Circular Reasoning: Understanding Self-Reinforcing Loops in Large Reasoning Models

1 Introduction

• Circular Reasoning traps inference in a self-reinforcing cycle where generated content acts as a logical premise for recurrence, unlike ordinary repetition.The cycle blocks access to a valid conclusion and prevents flexible backtracking.Source: §1 · p. 1
• LoopBench contains 700 samples across seven subtasks and shows severe numerical or statement loops across most evaluated LRMs.Numerical loops are similarly prevalent across model types, whereas statement loops are significantly more common in LRMs than in base or instruction-tuned models.Source: §1 · p. 1
• Loop entry produces an internal state collapse with clear boundaries, while semantic circularity consistently precedes explicit textual repetition.This identifies semantic repetition as a latent structural precursor to the visible loop.Source: §1 · pp. 1–2
• Reasoning impasses generally precipitate loop onset, but statement loops involve dense high-entropy minority tokens whereas numerical loops arise from context saturation during long-form expansions.The two loop types therefore have distinct triggering mechanisms.Source: §1 · p. 2
• A V-shaped attention pattern sustains repetition by focusing on attention sinks and recent tokens once the immediate context becomes repetitive.This self-reinforcement helps explain why the cycle persists after onset.Source: §1 · p. 2
• The proposed CUSUM strategy monitors hidden-state warning signals to anticipate statement loops across diverse models.The approach targets a surge of reflection sentences that typically precedes loop formation.Source: §1 · p. 2

2.1 Benchmark Construction

• The benchmark contains 700 instances across seven subtasks spanning high-precision arithmetic and complex recursive reasoning.Arithmetic tasks include square root, long division, and Newton’s iteration; recursive tasks include Tower of Hanoi, truth-teller puzzles, path planning, and logical paradoxes.Source: §2.1 · p. 2, Figure 2 · p. 2

2.2 Key Observations on LoopBench

• LoopBench measures two granularities of repetition through Numerical Loop Rate and Statement Loop Rate across diverse model families and decoding configurations.The evaluation includes base, instruction-tuned, and reasoning models, with open- and closed-source systems.Source: §2.2 · pp. 2–3
• LoopBench triggers substantially more repetition than AIME2025 and SuperGPQA, with balanced-setting NLR reaching 29.90% and SLR reaching 44.86%.These results indicate that the benchmark exposes model fragility under reasoning patterns overlooked by standard evaluations.Source: §2.2 · p. 3, Table 1 · p. 3
• Numerical loops occur across training paradigms, whereas statement loops are negligible in base and instruction-tuned models but prevalent in LRMs.The passage reports no significant correlation between repetition susceptibility and model scale or distillation status.Source: §2.2 · p. 3
• Closed-source models generally show lower repetition rates than open-source models, yet most proprietary models still have numerical-loop DLR above 15% except gemini-2.5-pro and claude-Opus-4.1.Even advanced systems remain vulnerable to high-precision arithmetic traps.Source: §2.2 · pp. 3–4
• Higher temperature and sampling reduce repetition frequency but do not eliminate it, while numerical loops persist at T = 1.0.Higher randomness improves diversity but can reduce reasoning accuracy and increase hallucinations, creating a loop–accuracy trade-off.Source: §2.2 · p. 4

3.1 Loop Behavior Analysis

• Loop states form distinct, rigid internal regimes: determinism rises as entropy falls, representations contract, and repetitive and normal states have separable boundaries.In deep cycles, cosine similarity approaches 1.0 while vector norm differences vanish.Source: §3.1 · p. 4, Table 2 · p. 5
• Semantic circularity precedes explicit textual repetition, with hidden-state trajectories entering recurrent cluster cycles while sentences remain lexically distinct.The reasoning graph therefore exposes a semantic attractor before verbatim repetition begins.Source: §3.1 · p. 5, Figure 5 · p. 5

3.2 Loop Triggering Mechanisms

• Reasoning impasses trigger loops, but statement loops pivot on reflective high-entropy tokens whereas numerical loops arise from long periodic expansions that saturate context.Statement-loop attention to high-entropy tokens peaks immediately before repetition; numerical loops persist across base and instruction-tuned models.Source: §3.2 · pp. 5–6
• Statement loops become self-reinforcing when dense high-entropy tokens monopolize attention and drive recursive self-reflection.The resulting attention imbalance destabilizes the reasoning path before explicit textual repetition.Source: §3.2 · pp. 5–6

3.3 Loop Persistence Mechanisms

• A V-shaped attention pattern sustains loops by concentrating attention on sinks and recent tokens, causing redundant recent content to increase confidence in its own recurrence.Once repetition saturates the recent context, this feedback confines generation to a nearly inescapable attractor.Source: §3.3 · p. 6

4 Early Prediction of Loop Onset

• A positive hidden-state prediction score signals likely loop entry because its probability rises sharply before repetition but not during normal reasoning.The score is computed from the average last-layer hidden state of each sentence.Source: §4 · p. 7
• CUSUM monitors sentence-level hidden-state scores and accumulates only systematic deviations above a calibrated reference value.Intervention requires the statistic to exceed a threshold for p consecutive steps, filtering transient spikes.Source: §4 · p. 7
• The method detects loop warning signals more than 40 sentences, or 1500 tokens, before repetition on average across diverse models.Results span Qwen, Llama, Phi, and GPT families.Source: §4 · pp. 7–8, Table 3 · p. 8
• Persistence lowers false-positive rates compared with the p = 1 baseline, which can exceed 50% despite marginally higher recall.The evaluation uses Early Detection Rate, False Positive Rate, and sentence- and token-earliness metrics on balanced LoopBench tests.Source: §4 · pp. 7–8, Table 3 · p. 8

5 Related Work

• Neural text degeneration has been addressed through corpus deduplication, repetition-penalizing objectives, isotropy improvements, and decoding methods balancing diversity and coherence.These approaches target repetition across data, training, and inference stages.Source: §5 · p. 8
• Circular reasoning re-emerges in LRMs because reflective trial-and-error chains create meaningful logic loops that standard n-gram deduplication cannot capture.The paper distinguishes this pathology from simple textual repetition.Source: §5 · p. 8
• Mechanistic interpretability studies model behavior through components including neurons, attention heads, FFNs, layers, and circuits, with repetition linked to induction heads and repetition neurons.This paper analyzes LRM loops primarily through hidden states and attention mechanisms.Source: §5 · p. 8

6 Conclusion

• Circular reasoning is characterized through LoopBench as a deterministic phase transition with semantic circularity preceding textual repetition.The analysis identifies task-specific triggers and a V-shaped attention mechanism that sustains loops.Source: §6 · p. 8
• A CUSUM-based strategy anticipates loop formation and supports reasoning stability.The conclusion presents this as the paper’s validated intervention direction.Source: §6 · p. 8

• The study focuses on high-precision arithmetic and recursive reasoning, leaving loop dynamics in other domains unexplored.Source: Limitations · p. 9
• Mechanistic analysis centers on DeepSeek and Qwen families, so universality across other architectures requires further verification.Source: Limitations · p. 9
• The intervention is optimized for statement loops at inference time rather than eliminating the underlying pathology through training.Source: Limitations · p. 9

Ethical Considerations

• Early detection and termination of infinite reasoning loops is presented as a way to reduce computational waste and energy consumption.Source: Ethical Considerations · p. 9
• LoopBench is derived exclusively from public benchmarks and involves no personally identifiable information or safety concerns.Source: Ethical Considerations · p. 9

A Detail of LoopBench

• The benchmark uses high-precision arithmetic and recursive iterative logic, covering seven sub-tasks generated as 100 samples per sub-task.Source: A Detail of LoopBench · p. 11
• Selected arithmetic tasks cap required precision at 500 digits, while all samples require extensive multi-step derivations typically exceeding 100 reasoning steps.Source: A Detail of LoopBench · p. 11
• A unified system instruction mandates explicit step-by-step derivation across LoopBench queries to prevent shortcut-based reasoning.Source: A Detail of LoopBench · p. 11, Figure 10 · p. 11
• The benchmark tests numerical fidelity through high-precision arithmetic and logical coherence through complex recursive reasoning.Source: A Detail of LoopBench · p. 11

B.1 Detailed Setup

• The evaluation spans open- and closed-source models across reasoning, base, instruction-tuned, and API-based systems.Source: §B.1 · pp. 11–13
• LoopBench contains high-precision arithmetic and recursive reasoning tasks, complemented by AIME2025 and a stratified SuperGPQA subset.Source: §B.1 · p. 13
• Numerical Loop Rate and Statement Loop Rate quantify the two observed repetition forms.Source: §B.1 · p. 13
• Open-weight models are tested under conservative, balanced, and explorative decoding with a fixed repetition penalty of 1.1.Source: §B.1 · p. 14

B.2 Case Study

• Case studies conducted locally, through official APIs, and on official web platforms show both numerical and statement loops.Source: §B.2 · p. 14

C Extended Analysis of Circular Reasoning

• Extended analyses on DS-Qwen-14B and Qwen-8B support circular reasoning as a consistent behavioral pattern rather than isolated anomalies.Source: C Extended Analysis of Circular… · p. 14

C.1 Additional Results on Phase Transition

• Across DS-Qwen-14B and Qwen3-8B, loop cases show a Determinism Surge, while non-loop baselines retain stochastic variation.Source: §C.1 · p. 14, Figure 11 · p. 15, Figure 12 · p. 15
• Hidden-state projections collapse into low-dimensional periodic orbits, and semantic circularity precedes explicit textual repetition.Source: §C.1 · p. 14
• Reflective markers such as “Wait,” “But,” and “Maybe” accompany recursive hesitation rather than resolving ambiguity.Source: §C.1 · pp. 14–15

C.2 Additional Results of ‘V-Shaped’ Attention Patterns

• V-shaped attention concentrates on attention sinks and recent redundant tokens, reinforcing repetition and blocking escape to broader context.Source: §C.2 · p. 15

C.3 Additional Results of Statement Triggers

• Statement-loop cases show attention spikes on high-entropy pivot tokens immediately before repetition, unlike relatively uniform non-loop attention.Source: §C.3 · p. 16
• Reasoning loops contain significantly more high-entropy tokens than non-loop paths and instruction-following outputs.Source: §C.3 · p. 16

D Experimental Setup for Prediction

• CUSUM detection statistics accumulate rapidly and cross the decision threshold in loop cases, while remaining low-amplitude for non-loop reasoning.This pattern is reported for both DS-Qwen-14B and Qwen3-8B and indicates resistance to false alarms.Source: D Experimental Setup for Prediction · p. 16

D.1 Evaluation Metrics

• Loop and non-loop figures distinguish a determinism surge at repetition onset from high-variance, stochastic baseline behavior.Numerical and statement loops show entropy collapse, whereas corresponding non-loop cases lack a distinct collapse pattern.Source: Figure 15 · p. 17, Figure 16 · p. 17, Figure 17 · p. 18
• Early Detection Rate measures the proportion of loops detected before their actual repetition onset.A detection is successful when t_alert < t_onset.Source: §D.1 · p. 17
• False Positive Rate measures the proportion of normal, non-repetitive samples incorrectly flagged as loops.The metric uses false alarms among normal samples as its numerator.Source: §D.1 · p. 17
• Average Sentence Earliness and Average Token Earliness quantify warning lead time at sentence and token granularity, respectively.Higher ASE values indicate earlier sentence-level warnings, while ATE measures the corresponding token-level margin.Source: §D.1 · p. 17

D.2 Hyperparameter Configuration.

• CUSUM hyperparameters are tuned separately for each model through validation-set grid search over persistence p and threshold scaling factor α.The selected configurations are summarized in Table 10.Source: §D.2 · p. 18

E Details of Intervention Strategy

• Upon detecting a numerical or statement loop, the method injects a conclusion-inducing token sequence at the breakpoint rather than truncating the reasoning trace.The example prompt is “Conclude now and provide the answer.”Source: E Details of Intervention Strategy · p. 18
• The injected sequence externally perturbs local context, shifts next-token probabilities toward conclusion, and breaks the deterministic deadlock.The intervention guides finalization from the augmented context instead of continued repetition.Source: E Details of Intervention Strategy · p. 18
• Looping hidden states become progressively rigid, with high cosine similarity and low L2 distance across repeated cycles compared with dynamic baselines.This information blockage is observed in DS-Qwen-14B and Qwen3-8B cases.Source: Figure 19 · p. 19, Figure 20 · p. 19, Figure 21 · p. 20
• Early intervention raises DS-Qwen-7B completion from 0.80 to 0.88 and improves restoration across most evaluated models.The reported gains support intervening before repetitive patterns become entrenched.Source: E Details of Intervention Strategy · p. 19, Table 12 · p. 27

F Declaration of AI Use

• Semantic circularity precedes exact textual repetition, while reasoning graphs collapse into stable periodic attractors or low-dimensional manifolds.These patterns appear across DS-Qwen-14B and Qwen3-8B statement-loop cases.Source: Figure 22 · p. 20, Figure 23 · p. 21, Figure 24 · p. 21
• V-shaped attention concentrates on recent tokens and suppresses global context, reinforcing numerical sequences and reflective phrases across architectures.The same local-history pattern is reported for numerical and statement loops.Source: Figure 28 · p. 25, Figure 29 · p. 25, Figure 30 · p. 25

E Details of Intervention Strategy

• CUSUM-based detection processes reasoning-step classifier scores with reference drift, threshold, and persistence parameters, returning a detection flag and trigger step.The algorithm triggers intervention when the cumulative statistic exceeds its threshold under the persistence condition.Source: F Declaration of AI Use · p. 26

F Declaration of AI Use

• High-entropy pivot tokens attract increased attention immediately before loops, whereas non-loop reasoning distributes attention more evenly.Loop states also contain a higher density of these uncertainty-inducing tokens.Source: Figure 32 · p. 27, Figure 33 · p. 27, Figure 34 · p. 28