Driving · 2026-10-14

Looped Reasoning Finishes Earlier Than You Think

Sumin Park, T. Konstantin Rusch

AI-assisted summary based on the paper. Consult the linked source for methods and limitations. Benchmark numbers appear after editor verification.

TL;DR

The authors show that slowly decaying oscillatory modes can become rate-limiting for convergence in recurrent reasoners by transferring their convergence rate to coupled complementary dynamics. Instead of damping oscillations, they propose QRR (Quotient Recurrent Reasoner), a gauge-equivariant recurrent architecture with exact token-local phase-rotation symmetry that leaves quotient (reasoning-relevant) dynamics unchanged by symmetry motion. Empirically, QRR converges stably and faster and matches or surpasses recurrent reasoning baselines on Sudoku, Maze, and Mini-ARC.

Why it matters

Looped reasoners can exhibit oscillations, cycles, or drift rather than settling to a fixed point; the paper argues that certain slow oscillatory modes can delay convergence. QRR addresses this by redefining convergence modulo a symmetry: motion along token-local gauge directions does not affect the quotient state. This clarifies when oscillatory latent motion is harmless and when it becomes a bottleneck, and offers an architectural way to avoid bottlenecks without suppressing reasoning-supporting dynamics.

Method

Analyze local dynamics near a solution root by separating an oscillatory spectral coordinate from complementary latent dynamics, and show how a slower-decaying oscillatory mode can dominate the convergence rate (rate inheritance).
Causally verify the mechanism in nonlinear recurrences (e.g., FPRM) by intervening on only the Jacobian-identified oscillatory coordinate while keeping the complementary state fixed, and comparing predicted vs measured response and settling-time gains.
Introduce QRR with token-local U(1)^S gauge-equivariant recurrence and a G-invariant readout; quotient dynamics are insensitive to symmetry-generated phase rotations, enabling earlier halting based on quotient residuals rather than raw-state stationarity.

Limitation

The paper notes that suppressing oscillatory components indiscriminately can disrupt reasoning, and therefore QRR does not assume any correspondence between its gauge directions and the slow modes identified in other reasoners.

Abstract (from arXiv)

Looped reasoning with dynamic depth can be viewed as relaxation toward solution attractors in a learned dynamical landscape. In practice, however, latent trajectories often oscillate, cycle, or drift rather than cleanly converge to stable fixed points. How these non-stationary dynamics contribute to reasoning and constrain convergence remains unclear. We show that slowly decaying oscillatory modes can impose their convergence rate on coupled latent dynamics, creating a bottleneck. Yet damping these modes indiscriminately can disrupt the reasoning computation they support. This tension motivates an architectural approach: introducing rotational degrees of freedom along which symmetry-generated rotational motion leaves the reasoning-relevant quotient dynamics unchanged by construction. We propose QRR (Quotient Recurrent Reasoner), a gauge-equivariant recurrent architecture with an exact per-token phase-rotation symmetry. Motion along these symmetry directions leaves the quotient representation unchanged and is dynamically decoupled from reasoning-relevant dynamics. Empirically, QRR achieves stable and faster convergence to solution roots in quotient space, without forming a dominant slow oscillatory bottleneck, while matching or surpassing recent recurrent reasoning models on Sudoku, Maze, and Mini-ARC.

Publication record

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First public release: 2026-10-13 · Indexed by Pub-AI: 2026-10-14 UTC

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