Port-Hamiltonian Latent Deliberation: Mitigating the Deliberation Drift Cliff in Test-Time Compute Scaling

2026-09-29

AI Summary

This study addresses the "deliberation drift cliff" and deep-thinking collapse arising from iterative reasoning in continuous latent spaces during test-time compute scaling. To this end, it introduces DG-HHD, a novel port-Hamiltonian latent deliberation architecture. The proposed method leverages Helmholtz–Hodge decomposition to orthogonally decouple rotational and gradient flows, integrating tangential projection tensor networks with an RK45 integrator to eliminate automatic differentiation dependencies and overcome accuracy bottlenecks inherent in conservative flows. Experimental results demonstrate that DG-HHD achieves monotonic compute scaling for multi-hop reasoning in small language models, improving peak accuracy by 25.94% and accelerating inference by over 1.8× while effectively suppressing out-of-distribution drift.

Abstract

Test-time compute scaling has emerged as a cornerstone of advanced machine reasoning, yet performing iterative deliberation directly within continuous latent representation spaces reveals a catastrophic pathology: the Deliberation Drift Cliff. While unconstrained recurrent latent models achieve initial reasoning gains at short horizons (K <= 4), their reasoning collapses when extrapolated to deeper thinking steps (K >= 16), dropping by 22% to 62% across standard logical benchmarks. We resolve the trilemma among expressivity, Lyapunov stability, and computational efficiency in test-time latent reasoning through a 22-round empirical and theoretical investigation. We demonstrate that strictly conservative scalar potential gradient flows suppress long-range drift (cliff 3.40%) but bottleneck peak reasoning accuracy at 32.73%, whereas unconstrained rotational flows achieve high symbolic expressivity (82.33%) but suffer a severe 36.87% drift cliff. To resolve this geometric duality, we establish Port-Hamiltonian Latent Deliberation (PH-LD) and propose the Direct-Gradient Pure-Tensor Helmholtz-Hodge Decomposition (DG-HHD). DG-HHD parameterizes the attracting flow as a tangent projection tensor network while orthogonally decoupling non-zero circulation (Hodge machine error 1.65e-17, contraction error 5.55e-17), eliminating runtime autograd dependencies to achieve 1.84x vector field and 2.09x RK45 rollout speedups. In a 15-arm symmetrical Pareto benchmark, DG-HHD achieves 58.67% peak accuracy (+25.94% absolute gain over conservative HHD) and retains 35.27% at K=32. Transferred to small language model (SLM) multi-hop causal reasoning, DG-HHD delivers monotonic compute scaling (49.33% to 51.56%) and suppresses out-of-distribution drift (cliff -0.66%). All 30 Level 0 deterministic invariants are certified.

Problem

• Test-time compute scaling
• Deliberation Drift Cliff
• Latent reasoning
• Lyapunov stability
• Expressivity-stability tradeoff

Innovation

• Port-Hamiltonian systems
• Helmholtz-Hodge decomposition
• test-time compute scaling
• latent deliberation
• tensor networks