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

Zeyu Jia

arXiv:2609.37351v1cs.LGcs.AIcs.CL

TL;DR

Unconstrained continuous latent deliberation can collapse when reasoning is extended beyond its training horizon. The paper introduces PH-LD and DG-HHD to combine constrained dynamics with orthogonal attracting and circulatory flows. DG-HHD improves peak reasoning accuracy, while the small-language-model transfer shows monotonic compute scaling and no drift cliff.

  • Problem

    Unconstrained latent reasoning encounters a Deliberation Drift Cliff when trajectories extend beyond the training horizon.

    Source: §1 · p. 2, §2.2 · p. 3

  • Method

    PH-LD models deliberation with constrained dynamical systems, and DG-HHD decomposes tangent flows into orthogonal attracting and circulatory components.

    Source: §1 · pp. 2–3, §4.2 · p. 5, §8 · p. 10

  • Results

    DG-HHD raises peak reasoning accuracy from 32.73% to 58.67%, while its SLM transfer shows monotonic compute scaling and zero drift cliff.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The investigation supports PH-LD as a mathematically principled and computationally efficient foundation for continuous test-time reasoning.

    Source: §1 · p. 3

    Language experiments use compact Dmodel = 128–256 models and a single continuous thought vector rather than multi-billion-parameter models or token-sequence deliberation.

    Source: §7.2 · p. 9

Abstract

from arXiv · show

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.

1 Introduction

Continuous latent deliberation can suffer a Deliberation Drift Cliff at extended horizons, while PH-LD combines geometric constraints with a flow decomposition to address stability, expressivity, and efficiency. DG-HHD improves peak reasoning accuracy, and its transfer to small language models shows monotonic compute scaling with no drift cliff.

1 Introduction

  • Beyond K = 4, unconstrained continuous deliberation encounters the Deliberation Drift Cliff, with reasoning collapsing by 22% to 62% at K = 16–32.The paper attributes the cliff to coupling between local numerical truncation drift and global vector-field divergence.Source: §1 · p. 2
  • PH-LD models latent trajectories as constrained dynamical systems, combining unit-sphere confinement, contact mechanics, and orthogonal attracting and circulatory flows.The sphere prevents norm explosion; contact mechanics adds dissipation without destroying phase-space symplectic structure.Source: §1 · pp. 2–3
  • 58.67% peak reasoning accuracy, up from 32.73%, is achieved by DG-HHD while preserving orthogonality to machine precision (1.65 × 10−17).Direct tangent-space parameterization avoids the scalar-potential Hessian constraint.Source: §1 · p. 3
  • DG-HHD parameterizes vpot directly in tangent space without scalar potential auto-differentiation, breaking the symmetric Hessian constraint and lifting peak reasoning accuracy from 32.73% to 58.67% while strictly bounding OOD drift.An embedded Runge-Kutta 4(5) (RK45) integrator with adaptive step-size selection and spherical retractions maintains geodesic tracking with < 10−6 norm drift across 16 steps.Source: §1 · p. 3
  • The SLM bridge shows monotonic compute scaling and zero drift cliff when continuous Riemannian flows are used for multi-hop deduction.The empirical investigation includes 22 research cycles, 30 deterministic invariant certifications, and a 15-arm Pareto benchmark.Source: §1 · p. 3

2 The Deliberation Drift Cliff: Analysis and Falsification

The section formulates latent deliberation as manifold-valued state evolution and identifies numerical discretization and out-of-distribution vector-field divergence as distinct sources of long-horizon drift. Unconstrained baselines show large accuracy declines as deliberation depth increases.

2.1 Mathematical Formulation

  • Deliberation maps input context x to an initial latent q0 on the unit sphere, integrates K steps, and decodes terminal state qK into answer predictions.The state transition for unconstrained recurrence is qk+1 = qk + h · fθ(qk, c).Source: §2.1 · p. 3, §2.2 · p. 3

2.2 Two-Fold Anatomy of the Drift Cliff

  • The drift cliff has two proposed sources: Euler truncation errors push states off-manifold, while unconstrained fields diverge on OOD trajectories beyond their K ≤ 4 training horizon.These mechanisms can produce norm growth or dimensional collapse, and non-zero positive divergence can lead to non-semantic limit cycles or infinity.Source: §2.2 · p. 3
  • 96.60% at K = 4 falls to 34.13% at K = 64 for B0-Naive (∆= −62.47%); unconstrained pool architectures fall from 45.3% to 23.1% (∆= −22.2%).These declines are reported on relational multi-hop reasoning graphs and the 15-arm Pareto benchmark, respectively.Source: §2.2 · p. 4

3 Port-Hamiltonian and Contact Mechanics on Manifolds

The paper uses port-Hamiltonian and contact mechanics to represent dissipation in latent dynamics, then applies RATTLE integration to keep trajectories on the sphere. The described invariant suite reports that RATTLE maintains the spherical constraint across repeated steps.

3.1 Port-Hamiltonian Systems on Phase Space

  • Port-Hamiltonian dynamics combine total Hamiltonian energy with a skew-symmetric interconnection matrix and positive-semidefinite dissipation, establishing unconditional Lyapunov stability.The interconnection preserves energy while the dissipation matrix is nonnegative.Source: §3.1 · p. 4

3.2 Contact Hamiltonian Systems in Extended Phase Space

  • Contact Hamiltonian mechanics extends phase space with an internal action variable that accumulates dissipation and provides a thermodynamic stopping criterion.The extended coordinates are (q, p, s), with s representing the internal action variable.Source: §3.2 · p. 4

3.3 Symplectic RATTLE Integration on Spheres

  • RATTLE enforces sphere-position and cotangent-velocity constraints; the invariant suite reports < 10−6 constraint drift across 64 consecutive integration steps.It solves for Lagrange multipliers to enforce both constraints.Source: §3.3 · p. 4

4 Direct-Gradient Helmholtz-Hodge Decomposition

The section contrasts conservative attraction with non-conservative circulation, introduces a directly parameterized orthogonal Hodge field, and describes how it avoids runtime autograd overhead.

4.1 The Duality: Conservative Potential vs. Non-Conservative Circulation

  • The formulation reduced OOD drift cliff to 3.40% versus 36.87%, but its peak reasoning accuracy was capped at 32.73% ± 1.10%; unconstrained spectral contraction flows reached 82.33% ± 13.61% peak accuracy.Source: §4.1 · p. 5
  • A scalar potential enforces a symmetric Riemannian-gradient Hessian, restricting reachable attractor-basin topology where logical branch traversal requires non-conservative circulation.Circulation can loop around energetic barriers without requiring a full energy climb.Source: §4.1 · p. 5

4.2 Direct-Gradient Pure-Tensor HHD Formulation

  • DG-HHD parameterizes attraction directly in tangent space and orthogonally projects circulation; its theorem asserts exact Hodge orthogonality and directional contraction.The formulation uses the tangent projection Pq and an orthogonal projection for the rotational field.Source: §4.2 · pp. 5–6

4.3 Resolution of the Autograd Latency Inversion

  • 1.84× per-call speedup and 3.82× wall-clock training acceleration follow from evaluating DG-HHD in pure tensor mode without forward-rollout autograd.The prior RK45 setup required 96 sequential backward graph evaluations across a 16-step rollout.Source: §4.3 · p. 6

5 Language Model Latent Deliberation Bridge

The bridge inserts continuous Riemannian deliberation into a Transformer: it maps an intermediate hidden state to the unit sphere, evolves it, then reinjects it for output generation.

5 Language Model Latent Deliberation Bridge

  • At the deliberation layer, an orthonormal adapter projects the final reasoning-token hidden state onto the unit sphere.The causal Transformer first processes tokens through intermediate layer Ldelib.Source: §5 · p. 6
  • The static context condition is formed by mean-pooling the prompt-prefix hidden states and applying LayerNorm.Source: §5 · p. 6
  • The state undergoes K DG-HHD deliberation steps with embedded RK45, then is unprojected and reinjected into the residual stream for subsequent Transformer layers.Those subsequent layers generate the final output tokens.Source: §5 · pp. 6–7

6 Comprehensive Empirical Evaluation

Across the 15-arm benchmark, DG-HHD combines higher peak reasoning accuracy with long-range drift mitigation and preserved representation rank. On natural-language deduction it scales monotonically with compute, while latency profiling and invariant tests assess efficiency and certification.

6.1 15-Arm Symmetrical Pareto Benchmark (EXP-12)

  • 58.67% ± 14.93% peak accuracy at K = 4 gives DG-HHD a +25.94% absolute gain over scalar-potential Autograd-HHD (32.73% ± 1.10%).The benchmark also reports a +46.53% compute-scaling gain, from 12.13% to 58.67%.Source: §6.1 · p. 7
  • 35.27% ± 3.07% accuracy at K = 32 leaves DG-HHD with a 23.40% drift cliff, versus the unconstrained flow’s 36.87% drop from 82.33% to 45.47%.At K = 32, DG-HHD also retains effective representation rank 15.15 ± 2.19, only 0.67 below 15.82; B0-Pool falls to 10.99.Source: §6.1 · p. 7

6.2 Natural Language Multi-Hop Deductive Reasoning (EXP-14)

  • 49.33% → 50.22% → 50.89% → 51.56% accuracy shows monotonic compute scaling for SLM-DG-HHD-RK45.On deep 4–5 hop problems, its accuracy improves from 47.78% to 48.44% between K = 4 and K = 16 (Δ = −0.66%), while unconstrained SLM-SC-RFM-RK45 degrades by 1.78%.Source: §6.2 · p. 8
  • 25.4% lower inference latency at K = 16 reduces per-forward evaluation from 66.30 ms for Autograd-HHD to 49.44 ms.The comparison is reported for the natural-language multi-hop reasoning evaluation.Source: §6.2 · p. 8

6.3 Microsecond-Level Computational Latency Profiling

  • 1.74×–1.84× faster vector-field calls and 1.70×–2.09× faster multi-step RK45 integration accompany a 3.82× end-to-end training acceleration.The latency evaluation profiled vector-field execution and RK45 integration across batch sizes 1, 8, 32, and 64 on an NVIDIA GPU.Source: §6.3 · p. 8

6.4 Level 0 Deterministic Physical and Algebraic Invariants Certification

  • 30 invariant tests passed unconditionally with zero manual tolerances.Table 4 identifies the suite as Level 0 deterministic algebraic and physical invariants certification.Source: §6.4 · p. 9, Table 4 · p. 9

7 Discussion and Limitations

The discussion presents DG-HHD as a way to combine non-conservative circulation with contraction and stability, and identifies the scope of the language-model experiments as a limitation.

7.1 Resolution of the Trilemma

  • DG-HHD bridges expressivity and drift control by preserving non-conservative circulation alongside exact Hodge orthogonality and analytical Lyapunov contraction.The discussion also states that structure-preserving RATTLE and Adaptive RK45 integration provide a mathematical barrier against manifold divergence; more data alone does not solve discretization error.Source: §7.1 · p. 9

7.2 Limitations and Frontier Trajectories

  • The natural-language experiments use compact Transformer deliberation models with D_model = 128–256 on synthetic deduction tasks; scaling to multi-billion-parameter models remains an engineering direction.Source: §7.2 · p. 9
  • The current bridge deliberates on one continuous thought vector before output generation, rather than over entire token sequences.Source: §7.2 · p. 9

8 Conclusion

The paper introduces PH-LD and DG-HHD to address the Deliberation Drift Cliff in test-time latent compute scaling. It presents the framework as a stable, computationally efficient geometric paradigm for continuous artificial reasoning.

8 Conclusion

  • PH-LD and DG-HHD decouple contractive directional dissipation from scalar Hessian integrability, preserving orthogonal Lyapunov stability while eliminating runtime autograd latency.The framework was evaluated across 30 Level 0 deterministic invariants, a 15-arm Pareto benchmark, and natural-language multi-hop deduction.Source: §8 · p. 10

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