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arxiv: 2603.10384 · v2 · submitted 2026-03-11 · 💻 cs.AI

Recognition: 2 theorem links

· Lean Theorem

Beyond Scalars: Evaluating and Understanding LLM Reasoning via Geometric Progress and Stability

Xinyan Jiang , Ninghao Liu , Di Wang , Lijie Hu
Authors on Pith no claims yet

Pith reviewed 2026-05-15 13:57 UTC · model grok-4.3

classification 💻 cs.AI
keywords LLM reasoninggeometric kinematicshallucination detectiontrajectory analysisprogress and stabilityTRACED framework
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The pith

Correct LLM reasoning follows high-progress stable geometric trajectories while hallucinations show low progress and high curvature instability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that scalar probabilities are insufficient for evaluating LLM reliability and instead introduces a geometric approach to analyze reasoning traces. It decomposes these traces into measures of progress, representing displacement along the reasoning path, and stability, based on curvature. Correct reasoning is shown to exhibit high progress with stability, contrasting with the stalled displacement and fluctuating curvature of hallucinations. This framework, called TRACED, provides competitive performance on benchmarks while offering a physical interpretation of reasoning dynamics through concepts like hesitation loops and certainty accumulation. A reader would care because it shifts evaluation from simple scores to structural dynamics that could improve detection of errors in AI outputs.

Core claim

By modeling LLM reasoning as geometric trajectories, the work establishes that factual correctness corresponds to trajectories with high displacement progress and low curvature fluctuations, while hallucinations correspond to low-progress paths with high curvature variations, enabling a probabilistic framework that maps these geometric signatures to reasoning quality.

What carries the argument

The TRACED framework that decomposes reasoning traces into Progress (displacement) and Stability (curvature) to reveal topological differences between correct and hallucinatory reasoning.

If this is right

  • Provides a more robust alternative to scalar probability-based evaluation across benchmarks.
  • Links geometric properties to cognitive-like concepts such as hesitation loops for high curvature and certainty accumulation for displacement.
  • Reveals distinct topological divergence between correct and hallucinatory reasoning patterns.
  • Achieves competitive performance with superior robustness in distinguishing correct outputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This geometric lens could extend to monitoring and intervening in ongoing reasoning processes to prevent hallucinations.
  • Similar trajectory analysis might apply to understanding decision-making in other AI models beyond language.
  • Connections could be drawn to human cognitive models where hesitation corresponds to high-curvature thinking paths.

Load-bearing premise

LLM reasoning traces can be meaningfully represented and decomposed as geometric trajectories whose progress and curvature properties reliably correlate with factual correctness versus hallucination.

What would settle it

A study showing no statistically significant difference in progress and stability metrics between correct and hallucinatory reasoning traces on a large benchmark would falsify the main claim.

Figures

Figures reproduced from arXiv: 2603.10384 by Di Wang, Lijie Hu, Ninghao Liu, Xinyan Jiang.

Figure 1
Figure 1. Figure 1: Topological Divergence of Reasoning Quality. Joint distribution of cumulative displacement (M) and curvature (K) across Structured and Open-Ended domains. The visualization confirms a consistent separation: correct reasoning traces (blue) exhibit a high-displacement, low-curvature pattern, while incorrect chains (red) are characterized by low-displacement stagnation and high-curvature oscillations. Our con… view at source ↗
Figure 3
Figure 3. Figure 3: Robustness and Efficiency. (Left) Class Imbalance: TRACED maintains discriminative stability against distributional shifts, specifically where the prior P(yn = 1) ∈ [0.3, 0.7]. (Right) Data Efficiency: The method achieves rapid geomet￾ric convergence, reaching a stability plateau with merely N ≈ 400 reference samples. 2 4 6 8 10 Dimension (k) 0.71 0.74 0.76 0.78 AUROC DeepSeek-R1-Llama-8B 2 4 6 8 10 Dimens… view at source ↗
Figure 6
Figure 6. Figure 6: Geometric Differences Across Domains. (Left) Cur￾vature Distribution: Structured reasoning (blue) exhibits a nar￾row peak, contrasting with the broad, heavy tail of open-ended reasoning (purple). (Right) Displacement Accumulation: Struc￾tured trajectories reveal step-wise growth driven by discrete break￾throughs, while open-ended tasks exhibit a smooth, continuous semantic flow. correct incorrect [PITH_FU… view at source ↗
Figure 5
Figure 5. Figure 5: Kinematic Scaling Laws of Reasoning. Log-log plot of Net Displacement D(t) = ||zT − z0||2 vs. reasoning length across six domains. Blue (Correct): Exhibits linear scaling (slope ≈ 0.82), characteristic of directed evolution (D ∝ T) where computation yields direct semantic progress. Red (Incor￾rect): Follows sub-linear scaling (slope ≈ 0.53), resembling ran￾dom walk (D ∝ √ T) and indicating progress stagnat… view at source ↗
Figure 8
Figure 8. Figure 8: Geometric Cost of State Transitions. (Left) Avg. curvature change (∆K). (Right) Avg. displacement change (∆M). Curvature encodes the cost of uncertaint directional reorientation, while displacement reflects accumulative semantic progress [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Geometric-Semantic Synchronization. Alignment between geometric displacement (gray) and cognitive states. token representations. Unlike prior works, we construct an evaluation signal based on theoretically grounded geo￾metric features of temporal reasoning, achieving consistent improvements and scalability across diverse tasks. Representation Analysis of Reasoning. Research on in￾ternal representations has… view at source ↗
Figure 10
Figure 10. Figure 10: Geometric Modes Across Different Models. Visualization of the reasoning geometric signatures for Qwen2.5, Llama-3.1, and Qwen3-Thinking. Consistent topological separation is observed across all architectures. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Sensitivity Analysis of Dimension k (AUPR). We evaluate the AUPR performance (↑) of TRACED across four models. 2 4 6 8 10 Dimension (k) 0.65 0.68 0.71 0.74 FPR@95 DeepSeek-R1-Llama-8B 2 4 6 8 10 Dimension (k) 0.54 0.58 0.62 0.65 Qwen3-4B-Thinking 2 4 6 8 10 Dimension (k) 0.67 0.71 0.75 0.78 Llama-3.1-8B-Instruct 2 4 6 8 10 Dimension (k) 0.69 0.73 0.76 0.80 Qwen2.5-7B-Instruct [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 12
Figure 12. Figure 12: Sensitivity Analysis of Dimension k (FPR@95). We evaluate the FPR@95 performance (↓) of TRACED across four models [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
read the original abstract

Evaluating LLM reliability via scalar probabilities often fails to capture the structural dynamics of reasoning. We introduce TRACED, a framework that assesses reasoning quality through theoretically grounded geometric kinematics. By decomposing reasoning traces into Progress (displacement) and Stability (curvature), we reveal a distinct topological divergence: correct reasoning manifests as high-progress, stable trajectories, whereas hallucinations are characterized by low-progress, unstable patterns (stalled displacement with high curvature fluctuations). Leveraging these signatures, our probabilistic framework achieves competitive performance and superior robustness across diverse benchmarks. Crucially, TRACED bridges geometry and cognition by mapping high curvature to ''Hesitation Loops'' and displacement to ''Certainty Accumulation'', offering a physical lens to decode the internal dynamics of machine thought.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces the TRACED framework, which represents LLM reasoning traces as geometric trajectories and decomposes them into Progress (quantified by displacement) and Stability (quantified by curvature). It claims that correct reasoning produces high-progress, stable trajectories while hallucinations produce low-progress, unstable patterns featuring stalled displacement and high curvature fluctuations (termed 'Hesitation Loops'), with displacement linked to 'Certainty Accumulation'. The framework is asserted to deliver competitive performance and superior robustness on benchmarks via a probabilistic model derived from these geometric signatures.

Significance. If the geometric embedding and kinematic computations are rigorously defined and the claimed separation between correct and hallucinated trajectories is empirically validated with controls, the work could supply a novel non-scalar lens for diagnosing LLM reasoning dynamics and reliability. The explicit mapping from curvature/displacement to cognitive interpretations is a distinctive feature that, if substantiated, would strengthen interpretability claims beyond standard logit-based metrics.

major comments (3)
  1. [Framework / Methods] The manuscript provides no definition of the ambient geometric space (hidden states, logit space, or otherwise) nor the discretization procedure that converts discrete token sequences into continuous trajectory points. Without these, the formulas for displacement (progress) and curvature (stability) cannot be evaluated or reproduced, rendering the central topological-divergence claim unverifiable.
  2. [Experiments / Results] The abstract asserts 'competitive performance and superior robustness' yet supplies no benchmark list, baseline comparisons, error bars, or statistical tests. Any quantitative claim that the geometric signatures improve over scalar-probability methods requires explicit tables or figures showing effect sizes and controls for prompt length or model scale.
  3. [Interpretation / Discussion] The mapping of high curvature to 'Hesitation Loops' and displacement to 'Certainty Accumulation' is presented as a cognitive bridge, but no derivation or validation links the kinematic quantities to these interpretations; the correspondence risks being post-hoc unless supported by controlled ablation or human-alignment studies.
minor comments (2)
  1. [Framework] Notation for the kinematic quantities (e.g., symbols for displacement vector and curvature scalar) should be introduced explicitly with equations rather than descriptive prose only.
  2. [Abstract] The phrase 'distinct topological divergence' is imprecise if the analysis remains strictly geometric (curvature and displacement) rather than invoking topological invariants; consider replacing with 'geometric divergence' or defining the topological aspect.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback, which highlights important areas for improving clarity, reproducibility, and empirical rigor. We will revise the manuscript to address each point as detailed below.

read point-by-point responses

  1. Referee: [Framework / Methods] The manuscript provides no definition of the ambient geometric space (hidden states, logit space, or otherwise) nor the discretization procedure that converts discrete token sequences into continuous trajectory points. Without these, the formulas for displacement (progress) and curvature (stability) cannot be evaluated or reproduced, rendering the central topological-divergence claim unverifiable.

    Authors: We agree that the current manuscript lacks sufficient detail on these foundational elements. In the revised version, we will explicitly define the ambient space as the hidden-state activations from the LLM's final transformer layer (prior to the language modeling head) and describe the discretization as mapping each generated token to its corresponding hidden-state vector, with trajectories constructed by connecting consecutive points. We will also include the exact formulas for displacement (as Euclidean norm of the net vector) and curvature (as the discrete second derivative approximating turning rate), along with pseudocode for the full computation pipeline. revision: yes

  2. Referee: [Experiments / Results] The abstract asserts 'competitive performance and superior robustness' yet supplies no benchmark list, baseline comparisons, error bars, or statistical tests. Any quantitative claim that the geometric signatures improve over scalar-probability methods requires explicit tables or figures showing effect sizes and controls for prompt length or model scale.

    Authors: We acknowledge that the main text currently under-reports the experimental details supporting the abstract claims. The revised manuscript will add a dedicated results table listing all benchmarks (arithmetic reasoning, commonsense QA, and hallucination detection tasks), direct comparisons against scalar baselines such as token probability and perplexity, mean performance with standard deviations across 5 random seeds, and paired statistical tests. We will further include an analysis subsection with controls for prompt length and model scale, reporting effect sizes where the geometric features yield measurable gains. revision: yes

  3. Referee: [Interpretation / Discussion] The mapping of high curvature to 'Hesitation Loops' and displacement to 'Certainty Accumulation' is presented as a cognitive bridge, but no derivation or validation links the kinematic quantities to these interpretations; the correspondence risks being post-hoc unless supported by controlled ablation or human-alignment studies.

    Authors: The interpretations are offered as kinematic analogies motivated by the observed divergence in our trajectory data, where high-curvature segments frequently coincide with repetitive or stalled generation. To mitigate the post-hoc concern, we will add an ablation experiment quantifying the performance drop when curvature features are removed from the probabilistic model. We will also revise the discussion to present these mappings as interpretive hypotheses rather than established cognitive equivalences and explicitly list controlled human-alignment studies as future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; framework introduces independent geometric decomposition

full rationale

The paper presents TRACED as a new framework that decomposes LLM reasoning traces into Progress (displacement) and Stability (curvature) to reveal topological differences between correct reasoning and hallucinations. No equations, definitions, or steps in the abstract reduce the central claims to fitted inputs, self-referential mappings, or self-citations by construction. The interpretive mappings (high curvature to Hesitation Loops, displacement to Certainty Accumulation) are offered as bridges from geometry to cognition rather than tautological redefinitions. Without explicit self-citation chains or ansatzes that presuppose the reported signatures, the derivation remains self-contained and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Abstract-only access limits visibility into exact parameters and axioms; the framework rests on the unproven premise that geometric kinematics meaningfully capture reasoning quality.

axioms (1)
  • domain assumption LLM reasoning traces can be represented as trajectories in a geometric space where displacement equals progress and curvature equals stability
    This representation is the foundational modeling choice stated in the abstract.
invented entities (2)
  • Hesitation Loops no independent evidence
    purpose: Cognitive interpretation of high-curvature unstable patterns in hallucinations
    New mapping introduced to link geometry to internal model dynamics
  • Certainty Accumulation no independent evidence
    purpose: Cognitive interpretation of high-displacement stable progress in correct reasoning
    New mapping introduced to link geometry to internal model dynamics

pith-pipeline@v0.9.0 · 5421 in / 1300 out tokens · 86467 ms · 2026-05-15T13:57:22.388728+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Semantic Step Prediction: Multi-Step Latent Forecasting in LLM Reasoning Trajectories via Step Sampling

    cs.LG 2026-04 unverdicted novelty 7.0

    Applying STP at consecutive semantic reasoning steps achieves 168x more accurate multi-step latent prediction on ProcessBench than frozen baselines, with trajectories forming smooth curves best captured by non-linear ...

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · cited by 1 Pith paper

  1. [1]

    URL https: //aclanthology.org/2025.acl-long.880/

    doi: 10.18653/v1/2025.acl-long.880. URL https: //aclanthology.org/2025.acl-long.880/. Zhao, Z., Koishekenov, Y ., Yang, X., Murray, N., and Can- cedda, N. Verifying chain-of-thought reasoning via its computational graph.arXiv preprint arXiv:2510.09312, 2025. Zhou, Y ., Wang, Y ., Yin, X., Zhou, S., and Zhang, A. R. The geometry of reasoning: Flowing logic...

    work page doi:10.18653/v1/2025.acl-long.880 2025

  2. [2]

    Temperature Sampling (T= 0.7 ):We set the temperature to0.7for all generation

    Generation Hyperparameters. Temperature Sampling (T= 0.7 ):We set the temperature to0.7for all generation. This setting strikes an optimal balance: it introduces sufficient stochasticity to reveal diverse reasoning paths (and potential hallucinations) for topological analysis, while maintaining enough coherence. Maximum Token Limit:(1)Standard Instruct Mo...

    work page

  3. [3]

    Answer: <NUMBER>

    Task-Specific Prompting.To ensure validity, we adopted task-specific templates derived from OpenCompass (Contribu- tors, 2023). We classify our benchmarks into four structural categories based on the required output format (see Table 4 for dataset mapping). Table 4.Overview of the Datasets.Statistics of the original source datasets.Note:Due to the cost of...

    work page 2023

  4. [4]

    Answer:”) exists. If found, we extract the subsequent text and identify all numeric values using the regexr

    Type A: Numeric Extraction (GSM8K).For arithmetic tasks, we employ a regex-based extraction pipeline robust to formatting noise. We verify if the specific answer prefix (e.g., “Answer:”) exists. If found, we extract the subsequent text and identify all numeric values using the regexr"\d+\.?\d*". We select thelast identified number as the prediction. Both ...

    work page

  5. [5]

    (?i)Answer:\s*?([A−D])?

    Type B: Multiple Choice Matching (GPQA, Social IQA, Fables).For option-selection tasks, we implement a parser to identify the predicted option letter. The parser scans the text following the answer delimiter for patterns matching r"(?i)Answer:\s*?([A−D])?" . We extract the final matching capture group, normalize it to uppercase, and perform an exact strin...

    work page

  6. [6]

    We extract the string content within the last \boxed{} tag

    Type C: Symbolic Matching (MATH).For complex mathematical expressions, we rely on the L ATEX \boxed{} format. We extract the string content within the last \boxed{} tag. To handle variability in L ATEX spacing, we normalize both the extracted content and the gold label by stripping all whitespace (e.g., x + y→x+y ) before comparison. 13 Evaluating and Und...

    work page

  7. [7]

    answer is

    Type D: Dynamic Extraction (TheoremQA).Given the heterogeneous output formats of TheoremQA, we implement a conditional parser guided by the sample’s Answer type. We first isolate the concluding segment following the phrase “answer is”. •Bool:We scan for case-insensitive occurrences of “True” or “False”. •Integer/Float:We apply the same regex extraction st...

    work page 2024

  8. [8]

    This limits the influence of any potential model bias to a small fraction of the dataset

    Minimal Dependence:The majority of samples are labeled by the deterministic parser, with the LLM judge required only for the remaining ambiguous instances. This limits the influence of any potential model bias to a small fraction of the dataset

    work page

  9. [9]

    Hesitation Loop

    Judge Audit:To verify the reliability of the semantic judge, we performed a manual audit on a random subset of 100 trajectories labeled by the judge. We observed an agreement rate of over 95% with human annotation, confirming that the instruction-tuned Llama-3-70B provides high-fidelity verdicts for these objective reasoning tasks. A.2.4. FINALLABELINGPRO...

    work page arXiv 2020

  10. [10]

    This indicates that the geometric signatures (Mn, Kn) are robust enough to separate classes even when the priors are not perfectly calibrated

    Stability Region (α∈[0.3,0.7] ):TRACED demonstrates remarkable stability when the data ratio fluctuates within the moderate range. This indicates that the geometric signatures (Mn, Kn) are robust enough to separate classes even when the priors are not perfectly calibrated

    work page

  11. [11]

    For example, at α= 0.1 (severe hallucination dominance), the AUROC forLlama-3.1-8B decreases by approximately6%compared to the balanced setting

    Performance Degradation at Extremes ( α <0.2 or α >0.8 ):As hypothesized, performance drops in extreme imbalance scenarios. For example, at α= 0.1 (severe hallucination dominance), the AUROC forLlama-3.1-8B decreases by approximately6%compared to the balanced setting. Table 6.Data Ratio Robustness.Average AUROC scores across six datasets under varying Pos...

    work page

  12. [12]

    For instance, at γ= 0.2 , the AUROC forDeepSeek-R1lags by approximately 4% compared to the full setting

    Sensitivity at Low Data Regime (γ <0.5 ):In the low-data regime (e.g., N= 80∼320 ), we observe a noticeable performance gap. For instance, at γ= 0.2 , the AUROC forDeepSeek-R1lags by approximately 4% compared to the full setting. This aligns with statistical theory: estimating the covariance matrix Σc in high-dimensional space requires sufficient samples ...

    work page

  13. [13]

    Beyond this point, increasing the data to 800 samples (γ= 1.0 ) yields only marginal gains

    Stability Plateau (γ≥0.5 ):Performance stabilizes significantly once the reference set size reaches approximately 400 samples (γ= 0.5 ). Beyond this point, increasing the data to 800 samples (γ= 1.0 ) yields only marginal gains. This suggests that N≈400 serves as a sufficient effective sample size to capture the converged geometric topology of reasoning, ...

    work page

  14. [14]

    This reflects astrict requirement for directness: in logical reasoning, the correct path is extremely narrow

    Strict Constraints (Structured):As shown in Figure 6 (Left), valid trajectories in GSM8K exhibit ahighly con- centrateddistribution. This reflects astrict requirement for directness: in logical reasoning, the correct path is extremely narrow. Any significant deviation (increased curvature) usually indicates a distraction or a logical error, rather than a ...

    work page

  15. [15]

    This reflects a high tolerance for variation: open-ended contexts allow the model to elaborate on details or use different sentence structures

    High Tolerance (Open-Ended):In contrast, SocialIQA displays abroad, heavy-tailed distribution. This reflects a high tolerance for variation: open-ended contexts allow the model to elaborate on details or use different sentence structures. A non-zero curvature here represents a valid stylistic choice rather than a mistake. Implication for Evaluation.This o...

    work page

  16. [16]

    The semantic distance often remains flat during intermediate calculations and showssharp, discrete jumpsat specific moments

    Step-wise Accumulation (Structured):The displacement curve for structured tasks resembles astaircase pattern. The semantic distance often remains flat during intermediate calculations and showssharp, discrete jumpsat specific moments. These jumps correspond to solving a distinct sub-problem (e.g., deriving a key variable value), which suddenly pushes the ...

    work page

  17. [17]

    The displacement increases steadily without sharp jumps

    Smooth Accumulation (Open-Ended):Conversely, open-ended tasks exhibit asmooth, continuous growth. The displacement increases steadily without sharp jumps. This reflects the nature of narrative construction, where under- standing and context are built up gradually and continuously as the description evolves, eventually saturating when the scenario is fully...

    work page

  18. [18]

    ”Let’s assume the probability isx/y...” Exploration Mn: LowKn: High (Diffusive)

    work page

  19. [19]

    ”Wait, this logic might double count the overlap...” Reflection (↑Curvature)

    work page

  20. [20]

    ”Let’s try a different approach using combinations...” Exploration

    work page

  21. [21]

    ”But does this satisfy the initial condition? I’m not sure...” Reflection (↑Curvature) Correct(Directed)

    work page

  22. [22]

    ”First, we calculate the total outcomes as63...” Exploration Mn: High Kn: Low(Ballistic)

    work page

  23. [23]

    ”This implies that the sum must be even...” Certainty (→Displacement)

    work page

  24. [24]

    ”Therefore, we can simplify the expression to...” Certainty (→Displacement)

    work page

  25. [25]

    geometric flow

    ”The calculation clearly leads to 216.” Certainty (→Displacement) Geometric Signature:Geometrically, this manifests as ahigh-curvature knot. Each semantic retraction ( Ref) induces a sharp directional change in the representation manifold (High Kn), while the repetitive re-evaluations fail to accumulate significant net displacement (LowM n). The trajector...

    work page 2023