Certified Access: Test-Time Compute on Conservative Physics-Structured AI Memory

Test-time compute mechanisms, such as retries, confidence-based escalation, and non-local routing, are conventionally deployed as black-box augmentations to trained models. These approaches typically introduce additional forward passes or learned halting heads without providing a structural account of the specific capabilities acquired by the additional compute, nor do they quantify the costs incurred against the model's fundamental stability guarantees. In this work, we propose a theoretical and empirical framework for test-time compute on a Latent Dynamics Model serving as the memory. When a latent state $(q,p)$ (produced by some encoder) is advanced by a damped symplectic step (velocity-Verlet) of a learned Hamiltonian, every compute mechanism that expands access has an exact phase-volume Jacobian, a bounded energy change, and an exact latch-transport law. We formalize latent reachability into two distinct failure modes. First, escape failures occur within $C_T$ when a state is trapped behind latent surface barrier. Second, reach failures occur when a target lies outside a kinematic causal box $C_T$. To resolve escape failures, we deploy a Lorentz squeeze, which injects bounded, re-absorbable energy that expands reach at test time. To resolve reach failures, we introduce a gated canonical translation (a one-step channel across $C_T$), which expands latent reach at $\detJ=1$ exactly. We verify this certificate stack on a designed analytic testbed, and show that reaching just beyond $C_T$ with the squeeze requires energy growing quadratically in the excess distance covered, whereas the gated channel's energy requirement is fixed and independent of distance. Furthermore, we separate transport from latent state-replacement, demonstrating that an untrained state-replacing map annihilates phase volume and erases stored charge, while the gated channel preserves it. Finally, we situate these mechanisms on a learned physics-structured memory, demonstrating calibrated compute-rationing and evaluating scaling boundaries for non-local routing against a physics-free baseline. The primary contribution of this work lies in the theoretical formalization and empirical verification of the certificate stack, prioritizing structural discipline over isolated benchmark performance.