Looped Reasoning Finishes Earlier Than You Think

Sumin Park; T. Konstantin Rusch

cs.LGOct 13, 2026

Looped Reasoning Finishes Earlier Than You Think

Authors: Sumin Park, T. Konstantin Rusch

Organizations: ELLIS Institute Tübingen · Max Planck Institute for Intelligent Systems · Tübingen AI Center · Liquid AI

Abstract

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.

Figures & tables

Figure 1: Oscillatory geometry and quotient convergence. FPRM and HRM exhibit distinct alternating and spiral dynamics near solution roots. In QRR, raw motion persists while quotient trajectory contracts.

Figure 2: Local oscillatory-coupling theory predicts the response and settling cost in the original FPRM recurrence. We control only the Jacobian-identified oscillatory coordinate once with $\alpha\in\{0,0.5,1\}$ and then resume unmodified recurrent updates. HRM results in Appendix A.4.1 .

Figure 3: Convergence and halting in quotient space.

Figure 4: Overview of the QRR recurrent block.

Figure 5: Spectral analysis and settling near solution roots for FPRM and QRR. Results for HRM in Appendix A.4.2

Model	${\tau_{\mathrm{root}}}$	Correct root found
	median [IQR] $\downarrow$	(%) $\uparrow$
HRM	1,032 [744, 1,272]	55.2
EqR	672 [546, 966]	86.1
FPRM	234 [152, 396]	90.4
QRR	122 [108, 148]	89.6

Table 2: Convergence and adaptive halting. (a) Root convergence on Sudoku-Extreme. (b) Adaptive halting of FPRM and QRR across tasks. Coverage is the fraction halted within finite cap.

Variant	Exact (%) $\uparrow$	Coverage (%) $\uparrow$	False halt (%) $\downarrow$	${\tau_{\mathrm{halt}}}$ [IQR] $\downarrow$
Training
QRR	97.9	98.0	0.1	27 [23, 63]
global $U(1)$	91.3	0.0	–	CAP (never fires)
first order w/o momentum	87.8	79.6	0.1	28 [22, 44]
FPRM trunk + momentum	77.0	100	23.0	12,650 [12,522, 12,871]

Table 3: QRR ablations on Sudoku-Extreme.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: The local oscillatory-coupling prediction extends to the two-dimensional spiral dynamics of HRM. We modify only the Jacobian-identified oscillatory coordinate once with $\alpha\in\{0,0.5,1\}$ and subsequently resume the original nonlinear HRM recurrence.

Figure 7: Spectral organization and settling in HRM and QRR. (a) Leading and complementary spectral rates near certified solution equilibria. QRR rates are measured in quotient space. (b) Distribution of the dominance score

D=\log r_{\mathrm{comp}}/\log r_{\mathrm{lead}}

. (c) Root-relative settling time before and after factoring out the leading-mode response.

Figure 8: Spectral separation of leading oscillatory modes. (a) Leading-mode modulus

r

and complementary spectral radius

\rho(A)

. (b) Spectral margin

r-\rho(A)

. (c) Oscillation angle

|\omega|

for FPRM and HRM.

Intervention	$\bm{\Delta}$ final accuracy	95% CI
Oscillatory suppression, early	$-5.9$ pp	$[-13.1,-0.6]$
Energy-matched control, early	$-1.1$ pp	$[-2.8,0.0]$
Oscillatory suppression, late	$0.0$ pp	$[0.0,0.0]$

Appendix

Table 4: Persistent suppression of the FPRM flip mode. Suppressing the alternating component before output stabilization reduces final exact accuracy, whereas an energy-matched non-oscillatory intervention does not show a significant effect.

Figure 9: Adaptive halting as a function of Sudoku difficulty. Difficulty as the number of blank cells (top) and the backtrack count (bottom). Left: mean halting time among certified examples. Middle: fraction of examples certified within the rollout budget. Right: exact-match accuracy.

Model	Layer-forward computation per native step	$\ell_{m}$ ( lfwd /step)
QRR	$1\times L_{\mathrm{layers}}=1\times 2$	$2$
FPRM	$1\times L_{\mathrm{layers}}=1\times 2$	$2$
HRM	$H_{\mathrm{cyc}}\bigl(L_{\mathrm{cyc}}L_{\mathrm{layers}}+H_{\mathrm{layers}}\bigr)=2(2\cdot 4+4)$	$24$
EqR	$H_{\mathrm{cyc}}(L_{\mathrm{cyc}}+1)L_{\mathrm{layers}}=3(6+1)2$	$42$

Appendix

Table 5: Conversion from native recurrent steps to recurrent layer-forward equivalents. An lfwd counts one forward evaluation of one recurrent network layer.

Data and optimization
Setting	Sudoku-Extreme	Maze-Hard	Maze-Unique	Mini-ARC
Training data	1K $\times$ 1000 aug.	1K, no aug.	1K, no aug.	149 tasks, 100 aug.
Training epochs	60K	60K	6K	100K
Global batch size	768	768	384	768
Learning rate	$3{\times}10^{-4}$	$1{\times}10^{-4}$	$2{\times}10^{-4}$	$6{\times}10^{-4}$
Weight decay	$10^{-3}$	$1.0$	$1.0$	$0.1$

Appendix

Table 6: Task-specific training and inference configurations for QRR. The core architecture is shared across tasks. The reported result is from the best in-run evaluation checkpoint.

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Reasoning models (e.g., DeepSeek-R1) generate long chains of thought to solve harder problems, but they often loop, repeating the same text at low temperatures or with greedy decoding. We study why this happens and what role temperature plays. With open reasoning models, we find that looping is common at low temperature. Larger models tend to loop less, and distilled students loop significantly even when their teachers rarely do. This points to mismatches between the training distribution and the learned model, which we refer to as errors in learning, as a key cause. To understand how such errors cause loops, we introduce a synthetic graph reasoning task and demonstrate two mechanisms. First, risk aversion caused by hardness of learning: when the correct progress-making action is hard to learn but an easy cyclic action is available, the model puts relatively more probability on the cyclic action and gets stuck. Second, even when there is no hardness, Transformers show an inductive bias toward temporally correlated errors, so the same few actions keep being chosen and loops appear. Higher temperature reduces looping by promoting exploration, but it does not fix the errors in learning, so generations remain much longer than necessary at high temperature; in this sense, temperature is a stopgap rather than a holistic solution. We end with a discussion of training-time interventions aimed at directly reducing errors in learning.

Abstract

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