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

Pratik Jawahar; Riccardo Maggioni; Jonas Petersen; Matteo Cozzi; Gian-Alessandro Lombardi; Marek Gajewski; Camilla Mazzoleni; Federico Martelli

cs.AIOct 29, 2026

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

Authors: Pratik Jawahar, Riccardo Maggioni, Jonas Petersen, Matteo Cozzi, Gian-Alessandro Lombardi, Marek Gajewski, Camilla Mazzoleni, Federico Martelli

Organizations: CERN · Forgis AG · ETH Zurich

Abstract

Test-time compute mechanisms, such as retries, confidence-based escalation and non-local routing, are conventionally deployed as black-box additions to trained models. They typically add forward passes or halting heads, but give no structural account of what the extra compute buys or what it costs against the model's stability guarantees. We therefore propose a theoretical and empirical framework for test-time compute on a Latent Dynamics Model used as the memory. When an encoder's latent state $(q,p)$ is advanced by a damped symplectic (velocity-Verlet) step of a learned Hamiltonian, every mechanism that expands access has an exact phase-volume Jacobian, a bounded energy change and an exact latch-transport law. We split reachability failures into escape failures and reach failures. A Lorentz squeeze resolves escape failures by injecting bounded, re-absorbable energy. A gated canonical translation (a one-step channel across $C_T$ ) resolves reach failures at $\detJ=1$ exactly. We verify this certificate stack on a designed analytic testbed. Reaching just beyond $C_T$ with the squeeze requires energy that grows quadratically in the excess distance. The gated channel's energy cost equals the potential difference between its endpoints, which is zero on the matched channel measured here, and so does not grow with distance in this testbed. An untrained state-replacing map annihilates phase volume and erases stored charge, while the gated channel transports it. On a learned physics-structured memory, we show calibrated compute rationing and evaluate the scaling limits of non-local routing against a physics-free baseline. The main contribution is the formalization and verification of the certificate stack, not benchmark performance.

Figures & tables

Mechanism	Energy Injected	Volume $\det J$	Governor Re-absorption	Latch Impact
Gated channel (translation)	$\Delta V{=}V(b){-}V(a)$ , discrete jump	1 exactly	$\approx(1/\gamma_{c})\ln(1{+}\Delta V/E^{\star})$	Transport via shift $p^{\top}X\Delta$
Gated channel (throat)	standing wave, no jump	$(1{-}\gamma)^{d}$ (fixed $\gamma$ )	continuous	adds $\mu^{2}{\propto}A/\ell^{2}$
Squeeze $S^{(M)}$	$\leq(e^{2\|\zeta\|}{-}1)H$	1 exactly	$\approx 2\zeta/\gamma_{c}$	preserved

Table 1: Certificate per mechanism. The

(1-\gamma)^{d}

entry holds for the damping value frozen within a step; the state-dependent governor is not covered by it.

Algorithm ( $r^{\ast}$ at $2T$ )	$b{=}1.0$	$2.0$	$3.0$	$3.6$	$4.0$	$5.0$
Gated channel with certificate	$1.01$	$2.01$	$3.01$	$0.09$	$0.09$	$0.09$
Gated channel, ablation	$1.01$	$2.01$	$3.01$	$102.13$	$103.43$	$104.83$
State-replacing map	$1.01$	$2.01$	$3.01$	$102.13$	$103.43$	$104.83$
Escape rate, certified	$0$	$0$	$0$	$0$	$0$	$0$
Escape rate, blind	$0$	$0$	$0$	$1.0$	$1.0$	$1.0$
Gated channel certificate logic	ADMIT	ADMIT	ADMIT	REJECT	REJECT	REJECT

Table 2: The BIBO battery: maximum excursion radius

r^{\ast}

2T

against the exit locus

b

, escape rates, and the admit/reject decision of each screen.

Figure 1: The BIBO battery. Maximum excursion radius $r^{\ast}=\max_{t}\|q_{t}\|$ at horizon $2T$ , against the requested exit locus $b$ . Exits inside the coercive component stay bounded for every arm. Beyond the coercive edge $x_{b}=3.54$ the unscreened arms grow as $r^{\ast}\propto T$ , while the screened gated channel holds at $r^{\ast}=0.09$ because it refuses the jump rather than making it safe. The state-replacing map coincides exactly with the screen-ignored ablation, so the screen, not the jump, provides boundedness. Note $b=5.0$ : the energy change is exactly zero and the exit still escapes.

Algorithm	$d{<}L$	$d{>}L$	Measured $\det J$	Diagnostic
Plain relaxation	$0$	$0$	$(1{-}\gamma)^{d}$	Escape-blocked
Squeeze $S^{(M)}$	$1$	$\to 0$	$1.000\pm 4\mathrm{e}{-6}$	Reach beyond $C_{T}$ needs energy $\leq e^{2\|\zeta\|}H$
Gated channel	$1$	$1$	$1.0$ exact	Flat access scaling; energy change $0.0$
Newtonian control	$1$	$1$	$1.0$	Confirms relativistic cap bounds flow
State-replacing map	$1$	$1$	$0.0$ exact	State overwritten; volume annihilated
Dense/ $V$ -throat	$\to 0$	$0$	$(1{-}\gamma)^{d}$	Modulates barrier; fails global reach

Table 3: Reach battery summary: landing rates inside (

d{<}L

) and beyond (

d{>}L

) the causal box, measured

\det J

, and diagnostic. The

(1{-}\gamma)^{d}

entries hold for fixed

\gamma

Figure 2: Verification of the certificate stack on the designed testbed (Sec. 3.2 ). (a) Reach: landing rate against basin distance $d$ on a symmetric double well (dim 2, 5 seeds, $L=2.5$ ). (b) The certificate in force: outgoing against incoming Goldstone charge for 16 states drawn from the capture ball. Dashed line in (a): causal box $L$ ; grey band: crossover bracket. In (b), coset-tangent and across-coset are gated-channel jumps; the dashed line is $Q_{\rm out}=Q_{\rm in}$ .

Mechanism	$\det J$	$\Delta Q$ mean	$\Delta Q$ std	Predicted $p^{\top}X\Delta$	Max Error	$\mathrm{std}(Q_{\rm out})$
Gated channel (coset-tangent)	$1.0000$	$-0.0000$	$0.0000$	$0.0000$	$1.2\mathrm{e}{-7}$	$0.0803$
Gated channel (across-coset)	$1.0000$	$0.2500$	$0.0000$	$0.2500$	$0.0$	$0.0803$
Random shift	$1.0000$	$0.0972$	$0.2465$	N/A	—	$0.2793$
State-replacing map	$0.0000$	$0.0379$	$0.0803$	N/A	—	$0.0$

Table 4: Latch transport on 16 states from the capture ball.

Figure 3: Epoch-scaling frontier: CLU-EBM storage fidelity and gated accuracy against training budget for kv $\in\{32,64,96,128\}$ . Required epochs scale with capacity rather than saturating at a fixed ceiling, and kv $32$ over-trains between $2000$ and $4000$ epochs. Mean over 3 seeds (42–44).

Cell (N,kv)	Epochs	CLU Fidelity	Gate Acc	Intra-CLU Step-Reduction
N128/kv32	500	$0.76\pm 0.09$	$0.31\pm 0.04$	$1.2\times$
N128/kv32	2000	$1.00\pm 0.00$	$1.00\pm 0.00$	$9.9\times$
N256/kv64	500	$0.43\pm 0.05$	$0.06\pm 0.02$	$1.0\times$
N256/kv64	2000	$1.00\pm 0.00$	$0.99\pm 0.00$	$9.5\times$
N384/kv96	500	$0.40\pm 0.03$	$0.02\pm 0.01$	$1.0\times$
N384/kv96	2000	$0.97\pm 0.01$	$0.91\pm 0.02$	$6.2\times$

Table 5: Gate across capacity and training budget.

Algorithm	$N{=}4$ Acc (L/D)	$N{=}4$ FLOP	$N{=}8$ Acc (L/D)	$N{=}8$ FLOP
Local only	$0.500$ ( $1.00$ / $0.00$ )	$5.88\mathrm{e}7$	$0.500$ ( $1.00$ / $0.00$ )	$5.88\mathrm{e}7$
Gated energy	$0.887\pm 0.139$ ( $0.90$ / $0.88$ )	$1.18\mathrm{e}8$	$0.715\pm 0.172$ ( $0.82$ / $0.61$ )	$1.18\mathrm{e}8$
Dense network	$0.500$ ( $0.00$ / $1.00$ )	$1.18\mathrm{e}8$	$0.448\pm 0.070$ ( $0.00$ / $0.90$ )	$1.18\mathrm{e}8$
Diffusion chain	$0.652\pm 0.159$ ( $0.90$ / $0.41$ )	$1.76\mathrm{e}8$	$0.548\pm 0.138$ ( $0.82$ / $0.28$ )	$2.94\mathrm{e}8$
Physics-free router	$1.000\pm 0.000$ ( $1.00$ / $1.00$ )	$8.81\mathrm{e}7$	$0.948\pm 0.070$ ( $1.00$ / $0.90$ )	$8.81\mathrm{e}7$

Table 6: Routing: local and distant accuracy (L/D) and floating-point operations per query over 5 seeds.

Inference-Time Compute Allocation

Inference-time compute allocation decides where, when, and how much computation models spend during prediction to improve quality under resource limits.

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Fewer Bits, More Tries: Recovering Capability Under Memory Constraints

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Texas A&M · Apple

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Open paper AskPDF arXiv

Large language models are generally more capable than smaller ones, but their memory requirements can make local deployment difficult. We ask whether part of this capability gap can instead be recovered by running a smaller or more compressed model multiple times and selecting among its outputs using public or model-generated tests. Single-user local inference can leave hardware underutilized, allowing parallel candidate generation to trade additional test-time computation for accuracy at modest latency cost. Across four code-generation benchmarks, verified retry recovers substantial accuracy lost to model size or compression. A 7B Qwen model with eight parallel attempts essentially matches a 32B model on MBPP+ and exceeds it on the other three benchmarks, while using about

4\times

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Abstract

Inference-Time Compute Allocation

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