds \(Kepler state/lift\), with deterministic kernels and pinned threads\. These systems are deliberately small: a known invariant and a clean readout are what let us isolate the certificate*object*and the representation gap exactly\. Scaling to realistic systems without known readouts is not the claim of this paper — exact invariants and readouts are precisely what make a certified\-object claim testable in the first place\.

#### 4\.2 World\-model pipeline

We use the standard latent\-rollout pipeline\. An observationoois encoded to a latentz=Eϕ​\(o\)z=E\_\{\\phi\}\(o\), advanced by a learned latent mapFθF\_\{\\theta\}, decoded byDψD\_\{\\psi\}, and read out to physical coordinates byΠ\\Pi; annn\-step rollout decodes to an observationo^n=Dψ​Fθn​Eϕ​\(o0\)\\hat\{o\}\_\{n\}=D\_\{\\psi\}F\_\{\\theta\}^\{\\,n\}E\_\{\\phi\}\(o\_\{0\}\)and reads out to the physical state

x^n=Π​o^n=Π​Dψ​Fθn​Eϕ​\(o0\)\.\\hat\{x\}\_\{n\}\\;=\\;\\Pi\\,\\hat\{o\}\_\{n\}\\;=\\;\\Pi\\,D\_\{\\psi\}\\,F\_\{\\theta\}^\{\\,n\}\\,E\_\{\\phi\}\(o\_\{0\}\)\.The three representations differ only in what plays the role ofooand how much of the pipeline is learned\. Instatethe model is handed canonical coordinates\(q,p\)\(q,p\)andΠ\\Piis the identity, so the readout term vanishes \(ϵΠ,num=0\\epsilon\_\{\\Pi,\\rm num\}=0\)\. Inliftan encoder learns a latent chart from low\-dimensional observations and a clean readout recovers\(q,p\)\(q,p\), soϵΠ,num≈0\\epsilon\_\{\\Pi,\\rm num\}\\approx 0and the certificate is dynamics\-limited\. Inpixelthe observation is a high\-dimensional image andDψ,ΠD\_\{\\psi\},\\Piform a learned decoder/readout that can be the dominant error source, soϵΠ,num\\epsilon\_\{\\Pi,\\rm num\}may dominate the budget\. Sliding the model along state→\\tolift→\\topixel while holding the*certified object*fixed is the representation\-robustness axis of the paper\.

#### 4\.3 The certified object

The object is the decoded physical invariantΨ​\(z\)=H⋆​\(Π​Dψ​z\)\\Psi\(z\)=H^\{\\star\}\(\\Pi D\_\{\\psi\}z\), and, for multi\-invariant systems, the decoded*vector*invariantΨI​\(z\)=I~​\(Π​Dψ​z\)\\Psi\_\{I\}\(z\)=\\tilde\{I\}\(\\Pi D\_\{\\psi\}z\)\(for Kepler, the standardized\(H~,L~\)\(\\tilde\{H\},\\tilde\{L\}\)\)\. We never certify the learned latent HamiltonianHθH\_\{\\theta\}or a learned scalar witnessCωC\_\{\\omega\}: both are evaluated only as*witnesses*, and a model can drive either to be conserved whileΨ\\Psidrifts\. Figure 1 draws this explicitly —Hθ,CωH\_\{\\theta\},C\_\{\\omega\}are a side branch with no path to the certified object, which is read off only where the state pipeline and the latent\-decoded pipeline meet onΨ\\Psi\. In practiceΨ\\Psiis evaluated step\-by\-step along the rollout: each latentzn=Fθn​Eϕ​\(o0\)z\_\{n\}=F\_\{\\theta\}^\{\\,n\}E\_\{\\phi\}\(o\_\{0\}\)is decoded and read out to\(q^n,p^n\)=Π​Dψ​zn\(\\hat\{q\}\_\{n\},\\hat\{p\}\_\{n\}\)=\\Pi D\_\{\\psi\}z\_\{n\}andH⋆H^\{\\star\}is evaluated there, so the per\-step decoded\-energy driftδphystube\\delta\_\{\\rm phys\}^\{\\rm tube\}and the initial defectϵ0\\epsilon\_\{0\}are estimated as suprema over the calibration tube — the representation \(ϵ0\\epsilon\_\{0\}\), readout \(ϵΠ,num\\epsilon\_\{\\Pi,\\rm num\}\), and dynamics \(δphystube\\delta\_\{\\rm phys\}^\{\\rm tube\}\) terms of the budget\.

#### 4\.4 Model variants

We compare three structural priors for the latent dynamicsFθF\_\{\\theta\}\. Within each representation level, variants use matched data, step size, latent dimension, training budget, and evaluation pipeline \(state has no image encoder/decoder; lift and pixel do\):

- •Plain\.An unconstrained residual latent map — the no\-structure baseline\.
- •Hard symplectic\.A latent integrator constrained to be symplectic \(a separable Störmer–Verlet step and, separately, a SympNet\), the standard inductive bias of Hamiltonian world models\.
- •Soft \(B\-self\)\.A self\-supervised witnessCωC\_\{\\omega\}— scalar for energy shells, vector\-valued for joint invariants \(e\.g\.Cω∈ℝ2C\_\{\\omega\}\\in\\mathbb\{R\}^\{2\}for Kepler\) — trained to be invariant along the latent rollout, with no access toH⋆H^\{\\star\}\. It can support a certificate for the decoded invariant only through the alignment bridge of §4\.6 — the witness itself is never the certified object\.

A*privileged*controller that uses the true\(H,L\)\(H,L\)appears only as a sanity reference / ablation, never as a method\. Every reported comparison fixes the certified object and varies only the prior and the representation, so a horizon difference is attributable to structure, not to a different evaluation target\.

#### 4\.5 The soft witness and its training

The B\-self witness is the load\-bearing “soft” object, so we are explicit about how it is trained — and about what it is*not*given\. Its objective combines \(i\) aninvariance\-along\-rolloutterm penalizing\|Cω​\(Fθ​z\)−Cω​\(z\)\|\\lvert C\_\{\\omega\}\(F\_\{\\theta\}z\)\-C\_\{\\omega\}\(z\)\\rvert; \(ii\) atrue\-temporal invarianceterm tyingCωC\_\{\\omega\}across real trajectory observations,ℒinv\-data=∑k\|Cω​\(Eϕ​ot\+k\)−Cω​\(Eϕ​ot\)\|2\\mathcal\{L\}\_\{\\text\{inv\-data\}\}=\\sum\_\{k\}\\lvert C\_\{\\omega\}\(E\_\{\\phi\}o\_\{t\+k\}\)\-C\_\{\\omega\}\(E\_\{\\phi\}o\_\{t\}\)\\rvert^\{2\}; and \(iii\) avariance / anti\-collapseterm that prevents the trivial constant solution\. Crucially,H⋆H^\{\\star\}never enters training: the witness is discovered self\-supervised and only*afterwards*aligned to decoded physical energy\. The ablation in §5\.3 confirms which terms are load\-bearing \(removing true\-temporal invariance drops the auditR2R^\{2\}from0\.920\.92to0\.490\.49; removing the anti\-collapse term collapses it to0\.250\.25; a stack decoder still recovers it,0\.950\.95\), and a battery of spurious controls — shuffled, random\-scalar, trajectory\-index, fresh\-head — collapse to negative held\-outR2R^\{2\}, showing the alignment is real rather than a fitting artifact\. The witness is a small scalar \(or, for joint invariants, vector\) head on the latent; the alignment audit fitsgCg\_\{C\}on the held\-out calibration seed and scores the auditR2R^\{2\}ofgC∘Cωg\_\{C\}\\circ C\_\{\\omega\}againstΨ\\Psion two held\-out shell splits \(interleave and extrapolate\-outer\)\.

#### 4\.6 Alignment\-bridge calibration

To turn the witness into a load\-bearing bridge to the certified decoded invariant we fit the calibrationgCg\_\{C\}as acontrolled\-Lipschitz monotone splineon held\-out energy shells, and contrast it with an affine baseline and an uncontrolled isotonic fit\. The certified condition of Proposition C is bi\-Lipschitz,0<κ≤gC′≤Lg0<\\kappa\\leq g\_\{C\}^\{\\prime\}\\leq L\_\{g\}; empirically the spline is genuinely bi\-Lipschitz \(finiteLg=0\.606L\_\{g\}=0\.606, strictly positiveκ=0\.242\\kappa=0\.242\), whereas the isotonic fit has a clipped Lipschitz constant∼102\\sim 10^\{2\}that drivesTalignT\_\{\\rm align\}to zero — so*monotone alone is insufficient*\. The bridgegC∘Cωg\_\{C\}\\circ C\_\{\\omega\}is a witness alignment; the certified object staysΨ=H⋆​\(Π​Dψ​z\)\\Psi=H^\{\\star\}\(\\Pi D\_\{\\psi\}z\), and we report the auditR2R^\{2\}ofgC∘Cωg\_\{C\}\\circ C\_\{\\omega\}againstΨ\\Psias the alignment quality\.

#### 4\.7 Tube\-restricted certification and reporting discipline

Every certificate is restricted to a tubeKc,ρK\_\{c,\\rho\}around the shell, with the certifierTshell=⌊\(mc,ρ​ϵ−ϵ0\)/δcert⌋T\_\{\\rm shell\}=\\lfloor\(m\_\{c,\\rho\}\\epsilon\-\\epsilon\_\{0\}\)/\\delta\_\{\\rm cert\}\\rfloormatching Propositions A/B, andwhen a sub\-tube is used the excluded fraction is reported\. For pixels we apply a frozen*readout\-stable*rule: with a readout\-Lipschitz scores=L⋅rs=L\\cdot r, calibration points withssaboveτs=10⋅medianCAL​\(s\)\\tau\_\{s\}=10\\cdot\\mathrm\{median\}\_\{\\rm CAL\}\(s\)are excluded, yielding a readout\-stable sub\-tube on which the decoded invariant is faithful \(excluded fraction≤2\.3%\\leq 2\.3\\%per seed\)\. For the joint invariant a rank guard \(rank⁡D​I~=2\\operatorname\{rank\}D\\tilde\{I\}=2,σ∗\>0\\sigma\_\{\*\}\>0\) prevents a degenerate direction from inflating the horizon and makes a metric re\-charting*legal*only if it does not buy horizon by collapsingσ∗\\sigma\_\{\*\}\. Two disciplines hold throughout: every certifier rule \(the stable\-tube rule, theσ∗\\sigma\_\{\*\}guard, the legal\-chart anti\-cheat; Appendix B\) isfrozen before the final run, so a certificate cannot be tuned to pass; and pilot/diagnostic quantities \(e\.g\. the four\-way defect decomposition\) show mechanism only and are never quoted as final\-table results\. Horizons are read off a frozenϵ\\epsilon\-grid; a certificate is*non\-vacuous*whenT≥1T\\geq 1for some gridϵ\\epsilon, and we report per\-seed non\-vacuous counts and the seed\-level win rateP​\(TB\>Tplain\)P\(T\_\{B\}\>T\_\{\\rm plain\}\)rather than a single tuned point\.

### 5\. Results

#### 5\.1 The state / lift ladder \(Figure 2\)

Figure 2is the whole\-paper results map: one representation per column, each with its primary result, certificate status, and caveat\.

![State / lift / pixel certificate ladder: hard symplectic works in state (WS3a 3/3 vs 0/3); the soft invariant survives the learned lift while the hard prior does not; pixel certification is recovered on a readout-stable sub-tube.](https://arxiv.org/html/2606.24945v1/x2.png)Figure 2:State / lift / pixel certificate ladder: hard symplectic works in state \(WS3a3/33/3vs0/30/3\); the soft invariant survives the learned lift while the hard prior does not; pixel certification is recovered on a readout\-stable sub\-tube\.*State\.*On state\-space Kepler the joint\-invariant shell behaves as the theory predicts once train/eval eccentricity bands are aligned to the certified tube: hard symplectic Verlet gives a robust, non\-vacuousTjointT\_\{\\rm joint\}—symplectic3/33/3non\-vacuous\(medianδI=0\.064\\delta\_\{I\}=0\.064\) versusplain0/30/3\(δI=0\.489\\delta\_\{I\}=0\.489\), a≈7\.6×\\approx 7\.6\\timessmaller drift\. A privileged controller using the true\(H,L\)\(H,L\)is reported only as a sanity reference\. The horizons are short \(Tjoint​\(ϵ=1\)=1→7T\_\{\\rm joint\}\(\\epsilon\{=\}1\)=1\\to 7overϵ≤4\\epsilon\\leq 4\): the result is a*robust structure separation*, not a long horizon\.

*Lift\.*The move to a learned lift is where the hard\-vs\-soft story appears\. Hard symplectic structure doesnotcross the learned chart — a separable Verlet integrator and a SympNet are statistically indistinguishable from plain — while the soft B\-self invariantdoescross, with a structure\-gain ratioRshell≈1\.30R\_\{\\rm shell\}\\approx 1\.30andP​\(B\>plain\)=0\.88P\(B\>\\mathrm\{plain\}\)=0\.88over88seeds, and a witness\-alignment audit that is positive \(meanR2≈0\.98R^\{2\}\\approx 0\.98, controls≈0\\approx 0\)\. The interpretation is Proposition D in action: symplecticity pins the form of the latent flow but not the identity of the conserved scalar, so under an arbitrary learned chart the symplectic prior no longer protectsH⋆H^\{\\star\}, whereas a witness trained to*be*an invariant can be aligned to it\.

#### 5\.2 The alignment bridge \(Figure 3\)

Figure 3shows how a soft witness becomes a load\-bearing bridge to the certified decoded invariant, and what kind of calibration that requires:controlled\-Lipschitz monotone, not merely monotone\. On the pendulum, a monotone\-spline calibrationgCg\_\{C\}beats an affine baseline on held\-out energy shells with no overfit: held\-outR2=0\.984R^\{2\}=0\.984\(interleave\) and0\.9880\.988\(extrapolate\-outer\) versus affine0\.950/0\.9670\.950/0\.967\. The calibration is genuinely bi\-Lipschitz — finiteLg=0\.606L\_\{g\}=0\.606and strictly positiveκ=0\.242\\kappa=0\.242— exactly Proposition C’s condition, and it yields a positive certified horizonTalign=5\.1T\_\{\\rm align\}=5\.1steps atϵ=1\.0\\epsilon=1\.0, turning on past a critical toleranceϵ⋆≈0\.76\\epsilon\_\{\\star\}\\approx 0\.76\.

![WS2 alignment bridge: monotone alone is insufficient --- a controlled-Lipschitz (\kappa=0.242, L_g=0.606) spline calibration makes the decoded-invariant certificate non-vacuous, whereas uncontrolled isotonic monotonicity drives T_{\rm align}\to0. The certified object remains Hˆ\star(\Pi D_\psi z).](https://arxiv.org/html/2606.24945v1/x3.png)Figure 3:WS2 alignment bridge: monotone alone is insufficient — a controlled\-Lipschitz \(κ=0\.242\\kappa=0\.242,Lg=0\.606L\_\{g\}=0\.606\) spline calibration makes the decoded\-invariant certificate non\-vacuous, whereas uncontrolled isotonic monotonicity drivesTalign→0T\_\{\\rm align\}\\to 0\. The certified object remainsH⋆​\(Π​Dψ​z\)H^\{\\star\}\(\\Pi D\_\{\\psi\}z\)\.Two controls make the claim load\-bearing\. The true witness beats every spurious control by a wide margin: shuffled, random\-scalar, trajectory\-index, and fresh\-head calibrations all collapse to*negative*held\-outR2R^\{2\}\(−0\.40\-0\.40,−0\.47\-0\.47,−68\-68,−0\.22\-0\.22\) — alignment is real, not a fitting artifact\. And the spline\-vs\-isotonic contrast is the lesson: an isotonic monotone fit has a clipped Lipschitz constant∼102\\sim 10^\{2\}and drivesTalignT\_\{\\rm align\}to zero, so*monotone alone is insufficient*\. Throughout,gC∘Cωg\_\{C\}\\circ C\_\{\\omega\}is the bridge; the certified object staysΨ=H⋆​\(Π​Dψ​z\)\\Psi=H^\{\\star\}\(\\Pi D\_\{\\psi\}z\)\.

#### 5\.3 Pixel decoded\-energy recovery \(Figure 4\)

![WS1 pixel decoded-energy recovery (full four-panel). (a) ladder attribution; (b) diagnostic \delta-decomposition (soft-witness \delta_{\rm decoder} 0.163 vs plain 2.16 below the frozen boundary \approx0.22 --- a diagnostic, not a final metric); (c) the 10-seed certificate: 8/10 beat plain at \epsilon=2.0, non-vacuous 9/10 vs plain 4/10, alignment-positive 10/10, \delta_{\rm tube}ˆ{\rm stable} 0.237 vs plain 0.371; (d) ablation: true-temporal invariance and anti-collapse load-bearing, stack decoder still recovers it. A certificate on the readout-stable sub-tube, not the global pixel tube.](https://arxiv.org/html/2606.24945v1/x4.png)Figure 4:WS1 pixel decoded\-energy recovery \(full four\-panel\)\. \(a\) ladder attribution; \(b\) diagnosticδ\\delta\-decomposition \(soft\-witnessδdecoder\\delta\_\{\\rm decoder\}0\.1630\.163vs plain2\.162\.16below the frozen boundary≈0\.22\\approx 0\.22— a diagnostic, not a final metric\); \(c\) the 10\-seed certificate:8/108/10beat plain atϵ=2\.0\\epsilon=2\.0, non\-vacuous9/109/10vs plain4/104/10, alignment\-positive10/1010/10,δtubestable\\delta\_\{\\rm tube\}^\{\\rm stable\}0\.2370\.237vs plain0\.3710\.371; \(d\) ablation: true\-temporal invariance and anti\-collapse load\-bearing, stack decoder still recovers it\. A certificate on the readout\-stable sub\-tube, not the global pixel tube\.Under pixel observation theϵΠ,num\\epsilon\_\{\\Pi,\\rm num\}/ decoder term can dominate, and the question is whether the decoded\-energy certificate survives a*visual*readout\. On the readout\-stable sub\-tube, the stabilized B\-self witness recovers a non\-vacuous pixel decoded\-energy certificate:P​\(TBstable\>Tplainstable\)=8/10P\(T\_\{B\}^\{\\rm stable\}\>T\_\{\\rm plain\}^\{\\rm stable\}\)=8/10atϵ=2\.0\\epsilon=2\.0\(and6/106/10atϵ=1\.5\\epsilon=1\.5\), B\-self non\-vacuous in9/109/10seeds versus plain4/104/10, the alignment audit positive in10/1010/10\(r2r^\{2\}0\.880\.88–0\.960\.96\), with the tube defect compressed to a median0\.2370\.237versus plain0\.3710\.371, and a B\-self excluded fraction of at most2\.3%2\.3\\%per seed\. The honest scope is in the name: this is a certificate on areadout\-stable sub\-tube, not the global pixel tube\. The ablation \(Figure 4d\) shows the drivers are the soft\-invariance losses and the certifier, not the decoder architecture: removing true\-temporal invariance drops the alignment auditR2R^\{2\}to0\.490\.49and removing anti\-collapse collapses it to0\.250\.25, whereas a stack decoder can recover the final certificate under the stabilized recipe \(R2≈0\.95R^\{2\}\\approx 0\.95\)\. So true\-temporal invariance and anti\-collapse are load\-bearing, while the temporal decoder improves decoder consistency and was instrumental in diagnosing the original pixel bottleneck rather than being the sole driver of recovery\.

#### 5\.4 The Kepler geometric boundary \(Appendix Figure 5\)

Kepler pushes the joint invariant\(H~,L~\)\(\\tilde\{H\},\\tilde\{L\}\)through a learned lift on the most nonlinear and only*singular*system; the result is a clean geometric boundary, established by a ladder that rules out each easy explanation \(full detail in Appendix Fig\. 5 and Table C\.5\)\. The state result already passes \(§5\.1: symplectic3/33/3vs plain0/30/3\)\. Under the learned lift the decoded\-I~\\tilde\{I\}reconstruction defect exceeds the budget, and neither capacity nor geometry rescues it: varying the latent widthdz∈\{8,16,32\}d\_\{z\}\\in\\\{8,16,32\\\}moves the floor only1\.21×1\.21\\times, with the residual concentrated at small radius where∂H/∂r∼r−2\\partial H/\\partial r\\sim r^\{\-2\}amplifies small\(q,p\)\(q,p\)errors; and legal\(H,L\)\(H,L\)re\-chartings cut the charted defect∼2\.8×\\sim 2\.8\\timesand even*raise*σ∗\\sigma\_\{\*\}\(0\.097→0\.1200\.097\\to 0\.120,cond≤50\\operatorname\{cond\}\\leq 50\) yet still giveTJ=0T\_\{J\}=0\. The conclusion is sharp and honestly attributed: the Kepler\-lift decoded joint\-invariant certificate is floor\-limited bynear\-periapsisH~\\tilde\{H\}\-stiffness, robust to dynamics, capacity, and legal metric geometry alike\.This is a system\-specific inverse\-square geometric boundary; it is not evidence that conservation\-aligned certificates fail under learned lifts in general, and it does not undercut the pendulum\-lift / pixel positives of §5\.1–5\.3\.

### 6\. Discussion and limitations

We show that physical conservation can be certified through learned representations only when the certified object is the decoded physical invariant — not a latent Hamiltonian or an unaligned learned witness\. The three settings tell one story about the*representation robustness*of certificates\. In known canonical coordinates hard symplectic structure is the clear winner: it buys the longest certified shell because the integrator’s conserved scalar*is*— up to backward error — the physical energy in those coordinates\. Move to a learned chart and that coincidence breaks, and Proposition D makes the reason precise: symplecticity is a constraint on the*form*of the flow \(it preserves a symplectic form\), not a guarantee about*which*function is conserved\. A symplectic learned map can therefore conserve a shadow quantity that has drifted fromH⋆H^\{\\star\}once the coordinates are learned — exactly what the lift nulls show \(Verlet and SympNet collapse to plain\)\. A*soft*invariant, by contrast, is trained to be an invariant of the learned latents and can then be aligned to physical energy through the bridge of §3\.3; geometrically it behaves like a foliation of the latent space by level sets that a monotone calibration can register, and so it is more robust to the representation change that defeats the hard prior\.

This is why the wording is careful: hard symplectic isnotuseless\. It is*coordinate\-sensitive*— it wins where the chart is right \(the longest state\-space horizons\) and degrades where the chart is learned, and it may remain essential for phase fidelity or other geometric observables a scalar invariant does not capture\. The recommendation is therefore not “drop symplectic structure” but “either certify in coordinates where the symplectic form is the physical one, or pair the structure with an explicit alignment of the conserved scalar\.” The pixel result adds an orthogonal axis: even with a working soft witness, visual observation introduces a decoding/readout bottleneck, and certification there is about stabilizing the readout enough to find a sub\-tube on which the decoded invariant is faithful — a perceptual problem, not a structural one\. Kepler closes the arc on the negative side: where the invariant itself is geometrically singular, no amount of representation, capacity, or legal re\-charting removes the near\-periapsis floor\.

Limitations and scope\.The positive certificates are demonstrated on low\-dimensional conservative systems and at modest horizons; the structure separations are statistically robust but the certified horizons themselves are short, and we make no claim about long\-horizon or high\-dimensional rollouts\. All certificates are*tube\-restricted*; the pixel certificate is explicitly on areadout\-stable sub\-tubewith a small but nonzero excluded fraction \(≤2\.3%\\leq 2\.3\\%\), not the global tube\. The soft\-invariant survival result is shown for the lift on the pendulum and for state\-space Kepler; we do not claim it for every system\. The Kepler\-lift result is asystem\-specific geometric boundarydriven by the inverse\-square periapsis singularity, reported with full ladder attribution rather than as a structure failure\. Finally, our contribution is a certified\-object*methodology*and a representation\-robustness study, not a benchmark\-scale vision world model; we make no benchmark\-leaderboard claim and ask to be read as the former\.

Future work is deliberately scoped*out*of this paper\. Engineering an observation/lift representation in which near\-periapsisH~\\tilde\{H\}is not stiff is a separate study; extending the joint\-shell certificate to a non\-singular multi\-invariant system would confirm the boundary is geometric\. The backward\-error, action\-drift, and shadowing results \(Propositions D–F\) remain conditional; making the alignment bridge a coordinate\-free level\-set stability theorem, and pinning down when a symplectic learned map’s shadow Hamiltonian aligns withH⋆∘Π∘DH^\{\\star\}\\circ\\Pi\\circ D, are the open theory questions\.

### References

## Appendix

*Numbers are parsed from frozen tagged artifacts\. Full anonymized configuration dictionaries, run identifiers, and code are provided in the supplementary material\.*

### Appendix A\. Proofs

Throughout,Ψ​\(z\)=H⋆​\(Π​Dψ​z\)\\Psi\(z\)=H^\{\\star\}\(\\Pi D\_\{\\psi\}z\);Kc,ρK\_\{c,\\rho\}is a tube around the shellΣc=\{x:H⋆​\(x\)=c\}\\Sigma\_\{c\}=\\\{x:H^\{\\star\}\(x\)=c\\\}; andmc,ρ=infx∈Kc,ρ∥∇H⋆​\(x\)∥\>0m\_\{c,\\rho\}=\\inf\_\{x\\in K\_\{c,\\rho\}\}\\lVert\\nabla H^\{\\star\}\(x\)\\rVert\>0is the shell sensitivity \(a regular\-level assumption on the tube\)\. A generic distance lemma is used repeatedly: if\|H⋆​\(x\)−c\|≤η\\lvert H^\{\\star\}\(x\)\-c\\rvert\\leq\\etaon the tube thendist​\(x,Σc\)≤η/mc,ρ\\mathrm\{dist\}\(x,\\Sigma\_\{c\}\)\\leq\\eta/m\_\{c,\\rho\}, since∥∇H⋆∥≥mc,ρ\\lVert\\nabla H^\{\\star\}\\rVert\\geq m\_\{c,\\rho\}\(mean value theorem\)\. Horizons are integer step counts, hence the floor\.

A\.1 Proposition A \(state shell\)\.Letδstate=supx∈K\|H⋆​\(fθ​x\)−H⋆​\(x\)\|\\delta\_\{\\rm state\}=\\sup\_\{x\\in K\}\\lvert H^\{\\star\}\(f\_\{\\theta\}x\)\-H^\{\\star\}\(x\)\\rvertandϵ0state=\|H⋆​\(x0\)−c\|\\epsilon\_\{0\}^\{\\rm state\}=\\lvert H^\{\\star\}\(x\_\{0\}\)\-c\\rvert\. Telescoping the one\-step invariant change,

\|H⋆​\(xn\)−H⋆​\(x0\)\|≤∑k=1n\|H⋆​\(fθ​xk−1\)−H⋆​\(xk−1\)\|≤n​δstate\.\\lvert H^\{\\star\}\(x\_\{n\}\)\-H^\{\\star\}\(x\_\{0\}\)\\rvert\\leq\\sum\_\{k=1\}^\{n\}\\lvert H^\{\\star\}\(f\_\{\\theta\}x\_\{k\-1\}\)\-H^\{\\star\}\(x\_\{k\-1\}\)\\rvert\\leq n\\,\\delta\_\{\\rm state\}\.By the distance lemma,dist​\(xn,Σc\)≤\(ϵ0state\+n​δstate\)/mc,ρ\\mathrm\{dist\}\(x\_\{n\},\\Sigma\_\{c\}\)\\leq\(\\epsilon\_\{0\}^\{\\rm state\}\+n\\,\\delta\_\{\\rm state\}\)/m\_\{c,\\rho\}\. Requiring the bound≤ϵ\\leq\\epsilonand solving for the largest integernngivesTshellstate=⌊\(mc,ρ​ϵ−ϵ0state\)/δstate⌋T\_\{\\rm shell\}^\{\\rm state\}=\\lfloor\(m\_\{c,\\rho\}\\epsilon\-\\epsilon\_\{0\}^\{\\rm state\}\)/\\delta\_\{\\rm state\}\\rfloor\.□\\square

A\.2 Proposition B \(latent decoded\-energy shell\)\.Letx^n=Π​Dψ​Fθn​Eϕ​\(o0\)\\hat\{x\}\_\{n\}=\\Pi D\_\{\\psi\}F\_\{\\theta\}^\{\\,n\}E\_\{\\phi\}\(o\_\{0\}\)\. TelescopingΨ\\Psialong the latent rollout bounds the decoded\-energy drift byn​δphystuben\\,\\delta\_\{\\rm phys\}^\{\\rm tube\}, withδphystube=supz∈Kz\|Ψ​\(Fθ​z\)−Ψ​\(z\)\|\\delta\_\{\\rm phys\}^\{\\rm tube\}=\\sup\_\{z\\in K\_\{z\}\}\\lvert\\Psi\(F\_\{\\theta\}z\)\-\\Psi\(z\)\\rvert\. The triangle inequality adds the initial embedding defectϵ0=\|H⋆​\(Π​Dψ​Eϕ​o0\)−H⋆​\(x0\)\|\\epsilon\_\{0\}=\\lvert H^\{\\star\}\(\\Pi D\_\{\\psi\}E\_\{\\phi\}o\_\{0\}\)\-H^\{\\star\}\(x\_\{0\}\)\\rvertand a readout/numerical termϵΠ,num\\epsilon\_\{\\Pi,\\rm num\}\(kept*separate*from the per\-step decoder\-consistency drift\):

\|H⋆​\(x^n\)−H⋆​\(x0\)\|≤ϵ0\+ϵΠ,num\+n​δphystube\.\\lvert H^\{\\star\}\(\\hat\{x\}\_\{n\}\)\-H^\{\\star\}\(x\_\{0\}\)\\rvert\\leq\\epsilon\_\{0\}\+\\epsilon\_\{\\Pi,\\rm num\}\+n\\,\\delta\_\{\\rm phys\}^\{\\rm tube\}\.The distance lemma and solving fornngiveTshelllatent=⌊\(mc,ρ​ϵ−ϵ0−ϵΠ,num\)/δphystube⌋T\_\{\\rm shell\}^\{\\rm latent\}=\\lfloor\(m\_\{c,\\rho\}\\epsilon\-\\epsilon\_\{0\}\-\\epsilon\_\{\\Pi,\\rm num\}\)/\\delta\_\{\\rm phys\}^\{\\rm tube\}\\rfloor\. The three additive terms are the*representation*\(ϵ0\\epsilon\_\{0\}\),*readout*\(ϵΠ,num\\epsilon\_\{\\Pi,\\rm num\}\), and*dynamics*\(δphystube\\delta\_\{\\rm phys\}^\{\\rm tube\}\) budget used throughout\.□\\square

A\.3 Proposition C \(alignment bridge\)\.LetCωC\_\{\\omega\}be a \(scalar or vector\) witness with per\-step driftδC=sup∥Cω​\(F​z\)−Cω​\(z\)∥\\delta\_\{C\}=\\sup\\lVert C\_\{\\omega\}\(Fz\)\-C\_\{\\omega\}\(z\)\\rVert,gCg\_\{C\}anLgL\_\{g\}\-Lipschitz calibration, andηD=supz\|Ψ​\(z\)−gC​\(Cω​z\)\|\\eta\_\{D\}=\\sup\_\{z\}\\lvert\\Psi\(z\)\-g\_\{C\}\(C\_\{\\omega\}z\)\\rvertthe decoder\-side alignment defect\.*\(C1, decoder\-side\.\)*InsertinggC∘Cωg\_\{C\}\\circ C\_\{\\omega\}twice,

\|Ψ​\(Fn​z\)−Ψ​\(z\)\|≤\|Ψ​\(Fn​z\)−gC​\(Cω​Fn​z\)\|⏟≤ηD\+\|gC​\(Cω​Fn​z\)−gC​\(Cω​z\)\|⏟≤Lg​n​δC\+\|gC​\(Cω​z\)−Ψ​\(z\)\|⏟≤ηD,\\lvert\\Psi\(F^\{n\}z\)\-\\Psi\(z\)\\rvert\\leq\\underbrace\{\\lvert\\Psi\(F^\{n\}z\)\-g\_\{C\}\(C\_\{\\omega\}F^\{n\}z\)\\rvert\}\_\{\\leq\\,\\eta\_\{D\}\}\+\\underbrace\{\\lvert g\_\{C\}\(C\_\{\\omega\}F^\{n\}z\)\-g\_\{C\}\(C\_\{\\omega\}z\)\\rvert\}\_\{\\leq\\,L\_\{g\}\\,n\\,\\delta\_\{C\}\}\+\\underbrace\{\\lvert g\_\{C\}\(C\_\{\\omega\}z\)\-\\Psi\(z\)\\rvert\}\_\{\\leq\\,\\eta\_\{D\}\},where the middle term uses Lipschitzness then telescoping\. Hence\|Ψ​\(Fn​z\)−Ψ​\(z\)\|≤2​ηD\+Lg​n​δC\\lvert\\Psi\(F^\{n\}z\)\-\\Psi\(z\)\\rvert\\leq 2\\eta\_\{D\}\+L\_\{g\}\\,n\\,\\delta\_\{C\}andTalign=⌊\(mc,ρ​ϵ−2​ηD\)/\(Lg​δC\)⌋T\_\{\\rm align\}=\\lfloor\(m\_\{c,\\rho\}\\epsilon\-2\\eta\_\{D\}\)/\(L\_\{g\}\\,\\delta\_\{C\}\)\\rfloor\.*\(C2, physical\-initial\.\)*Replacing one decoder\-sideηD\\eta\_\{D\}by an encoder\-side defectηE\\eta\_\{E\}\(comparinggC​\(Cω​Eϕ​o0\)g\_\{C\}\(C\_\{\\omega\}E\_\{\\phi\}o\_\{0\}\)againstH⋆​\(x0\)H^\{\\star\}\(x\_\{0\}\)\) gives the physical\-initial variant withηD\+ηE\\eta\_\{D\}\+\\eta\_\{E\}in place of2​ηD2\\eta\_\{D\}\.*\(Bi\-Lipschitz role\.\)*The condition0<κ≤gC′≤Lg0<\\kappa\\leq g\_\{C\}^\{\\prime\}\\leq L\_\{g\}is load\-bearing:LgL\_\{g\}controls the bound whileκ\>0\\kappa\>0makesgCg\_\{C\}invertible, puttingCωC\_\{\\omega\}’s level sets in bijection with the decoded\-energy level sets — without it, a monotone\-but\-flat calibration satisfies the inequality while certifying nothing \(the isotonic vacuity of §5\.2\)\. Throughout,gC∘Cωg\_\{C\}\\circ C\_\{\\omega\}is a*bridge*; the certified object remainsΨ\\Psi\.□\\square

A\.4 Joint\-invariant shell\.For a vector invariantI~\\tilde\{I\}with per\-step driftδI=sup∥I~​\(F​z\)−I~​\(z\)∥\\delta\_\{I\}=\\sup\\lVert\\tilde\{I\}\(Fz\)\-\\tilde\{I\}\(z\)\\rVert, telescoping gives∥I~​\(zn\)−I~​\(z0\)∥≤ϵ0\+n​δI\\lVert\\tilde\{I\}\(z\_\{n\}\)\-\\tilde\{I\}\(z\_\{0\}\)\\rVert\\leq\\epsilon\_\{0\}\+n\\,\\delta\_\{I\}\. The guardrank⁡D​I~=2\\operatorname\{rank\}D\\tilde\{I\}=2on the tube makesI~\\tilde\{I\}a submersion onto its joint level set, soσ∗=infx∈Kσmin​\(D​I~​\(x\)\)\>0\\sigma\_\{\*\}=\\inf\_\{x\\in K\}\\sigma\_\{\\min\}\(D\\tilde\{I\}\(x\)\)\>0and the joint\-shell distance is controlled by∥drift∥/σ∗\\lVert\\text\{drift\}\\rVert/\\sigma\_\{\*\}\. Solving≤ϵ\\leq\\epsilongivesTjoint=⌊\(σ∗​ϵ−ϵ0\)/δI⌋T\_\{\\rm joint\}=\\lfloor\(\\sigma\_\{\*\}\\epsilon\-\\epsilon\_\{0\}\)/\\delta\_\{I\}\\rfloor\. The guard makes a metric re\-charting*legal*: a chart cannot inflateTjointT\_\{\\rm joint\}by collapsingσ∗\\sigma\_\{\*\}, sinceσ∗J\\sigma\_\{\*\}^\{J\}is recomputed throughD​φD\\varphi\(Appendix B\.3\)\.□\\square

A\.5 Conditional statements \(D–F\)\.Not load\-bearing for the empirical claims\.*\(D, backward error / shadow\.\)*For a near\-identity analytic symplecticFhF\_\{h\}, a modified HamiltonianH~h\(N\)\\widetilde\{H\}\_\{h\}^\{\(N\)\}exists withϕH~h≈Fh\\phi^\{h\}\_\{\\widetilde\{H\}\}\\approx F\_\{h\}toO​\(hN\+1\)O\(h^\{N\+1\}\);*if*sup\|H~h\(N\)−H⋆∘Π∘D\|≤ηBEA\\sup\\lvert\\widetilde\{H\}\_\{h\}^\{\(N\)\}\-H^\{\\star\}\\circ\\Pi\\circ D\\rvert\\leq\\eta\_\{\\rm BEA\}, a decoded\-shell horizon follows \(program\-level\)\.*\(E, action\.\)*Tact​\(η\)T\_\{\\rm act\}\(\\eta\)targets the KAM/Nekhoroshev shape \(a shape prior, not a first\-version theorem\)\.*\(F, shadowing\.\)*Theorem\-faithful only on a uniformly hyperbolic toy; the double\-pendulum case is empirical\-appendix only\. Proposition D supplies the discussion’s intuition:*symplecticity constrains the form of the flow, not which scalar equals physical energy\.*

### Appendix B\. Frozen certifier rules

Each rule is fixed*before*the final run, so a certificate cannot be tuned to pass\.

B\.1 Pixel stable\-tube certifier \(v1\.1\)\.With readout\-Lipschitz scores=L⋅rs=L\\cdot r, exclude calibration points whosesis\_\{i\}exceedsτs=10⋅medianCAL​\(s\)\\tau\_\{s\}=10\\cdot\\mathrm\{median\}\_\{\\rm CAL\}\(s\), giving the readout\-stable sub\-tube\. Frozen: theτs\\tau\_\{s\}rule, theϵ\\epsilon\-grid, the certifierTshell=⌊\(ϵ​m−ϵ0\)/δcert⌋T\_\{\\rm shell\}=\\lfloor\(\\epsilon m\-\\epsilon\_\{0\}\)/\\delta\_\{\\rm cert\}\\rfloor\(matching Prop A/B\), the data splits, and the loss weights\. Coverage: the soft\-witness excluded fraction is≤2\.3%\\leq 2\.3\\%per seed \(median0\.0000\.000, max0\.02340\.0234\), re\-pinnable from the per\-seed certifier JSON; the*plain*variant’s excluded fraction runs higher \(up to0\.0310\.031\), and the certificate uses the soft\-witness coverage\. \(The seed\-0 certifier\-definition figure0\.0080\.008is a different quantity\.\) The certificate is on thisreadout\-stable sub\-tube, not the global pixel tube\.

B\.2 Joint\-shell rank guard\.σ∗=infx∈Kσmin​\(D​I~\)\\sigma\_\{\*\}=\\inf\_\{x\\in K\}\\sigma\_\{\\min\}\(D\\tilde\{I\}\)withrank⁡D​I~=2\\operatorname\{rank\}D\\tilde\{I\}=2on the tube; normalizedI~=\(\(H⋆−μH\)/sH,\(L−μL\)/sL\)\\tilde\{I\}=\\big\(\(H^\{\\star\}\-\\mu\_\{H\}\)/s\_\{H\},\\,\(L\-\\mu\_\{L\}\)/s\_\{L\}\\big\); theI~\\tilde\{I\}\-calibration seed is frozen\.

B\.3 Legal\-chart anti\-cheat\.A re\-chartingφ​\(H,L\)\\varphi\(H,L\)passes only if \(frozen\)TJ\>0T\_\{J\}\>0andσ∗J≥0\.3​σ∗A​0\\sigma\_\{\*\}^\{J\}\\geq 0\.3\\,\\sigma\_\{\*\}^\{A0\}andcond⁡\(D​φ\)≤50\\operatorname\{cond\}\(D\\varphi\)\\leq 50\. Guards: HL\-only inputs; model\-independent;σ∗J\\sigma\_\{\*\}^\{J\}recomputed viaD​φD\\varphi; raw baseline reported side\-by\-side; condition number capped\.

### Appendix C\. Seed\-level tables

C\.1 State Kepler joint shell\(3 seeds\)\.

variantδI\\delta\_\{I\}medianδI\\delta\_\{I\}per\-seednon\-vacuousTjointT\_\{\\rm joint\}atϵ=1\.0/1\.5/2\.0/2\.5/3\.0/4\.0\\epsilon=1\.0/1\.5/2\.0/2\.5/3\.0/4\.0plain0\.4890\.388 / 0\.489 / 0\.7320/30/0/0/0/0/0symplectic0\.0640\.094 / 0\.052 / 0\.0643/31/2/3/4/5/7conservation\_reg \(privileged\)0\.0820\.127 / 0\.061 / 0\.0822/31/2/2/3/4/5C\.2 Lift sweep\(pendulum\)\.

ϵ\\epsilonRshellBR\_\{\\rm shell\}^\{B\}P​\(B\>p\)P\(B\{\>\}p\)RshellsymR\_\{\\rm shell\}^\{\\rm sym\}P​\(s\>p\)P\(s\{\>\}p\)TpT\_\{p\}/TBT\_\{B\}/TsT\_\{s\}0\.301\.410\.881\.030\.384\.5/6\.2/4\.60\.501\.340\.881\.090\.509\.5/12\.5/10\.10\.701\.320\.881\.090\.5014\.5/19\.0/15\.21\.001\.300\.881\.100\.3822\.2/28\.8/23\.2Verdict: structural symplectic doesnotsurvive the lift \(sympl/plain∼1\.0\\sim\\\!1\.0\); the soft witnessdoes\(∼1\.3×\\sim\\\!1\.3\\timeshorizon,P=0\.88P\{=\}0\.88, 8 seeds\)\. Thewitness\-establishment audit\(10 seeds —*distinct*from the WS2 calibration audit of C\.3\) gives meanR2=0\.982R^\{2\}=0\.982\(per\-seed0\.9250\.925–0\.9990\.999\),\|ρ\|\\lvert\\rho\\rvert0\.9950\.995–0\.9990\.999, with shuffled/random controls≈0\\approx 0; the WS2 fidelity gate reuses seed\-0 of this audit \(R2=0\.99269R^\{2\}=0\.99269\)\.

C\.3 WS2 alignment\-bridge calibration\(8 seeds, interleave split — the*calibration*audit, not C\.2’s witness\-establishment audit\)\.

seedaffineisotonicsplineLgL\_\{g\}κ\\kappaTalignT\_\{\\rm align\}\(spline\)00\.9890\.9230\.9800\.5090\.178710\.9950\.9840\.9910\.4730\.177720\.9600\.9310\.9880\.5990\.233130\.7800\.9240\.9610\.9170\.125040\.9040\.9710\.9730\.7650\.168050\.9940\.9240\.9860\.5660\.366160\.9990\.9490\.9990\.4570\.3641870\.9840\.9590\.9970\.5620\.3217mean0\.9500\.9460\.9840\.6060\.2425\.1Extrapolate\-outer: spline0\.9880\.988/ affine0\.9670\.967/ isotonic0\.9440\.944\.Talign​\(ϵ\)T\_\{\\rm align\}\(\\epsilon\)sweep \(interleave mean\) overϵ∈\{0\.2,0\.3,0\.5,0\.7,1\.0,1\.5,2\.0\}\\epsilon\\in\\\{0\.2,0\.3,0\.5,0\.7,1\.0,1\.5,2\.0\\\}: spline0/0/0\.6/1\.5/5\.1/13\.0/20\.90/0/0\.6/1\.5/\\mathbf\{5\.1\}/13\.0/20\.9; isotonic0/0/0/0/0/0\.4/1\.10/0/0/0/0/0\.4/1\.1; affine0/0/1\.1/4\.4/10\.1/19\.6/29\.50/0/1\.1/4\.4/10\.1/19\.6/29\.5\.ϵ⋆\\epsilon\_\{\\star\}: spline0\.7550\.755, isotonic1\.1881\.188, affine0\.9620\.962\. Controls \(held\-outR2R^\{2\}\): shuffled−0\.40\-0\.40, random−0\.47\-0\.47, trajectory\-index−68\-68, fresh\-head−0\.22\-0\.22\. Fidelity0\.992692980\.99269298vs0\.992693010\.99269301\. Affine’s largerTalignT\_\{\\rm align\}comes from a small constantLgL\_\{g\}on a worse fit; the certificate\-grade contrast is spline \(controlledLgL\_\{g\}\) vs isotonic \(uncontrolled, vacuous\)\.

C\.4 Pixel recovery\(10 seeds; regenerated from per\-seed JSONs\)\.

seedTB​\(1\.5\)T\_\{B\}\(1\.5\)Tp​\(1\.5\)T\_\{p\}\(1\.5\)TB​\(2\.0\)T\_\{B\}\(2\.0\)Tp​\(2\.0\)T\_\{p\}\(2\.0\)Bδtubestable\\delta\_\{\\rm tube\}^\{\\rm stable\}plδtubestable\\delta\_\{\\rm tube\}^\{\\rm stable\}ηC\\eta\_\{C\}r2r^\{2\}010210\.2860\.3750\.962110200\.2260\.2860\.884200100\.2300\.3890\.928321310\.1850\.3660\.938400000\.2500\.2590\.913510200\.3050\.3810\.905610200\.2200\.3130\.900710200\.2310\.5860\.893801110\.2430\.2410\.951911210\.2800\.4220\.918Aggregates:P​\(TB\>Tp\)=8/10P\(T\_\{B\}\>T\_\{p\}\)=8/10@ϵ=2\.0\\epsilon\{=\}2\.0\(6/106/10@1\.51\.5;P​\(B≥p\)=10/10P\(B\\geq p\)=10/10\); B non\-vacuous \(ϵ≥1\.5\\epsilon\\geq 1\.5\)9/109/10\(plain4/104/10\);ηC\\eta\_\{C\}positive10/1010/10;δtubestable\\delta\_\{\\rm tube\}^\{\\rm stable\}B median0\.2370\.237vs plain0\.3710\.371\. Diagnostic four\-way decomposition \(not a final metric\): plainδdecoder\\delta\_\{\\rm decoder\}2\.162\.16/ soft\-witness0\.1630\.163\. AblationηC\\eta\_\{C\}R2R^\{2\}\(3 seeds\): full0\.920\.92/−\-true\-temporal0\.490\.49/−\-variance0\.250\.25/ stack decoder0\.950\.95\.

C\.5 Kepler\-lift rescue ladder\.Tier\-2 \(q,pq,p\-only weighting \+ channel norm\), per\-seedϵ0,Hq​95\\epsilon\_\{0,H\}^\{q95\}4\.63/5\.90/4\.884\.63/5\.90/4\.88\(median4\.884\.88;\(q,p\)\(q,p\)recon∼0\.066\\sim 0\.066\)\. Tier\-2b capacity:dz∈\{8,16,32\}→ϵ0,Hq​95d\_\{z\}\\in\\\{8,16,32\\\}\\to\\epsilon\_\{0,H\}^\{q95\}2\.82/2\.49/2\.322\.82/2\.49/2\.32\(1\.21×1\.21\\times\); periapsis bin \(dz=32d\_\{z\}\{=\}32\)3\.873\.87vs1\.22/1\.171\.22/1\.17at mid/large radius\. Legal charts: baselineϵ0,J\\epsilon\_\{0,J\}11\.6911\.69/log⁡\(−H\)\\log\(\-H\)5\.635\.63/orbital\(a,λ\)\(a,\\lambda\)4\.214\.21\(2\.8×2\.8\\times\), withσ∗\\sigma\_\{\*\}0\.097→0\.1200\.097\\to 0\.120,cond≤50\\operatorname\{cond\}\\leq 50, and allTJ=0T\_\{J\}=0\.

### Appendix D\. Reproducibility

- •Artifacts\.The integration branch and per\-phase release tags are listed in the supplementary material; 132 tests were green at the final merge\.
- •Seeds / splits\.Disjoint seed triples per run: trainseed, evalseed\+99, calibrationseed\+777\. Seed counts: 8 \(alignment bridge\), 10 \(pixel\), 3 \(Kepler state/lift/rescue\)\.use\_deterministic\_algorithms=True; BLAS/OMP threads pinned\.
- •Environments\(two per\-run\)\. \(A\) macOS\-arm64: Python 3\.11\.15, PyTorch 2\.12\.0, NumPy 2\.4\.6, scikit\-learn 1\.9\.0, SciPy 1\.17\.1\. \(B\) Linux\-x86\_64: Python 3\.12\.3, PyTorch 2\.8\.0\+cu128, NumPy 2\.1\.2\.
- •Provenance\.Each run stores a config hash, commit, and full config in its per\-seed JSON; the full anonymized configuration dictionaries are provided in the supplementary material\. No filesystem paths, usernames, or branch names are embedded in the released artifacts\.

### Appendix Figure 5 — Kepler boundary

![Kepler exposes the geometric price of decoded joint-invariant certification. (a) state joint shell: symplectic 3/3 vs plain 0/3 (\delta_I 0.064 vs 0.489). (b) the autoencoder rungs as raw \epsilon_{0,H}ˆ{q95} and the legal-chart rung as the charted \epsilon_{0,J} --- the same quantity class on different chart metrics, separate budgets (not one axis). (c) the residual concentrates at small radius (periapsis 3.87 vs 1.22/1.17 at mid/large radius), exactly where \partial H/\partial r\sim rˆ{-2} amplifies small (q,p) errors. (d) legal (H,L) re-chartings improve the charted defect (2.8\times) and even raise \sigma_*, yet all give T_J=0. This is a system-specific inverse-square geometric boundary, not a structure failure.](https://arxiv.org/html/2606.24945v1/x5.png)Figure 5:Kepler exposes the geometric price of decoded joint\-invariant certification\. \(a\) state joint shell: symplectic3/33/3vs plain0/30/3\(δI\\delta\_\{I\}0\.0640\.064vs0\.4890\.489\)\. \(b\) the autoencoder rungs as rawϵ0,Hq​95\\epsilon\_\{0,H\}^\{q95\}and the legal\-chart rung as the chartedϵ0,J\\epsilon\_\{0,J\}— the same quantity class on different chart metrics, separate budgets \(not one axis\)\. \(c\) the residual concentrates at small radius \(periapsis3\.873\.87vs1\.22/1\.171\.22/1\.17at mid/large radius\), exactly where∂H/∂r∼r−2\\partial H/\\partial r\\sim r^\{\-2\}amplifies small\(q,p\)\(q,p\)errors\. \(d\) legal\(H,L\)\(H,L\)re\-chartings improve the charted defect \(2\.8×2\.8\\times\) and even raiseσ∗\\sigma\_\{\*\}, yet all giveTJ=0T\_\{J\}=0\. This is a system\-specific inverse\-square geometric boundary, not a structure failure\.

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