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

1 Introduction

• The paper defines certified access for latent dynamics and separates reach failures outside C_T from escape failures in which a potential barrier shields an in-range target.The certificate comprises phase-volume Jacobian, bounded energy change, and latch-transport law; its theorems depend on stated assumptions and are not guarantees for the full learned system.Source: §1 · p. 2
• On the analytic testbed, the gated channel reaches continuously with det J = 1, exact latch transport, and zero energy change on the matched channel.The Lorentz squeeze resolves escape with bounded injection; its energy cost grows exponentially in rapidity.Source: §1 · p. 2
• An untrained state-replacing map has det J = 0 and erases stored Goldstone charge, while the gated channel transports charge with preserved spread.Phase-volume preservation alone is insufficient for latch transport; a matched channel is required.Source: §1 · p. 2
• At capacity kv16 with a 400-epoch budget, the compute-rationing gate reduces relaxation steps by 4.81 ± 0.44×.Its value is escalatability and rationing, not a better baseline energy signal.Source: §1 · p. 2
• A one-hop non-local edge keeps FLOP count flat as distance grows, but a parameter-matched physics-free router outperforms the energy-gated edge in FLOPs and accuracy.A learned decision router may choose whether to use the edge while data still move through the certified det J = 1 channel.Source: §1 · p. 2
• Across kv32, kv64, and kv96, measured intra-CLU step reductions are 9.9×, 9.5×, and 6.2×; beyond kv64, epoch budget constrains performance.Gate accuracy degrades sharply under evaluation cue noise, while storage fidelity stays at 1.0.Source: §1 · p. 2

2 Setup: A Conservative Memory and its Reachable Set

• A Causal Learning Unit is a latent dynamics model whose encoder writes phase-space state (q, p) and whose learned Hamiltonian retrieves by damped-dynamics relaxation.The empirical results use the CLU reference training objective while measuring general properties of the learned Hamiltonian.Source: §2 · p. 3
• One CLU step uses damped velocity-Verlet integration of H = T(p) + Vθ(q); with fixed damping γ, its Jacobian determinant is (1 − γ)^d.The state-dependent governor's derivative adds Jacobian terms, so the fixed-damping certificate does not cover the governed map.Source: §2 · p. 3
• Because each drift advances coordinate i by less than εc/√Mi, T steps bound displacement by Li = Tεc/√Mi, independent of momentum and potential.The latent reachable set is defined by iterating the governed map, with its position shadow giving the reachable positions.Source: §2 · p. 3
• A kinematic reach failure places the target outside C_T, whereas an energetic escape failure places it inside C_T but behind a barrier the latent energy cannot cross.Any bounded energy injection acting faster than governor dissipation resolves the stated escape failure.Source: §2 · p. 3

3 Mechanism Certificates and Limitations

• The mass-weighted Lorentz squeeze has det J = 1 and an energy bound H(Sζz) ≤ e^(2|ζ|)H(z) for a quadratic matched Hamiltonian.The bound and the causal-box reach result use different kinetic energies, so they are not one joint certificate.Source: §3 · p. 4, Table 1 · p. 4
• A frozen-gate canonical translation has det J = 1 exactly, with energy cost equal to the endpoint potential difference.If the gate varies during the jump, its Jacobian acquires a first-order defect; freezing it restores the exact certificate, but only per branch.Source: §3 · p. 4

3.1 BIBO Stability Limits

• At b = 5.0, the zero-energy jump still escapes; boundedness requires screening exits to the coercive connected component.The figure reports that the screened gated channel refuses the jump rather than making it safe.Source: §3.1 · pp. 4–5, Figure 1 · p. 5, Table 2 · p. 5

3.2 Differentiating Access Mechanisms

• At d ≥ 4.0, the squeeze misses under the swept rapidity budget, while the gated channel’s matched symmetric-well jump has zero energy cost.The squeeze misses because the required rapidity exceeds the budget; its energy grows quadratically in excess distance.Source: §3.2 · p. 6
• On the designed reach testbed, the squeeze misses outer distances when they require more than its rapidity budget, whereas the channel translates to q = +1.000 with det J = 1.000 and zero energy change.The squeeze’s energy grows quadratically in excess distance; the channel’s endpoint potential difference is zero on this symmetric testbed.Source: §3.2 · pp. 5–6

3.2.1 The Certificate’s Consequence: State Erasure vs. Transport

• The matched gated channel transports latch charge within a fixed branch, whereas the state-replacing map has det J = 0 and irreversibly erases coset content.The channel preserves charge spread; volume preservation alone is insufficient, since a random det J = 1 shift can scramble the latch.Source: §3.2.1 · p. 7, Table 4 · p. 7

4.1 Calibrated Compute-Rationing

• At 400 epochs, the gate reduces relaxation steps by 4.81 ± 0.44× at kv16 relative to an always-full budget, with higher accuracy; the gain shrinks as capacity grows.At 2000 epochs, the gate matches rather than exceeds the full-budget baseline.Source: §4.1 · p. 7

4.2 Capacity and Training-Budget Map

• Required training epochs grow with capacity: kv32 saturates by 1000 epochs, while kv96 needs 4000 epochs to reach 0.975 accuracy.Beyond kv64, the epoch budget rather than a hard capacity limit constrains performance; kv32 over-trains between 2000 and 4000 epochs.Source: §4.2 · p. 7, Figure 3 · p. 8, Table 5 · p. 8
• At evaluation cue noise σ = 0.6 on kv32, gate accuracy is 0.36 while CLU storage fidelity stays at 1.0.The paper identifies the gate’s sharp degradation under cue noise as an open problem.Source: §4.2 · p. 8

4.3 Non-Local Routing

• The parameter-matched, physics-free router outperforms the energy-gated edge in both FLOPs and accuracy, while the one-hop edge keeps FLOPs flat as distance grows.The comparison is on the lattice routing task.Source: §4.3 · p. 8, Table 6 · p. 8

5 Discussion

• The gated channel reaches globally at det J = 1 exactly, with energy cost equal to the endpoint potential difference and exact latch transport at the jump; the matched channel has zero energy change in this testbed.The squeeze resolves escape with bounded injection, at a cost exponential in rapidity and quadratic in excess distance.Source: §5 · p. 8
• The certificates are conditional: the volume certificate assumes damping frozen within a step, latch transport holds only at the jump, and BIBO stability requires coercive-component screening.The guarantees do not bound trajectories or cover the state-dependent governor.Source: §5 · p. 9
• On learned memories, the mechanisms ration compute on clean retrieval, while a physics-free router is stronger for non-local routing; the paper claims no general leaderboard advantage.The main contribution is formalizing and verifying the certificate stack and mapping its limits empirically.Source: §5 · p. 9
• With a quadratic payload channel and fixed read budget, this implementation cannot achieve exact value fidelity, amplitude-independent address hold, and amplitude-independent read latency all at once.The authors describe this as an implementation trade-off, not a proved incompatibility; faded memory requires more integration steps to read.Source: §5 · p. 9

A Flag-Provenance Tables

• Computational experiments use non-default configurations and standard laptop CPUs.Source: A Flag-Provenance Tables · p. 10

A.1 Certificate Stack Configuration

• The analytic testbed uses relativistic kinetics for reach, gated-channel, and throat modes, with reach-rollout damping γ = 0 (sharp box), and evaluates reach seeds {0, 1, 2, 3, 4}.Source: §A.1 · p. 10

A.2 Calibrated Gate Configuration

• The calibrated-gate MQAR setup uses capacity kv ∈ {16, 24, 32}, five seeds, and a 400-epoch training budget.The gate is calibrated from residual-margin features with a learned write-time threshold and pexit = 0.5.Source: §A.2 · p. 10

A.3 Routing Configuration

• The routing configuration uses lattice distances N ∈ {4, 8}, a 12-dimensional embedding, 250 relaxation steps, and a 449-parameter router.Source: §A.3 · p. 10

A.4 Regime Map Configuration

• The regime map sweeps 500, 1000, 2000, and 4000 training epochs across five seeds.It uses a 32-dimensional CLU, 300 retrieval relaxation steps, and a coercive MLP potential.Source: §A.4 · p. 11

A.5 Latch Transport and BIBO Battery Configuration

• Latch transport uses 16 incoming states, while BIBO tests use damping γ = 0.02 and horizon T = 2000.The passage also specifies analytic potentials, oracle channel placement, and five seeds for reach and BIBO.Source: §A.5 · p. 11
• The calibrated-gate evaluation uses γ = 0.3 and 400 epochs, an under-converged regime; the regime map sweeps to 4000 epochs to isolate converged behavior.Source: §A.5 · p. 11

B Reach Battery Full Grid

• The reach battery uses five seeds and causal box L = 2.5.Source: B Reach Battery Full Grid · p. 11

• Squeeze retries recover no performance when selecting among already reachable attractors, so they address kinematic reach failure rather than local selection.The passage identifies local selection as outside the squeeze mechanism’s demonstrated use.Source: C Documented Negative Results · p. 11
• Under evaluation cue noise, gate accuracy degrades sharply while storage fidelity remains 1.0, indicating over-commitment to corrupted cues rather than storage failure.Noise-aware thresholds, longer relaxation budgets, and denoising initialization remain open work.Source: C Documented Negative Results · p. 11
• The raw energy signal adds negligible predictive value over a standard readout margin, so the framework relies on calibrated rationing rather than a superior physics signal.Source: C Documented Negative Results · p. 11

• 2.0×10−12 relative error: the causal cap of the reach box matches its prediction.Source: D Analytic Verification Protocols · p. 11
• 1.000: the Jacobian determinant matches its predicted value, with state-dependent deviations matching the analytical prediction to 2.2×10−16.Source: D Analytic Verification Protocols · p. 11
• Below 1.8 × 10−15: the Goldstone latch transit error; governor re-absorption also matches predicted exponential injections at every swept rapidity.Source: D Analytic Verification Protocols · p. 11

• The symmetric Lorentz squeeze has det J = 1, but the governed composite is Metropolis-within-annealing, converging to a colder MAP state rather than acting as a Gibbs sampler.The paper does not claim stationarity for the governed composite.Source: E Certified Markov Kernels:… · p. 12
• The squeeze orbit is a single hyperbola and is not ergodic; mixing across state space requires adding a MALA momentum-refreshment step.Source: E Certified Markov Kernels:… · p. 12
• 0.0065 versus 0.0995: calibrating Langevin noise with the discrete fluctuation-dissipation relation reduces finite-budget shadow-bias error.Relativistic kinetic modes require Maxwell-Jüttner targets and Metropolis-Hastings corrections; fixed-covariance Gaussian refreshes are invalid.Source: E Certified Markov Kernels:… · p. 12
• Unprojected coset-tangent proposals are always accepted and diffuse stored registers at a rate proportional to proposal-scale variance; orthogonal projection suppresses this drift.Without projection, certified MCMC retries progressively erase stored memory contents.Source: E Certified Markov Kernels:… · p. 12

• The energy inequality is proved for matched quadratic H, whereas the causal box uses relativistic kinetic energy, so these are not one joint statement.On a quartic well, higher powers of cosh ζ can make the raw ratio exceed e2|ζ|.Source: F Derivation of the Bounded-Injection… · pp. 12–13
• The squeeze acts on mass-weighted coordinates u = √Mi qi and v = pi/√Mi, mixing position and momentum through cosh ζ and sinh ζ while leaving other coordinates unchanged.Source: F Derivation of the Bounded-Injection… · p. 12
• At p0 = 1.2, reaching d = 4.0 requires ζ ≥2.0105 and d = 5.0 requires ζ ≥2.64, both beyond the swept ζ ≤2.0 budget.The squeeze battery drops to zero at d ≥4.0, while d ≤3.2 remains covered; the misses reflect the budget, not an absolute reach limit.Source: F Derivation of the Bounded-Injection… · p. 12
• The active-pair quadratic Hamiltonian is isotropic in those coordinates, Hq = 1/2κ (u2 + v2), the matched-quadratic condition underlying the energy bound.Source: F Derivation of the Bounded-Injection… · p. 12
• H(Sζz) ≤ e2|ζ|H(z), with equality only on u = ±v; the injected energy is bounded by (e2|ζ| −1)H, and spectator coordinates tighten the ratio.Source: F Derivation of the Bounded-Injection… · p. 13
• The launch-state energy ratio is cosh 2ζ = 1 + 2(M0δ/p0)2, so it is quadratic in excess distance δ; the certificate is not exponential in distance.The certificate is exponential in rapidity and quadratic in δ at large rapidity.Source: F Derivation of the Bounded-Injection… · p. 13