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 α∈{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
τroot
Correct root found
median [IQR] ↓
(%) ↑
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 (%) ↑
Coverage (%) ↑
False halt (%) ↓
τhalt [IQR] ↓
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 α∈{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=logrcomp/logrlead . (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 ρ(A) . (b) Spectral margin r−ρ(A) . (c) Oscillation angle ∣ω∣ for FPRM and HRM.
Intervention
Δ 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
ℓm ( lfwd /step)
QRR
1×Llayers=1×2
2
FPRM
1×Llayers=1×2
2
HRM
Hcyc(LcycLlayers+Hlayers)=2(2⋅4+4)
24
EqR
Hcyc(Lcyc+1)Llayers=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.
Setting
Sudoku-Extreme
Maze-Hard
Maze-Unique
Mini-ARC
Data and optimization
Training data
1K × 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
3×10−4
1×10−4
2×10−4
6×10−4
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.
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.
Charilaos Pipis, Shivam Garg, Vasilis Kontonis +3
MIT · Microsoft Research · University of Wisconsin–Madison
Recurrent models, which repeatedly update latent states with shared computation blocks, have emerged as powerful architectures for solving complex reasoning tasks. Existing inference-time methods scale computation by running more steps or sampling more trajectories, but ignore information revealed within each trajectory. Here we show that recurrent models can be improved at inference time by using their own readout probabilities to steer latent dynamics without retraining. We introduce Readout Feedback (RoFB), a test-time intervention that converts intermediate predictions into token-wise pairwise coupling forces injected into the latent dynamics. Across three recurrent models (AKOrN, ItrSA++, TRM) on Sudoku and Maze, RoFB yields clear gains in four of six model-task pairs and a small gain in one. The four positive pairs achieve performance unattainable by merely running more steps or selecting from multiple trajectories, at comparable or lower computational cost. These results suggest that closed-loop steering of latent dynamics can serve as a complementary inference-time control mechanism for recurrent reasoning models.
Looped architectures provide an inductive bias toward learning step-by-step procedures for tasks that require compositional reasoning. The number of effective layers reached by looping determines the quality of the solution these models find. Like deep architectures, looped architectures are prone to a signal propagation problem induced by depth as the halting decision is postponed. In this paper, we address this signal propagation issue using pre-norm layers and residual scaling. Building on these architectural modifications, we propose FPRM, a Transformer-based Fixed-Point Reasoning Model that uses fixed-point convergence as an end-to-end halting mechanism in a looped architecture. We show that fixed-point halting allows FPRM to adapt its compute to task difficulty. FPRM is effective on common reasoning benchmarks, namely Sudoku, Maze, state-tracking, and ARC-AGI.
Sajad Movahedi, Vera Milovanović, Shlomo Libo Feigin +5
ELLIS Institute Tübingen, Max Planck Institute for Intelligent Systems, Tübingen AI Center · ETH Zurich · Swiss Institute of Bioinformatics +2