State, Physics, Causality, and Memory Evaluation › Physical Rule and Constraint Tests

The previous section asked whether the hidden state is coherent. This one asks whether the dynamics obey a specified physical law under a named regime. A model can produce a smooth, coherent, plausible trajectory that travels at an unphysical speed through a dissipative medium, or that passes through a wall, or that gains energy at every bounce. Constraint tests are how you catch that. They are the natural complement to the trajectory-accuracy tests above: accuracy asks whether the model is close to the truth on average, and constraint tests ask whether the model respects the mechanisms that produce the truth.

Specify the regime before testing a rule. In this force-free, elastic toy, kinetic energy is conserved. Applied forces can do work; damping and lossy contacts can dissipate energy, so a driven dissipative system needs both contributions in its energy balance. A conservation test without a regime can penalize correct dynamics. Here a lossy contact reduces normal kinetic energy only when the incident normal velocity is nonzero.

We will evaluate three regimes, all with the same dynamics code but different parameters:

The damping factor multiplies both velocity components by $0.9$ each step, so total kinetic energy is multiplied by $0.81$. At a wall, restitution multiplies only the normal velocity by $-e$; its energy component is multiplied by $e^2=0.49$, while tangential energy is unchanged. Total energy therefore does not generally fall by a factor of $0.49$. These regimes test different mechanisms, but a residual alone cannot identify which mechanism is wrong.

For a rollout of length $H = 30$ starting from the true initial state $s_0$, define the predicted energy ratio $\hat\rho = \hat{E}_H / \hat{E}_0$ and the true ratio $\rho = E_H / E_0$, where $E$ is the kinetic energy

$$E_t = \tfrac{1}{2} m \|\mathbf{v}_t\|_2^2$$

where $m = 1$ kg is the mass and $\mathbf{v}_t$ is the velocity at step $t$. The energy ratio residual is

$$\mathrm{ERR} = \big| \hat\rho - \rho \big|$$

with the predicted and true ratios defined as

$$\hat\rho = \frac{\hat{E}_H}{\hat{E}_0}, \qquad \rho = \frac{E_H}{E_0}$$

where:

• $H$: the rollout horizon, $30$ steps, so $H\Delta t = 3.0$ s.
• $\hat{E}_t$: the kinetic energy of the model's predicted state at step $t$.
• $E_t$: the kinetic energy of the simulator's true state at step $t$.
• $\hat\rho$: the model's predicted energy ratio over the rollout.
• $\rho$: the true energy ratio produced by the simulator's own dynamics.

These ratios require positive initial energy. The fixtures satisfy that condition; the implementation floors denominators at $10^{-12}$ joules as a numerical safeguard, which changes the ratio convention near zero. In the elastic case $\rho=1$; with damping and no clipping $\rho=0.81^H$; with lossy contact the ratio depends on each impact. The violation rate uses the declared absolute ratio-error tolerance $0.05$, not a universal physical threshold.

We evaluate four predictors in each regime to compare regime dependence and model-class limitations. Simulator truth is reused as an identity reference; evaluating numerical integration drift would require an independent higher-accuracy reference or a time-step convergence study, which this example does not provide.

• True integrator. Simulator truth reused as the reference prediction. Its zero residual follows by identity, not from an independent numerical-accuracy check.
• Linear model fitted on force-free data. A one-step linear map $\hat{s}_{t+1} = A s_t + b$ estimated by ridge regression on force-free trajectories.
• Linear model fitted on damped data. The same estimator, fitted on damped trajectories.
• No-change control. The deliberately wrong negative control: $\hat{s}_{t+1} = s_t$. It preserves initial velocity and kinetic energy but freezes position, illustrating that passing an energy check is not sufficient for correct dynamics.

A non-zero constraint residual can come from at least five places, and attributing it correctly is most of the interpretive work:

• Learned dynamics error. The model misrepresents the mechanism.
• Numerical integration drift. Under-resolved integration can introduce error. Diagnosing it needs an independent numerical reference; our identity comparison does not measure it.
• Observation error. If the rollout is initialised from noisy observations rather than the true state, the initial energy estimate itself is noisy.
• Coordinate and unit errors. Inconsistent velocity conversion across initial and final states can distort ratios. A common multiplicative conversion of all velocities cancels in $E_H/E_0$; ratio agreement cannot detect every unit error.
• A mismatched regime. The model may be perfectly correct for a different regime than the one under test.

The experiment includes regime mismatch and model-class misspecification. A systematic signed residual can arise from either, from calibration bias, or from other errors; its sign does not uniquely diagnose the cause. Compare matched-regime data, held-out mechanisms and an appropriate numerical reference before assigning responsibility.

The next figure compares the free-trained and damped-trained linear fits on the same 20 elastic held-out rollouts, plotting each unit's signed energy-ratio residual instead of its average. The fits also use different realized initial-state draws in their training trajectories, so this comparison does not isolate a change in training regime alone.