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

arXiv: https://arxiv.org/abs/2609.37351 · HTML: https://mlsanity.com/paper/2609.37351/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 5 of its 11 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 5 of 5)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Training Large Language Models to Reason in a Continuous Latent Space](https://mlsanity.com/paper/2412.06769/) (2024)
- [Training Language Models to Self-Correct via Reinforcement Learning](https://mlsanity.com/paper/2409.12917/) (2024)

## Abstract (arXiv)

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.