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

Zeyu Jia

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…