Technical Summary: When Do Conservation Laws Survive Learned Representations?

Problem Statement

Modern world models achieve scalability by learning latent representations and rolling forward in that latent space rather than in observation space. However, this shift complicates the certification of physical trust. While conservation laws (e.g., energy conservation) provide a natural currency for trust in physical systems, a latent model replaces physical coordinates with learned ones. Consequently, a model might perfectly conserve a learned internal scalar (a latent Hamiltonian or witness) while the true physical energy drifts significantly.

The central question addressed is: When does a conservation law remain certifiable after a model learns a latent representation? Specifically, the paper seeks to bound, in advance and from measurable model defects, how many steps a rollout provably stays on a physical invariant's level set. The critical challenge is defining what is being certified: not the learned latent quantities, but the decoded physical invariant.

Methodology

The Certified Object

The paper fixes the certified object to the decoded physical invariant, denoted as $\Psi(z) = H^*(\Pi D_\psi z)$.

• $z$: Latent state.
• $D_\psi$: Decoder mapping latent to observation.
• $\Pi$: Readout mapping observation to physical coordinates $(q, p)$.
• $H^*$: The known physical invariant (e.g., true mechanical energy).

Crucially, the paper does not certify the learned latent Hamiltonian $H_\theta$ or a learned scalar witness $C_\omega$ directly. These are treated as "witnesses" or proxies. A model can conserve these internal quantities while the decoded physical invariant drifts. Certification is only valid where the state pipeline and the latent-decoded pipeline converge on $H^*$.

Certificate Framework

The authors derive "shell-horizon certificates" that bound the number of steps $T$ a rollout stays within a tolerance $\epsilon$ of the energy shell. The budget for this horizon decomposes into three distinct defect sources:

1. Representation Defect ($\epsilon_0$): Initialization error in the embedding/encoding.
2. Readout Defect ($\epsilon_{\Pi, \text{num}}$): Error introduced by the decoder and readout mapping (dominant in pixel observations).
3. Latent-Dynamics Defect ($\delta_{\text{tube}}^{\text{phys}}$): Per-step drift of the decoded invariant in the latent dynamics.

The horizon is calculated as:
$$T = \left\lfloor \frac{m_{c,\rho}\epsilon - \text{initialization defects}}{\text{per-step drift}} \right\rfloor$$
where $m_{c,\rho}$ is the shell's sensitivity (minimum gradient norm).

Alignment Bridge (Soft Witnesses)

To utilize a learned "soft" invariant (a scalar witness $C_\omega$ trained to be conserved in latent space), the paper introduces a controlled-Lipschitz alignment bridge.

• A soft witness is not inherently the physical energy. It must be aligned via a calibration function $g_C$.
• The condition for a valid certificate is bi-Lipschitz calibration ($0 < \kappa \le g'_C \le L_g$).
• This bridge transforms the soft witness into a theorem-shaped link to the decoded invariant. The paper demonstrates that simple monotonicity is insufficient; uncontrolled monotonicity (e.g., isotonic regression) can have infinite Lipschitz constants, collapsing the certified horizon to zero.

Experimental Setup

The framework is tested on three conservative systems: the simple pendulum, the harmonic oscillator, and the Kepler two-body problem (a singular system).

• Representations: The study varies the input representation while holding the certified object fixed:
  1. State: Canonical coordinates $(q, p)$ are input; readout is identity.
  2. Lift: Low-dimensional observations are encoded into a learned latent chart; readout is clean.
  3. Pixel: High-dimensional images are input; a learned decoder/readout introduces significant error.
• Model Variants:
  • Plain: Unconstrained residual latent map.
  • Hard Symplectic: Latent integrators constrained to be symplectic (Störmer–Verlet, SympNet).
  • Soft (B-self): Self-supervised scalar/vector witness trained to be invariant, aligned via the bridge.

Key Results

1. State Space (Canonical Coordinates)

In known phase coordinates, hard symplectic structure is the clear winner.

• On the Kepler system, hard symplectic models achieved non-vacuous joint-invariant certificates in 3/3 seeds, whereas plain models achieved 0/3.
• The per-step invariant drift was $\sim 7.6\times$ smaller for symplectic models.
• Conclusion: Hard priors work effectively when the chart is known and matches the physical symplectic form.

2. Learned Lift (Latent Coordinates)

When the model learns a chart from low-dimensional observations, hard symplectic priors collapse, while soft invariants survive.

• Hard symplectic models (Verlet, SympNet) became statistically indistinguishable from plain baselines. The symplectic constraint preserves the form of the flow but not the identity of the conserved scalar relative to physical energy under an arbitrary learned chart.
• The soft B-self invariant successfully crossed the learned chart, achieving a structure-gain ratio of $\approx 1.30$ and a win rate $P(\text{soft} > \text{plain}) = 0.88$ over 8 seeds.
• Conclusion: Hard geometric priors are coordinate-sensitive and do not automatically survive representation learning; soft invariants aligned via a bi-Lipschitz bridge are more robust.

3. Pixel Observations

Certification under pixel inputs is possible but restricted to a readout-stable sub-tube.

• The decoder/readout term often dominates the error budget.
• By applying a "frozen readout-stable rule" (excluding points where the readout Lipschitz score is too high), the stabilized soft witness recovered non-vacuous certificates.
• Results: 9/10 seeds were non-vacuous (vs. 4/10 for plain), with an excluded fraction of $\le 2.3\%$.
• Conclusion: Pixel certification is a perceptual problem; it requires stabilizing the readout to find a sub-tube where the decoded invariant is faithful.

4. The Kepler Geometric Boundary

The Kepler lift revealed an intrinsic geometric limit.

• Despite varying latent capacity and applying legal metric re-chartings, the decoded joint-invariant certificate failed ($T=0$) near the periapsis.
• This failure is attributed to the near-periapsis stiffness of the inverse-square potential, which amplifies small $(q, p)$ errors.
• Conclusion: This is a system-specific geometric boundary, not a failure of the conservation certification methodology itself.

Significance and Claims

The paper's primary contribution is a representation-robustness framing for conservation certificates. It argues that:

1. The Certified Object Matters: Certificates must be attached to the decoded physical invariant, not the latent Hamiltonian or unaligned witness. A model can conserve its internal proxy while failing physically.
2. Hard vs. Soft Priors: Hard canonical symplectic structure is coordinate-sensitive. It yields the longest horizons in known phase coordinates but fails to cross learned charts. Conversely, a soft invariant, trained to be conserved in latent space and aligned via a controlled-Lipschitz bridge, survives representation changes.
3. Monotonicity is Insufficient: Aligning a soft witness to physical energy requires bi-Lipschitz calibration. Mere monotonicity is too loose and leads to vacuous certificates.
4. Geometric Limits: Conservation-aligned certificates can fail due to intrinsic geometric singularities (e.g., Kepler periapsis) that no amount of representation learning or capacity scaling can overcome.

The authors position this work not as a benchmark for long-horizon vision world models, but as a methodological study on what can be certified when physical invariants are evaluated through learned representations. The findings suggest that for learned world models, one must either certify in coordinates where the symplectic form is physical or pair structural priors with explicit alignment of the conserved scalar.