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arxiv: 2605.03196 · v1 · submitted 2026-05-04 · 💻 cs.CL · cs.LG

Recognition: 3 theorem links

· Lean Theorem

Geometric Deviation as an Unsupervised Pre-Generation Reliability Signal: Probing LLM Representations for Answerability

Yucheng Du
Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:52 UTC · model grok-4.3

classification 💻 cs.CL cs.LG
keywords geometric deviationanswerabilityLLM representationspre-generation signalhidden statesmathematical promptsunsupervised detection
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The pith

Unanswerable mathematical prompts produce hidden-state deviations from answerable ones that can be detected before any generation occurs.

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

The paper tests whether the internal geometry of language model representations can flag unanswerable queries ahead of time. It shows that for math problems, unanswerable inputs move away from the center of answerable inputs in representation space. This movement gives a signal usable in one pass without any labeled failure data or output inspection. The signal works better than basic refusal checks and catches some cases the model does not refuse outright. However, the same pattern does not appear for factual questions, limiting the approach to tasks with clear formal structure.

Core claim

The authors establish that measuring how far a prompt's hidden states lie from the centroid computed over answerable prompts provides an unsupervised way to assess answerability prior to generation. This geometric deviation is reliable for mathematical prompts across multiple models, separates the two classes effectively, and remains useful even when models do not produce explicit refusals. The effect originates in early layers and fades later, and it does not generalize to factual prompts while showing partial presence in code prompts.

What carries the argument

Deviation of hidden states from the answerable reference centroid, which serves as a proxy for whether the model can answer the query.

Load-bearing premise

That the observed deviation encodes answerability rather than incidental features of the prompts such as their length, style, or complexity.

What would settle it

A dataset of unanswerable mathematical prompts where the hidden states fall within the same distribution as answerable ones after matching for length and style would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.03196 by Yucheng Du.

Figure 1
Figure 1. Figure 1: PCA of mean-centred last-layer representations (controlled experiments) for all three models. view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of own dist (cosine distance to answerable centroid) for all three models and all three prompt forms. MATH-U and CODE-U distributions are substantially elevated relative to their answerable counterparts, consistent across all three models. FACT-A and FACT-U distributions largely overlap, consistent with the non￾significant permutation tests. CODE distances are not directly comparable to MATH/F… view at source ↗
Figure 3
Figure 3. Figure 3: Top: Layer-wise answerability gap δl for MATH matched pairs (n = 20). All three models peak at layers 2–5 and generally decrease toward the last layer. Bottom: Absolute own dist traces for MATH-U (solid) and MATH-A (dashed). The gap narrows because MATH-A rises in deeper layers, not because the MATH-U signal decays. throughout the network. The overall decrease in δl is driven by the answerable class drifti… view at source ↗
read the original abstract

A reliable language model should be able to signal, prior to generation, when a query falls outside its knowledge. We investigate whether representation geometry can provide such a pre-generation signal by measuring the deviation of hidden states from an answerable reference set, requiring no labeled failure data and no access to model outputs. Across three instruction-tuned models (Llama 3.1-8B, Qwen 2.5-7B, and Mistral-7B-Instruct) and three prompt forms (Math, Fact, Code), we find that geometry primarily encodes task form. Within mathematical prompts, unanswerable inputs consistently deviate from the answerable centroid, yielding strong separation (ROC-AUC 0.78-0.84). This single-pass pre-generation signal outperforms a simple refusal baseline and compares favorably to self-consistency. It also captures cases where models do not explicitly refuse. In contrast, no reliable geometric signal emerges for factual prompts, indicating that the effect is form-conditional rather than universal. Code prompts show large effect sizes with higher variance, suggesting partial generalization beyond mathematical form. A layer-wise analysis reveals that the signal arises in early layers and gradually attenuates toward the output. These results suggest that answerability-related geometry is established before the final stages of generation. Together, these findings indicate that geometric deviation can serve as a lightweight pre-generation signal that is reliable in structured domains with formal answerability constraints, with clear boundaries on where it generalizes.

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 manuscript claims that the geometric deviation of LLM hidden states from the centroid of an answerable reference set can serve as an unsupervised, single-pass, pre-generation signal for prompt answerability. Experiments across Llama 3.1-8B, Qwen 2.5-7B, and Mistral-7B-Instruct on math, fact, and code prompts show that unanswerable mathematical inputs deviate consistently from the answerable centroid, producing ROC-AUC scores of 0.78-0.84 that outperform a simple refusal baseline and compare favorably to self-consistency; the signal also detects cases without explicit refusal. No reliable signal appears for factual prompts, while code prompts exhibit large but high-variance effects. The signal originates in early layers and attenuates toward the output, indicating that answerability-related geometry is established early and is form-conditional rather than universal.

Significance. If the deviation metric specifically encodes answerability (rather than surface features), the work supplies a lightweight, label-free reliability signal that operates before any tokens are generated. This is particularly valuable in structured domains with formal constraints such as mathematics, where it could complement or replace post-hoc consistency checks. The form-conditional results also advance understanding of how LLMs internally represent knowledge boundaries, with the early-layer localization offering a concrete mechanistic clue.

major comments (3)
  1. [Experimental Setup] The experimental setup provides no controls or matching for prompt length, syntactic complexity, or lexical diversity between answerable and unanswerable examples within each form. Because the reported separation is form-conditional and absent for factual prompts, the observed geometric deviation could be driven by these surface properties rather than answerability per se; this directly affects the interpretation of the ROC-AUC 0.78-0.84 results on mathematical prompts.
  2. [Results] Section 4 reports concrete ROC-AUC values and baseline comparisons but supplies no statistical significance tests, confidence intervals, or details on reference-set size and sampling procedure. Without these, it is impossible to judge whether the separation is robust or sensitive to the particular choice of answerable centroid.
  3. [Layer-wise Analysis] The layer-wise analysis states that the signal arises early and attenuates, yet no quantitative comparison (e.g., layer-by-layer AUC curves or statistical tests across models) is given to support the claim that answerability geometry is established before final generation stages.
minor comments (2)
  1. The abstract and results sections would benefit from an explicit statement of the total number of prompts per category and per model to allow readers to assess statistical power.
  2. [Methods] Notation for the deviation metric and centroid computation could be formalized in an equation early in the methods to improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback, which identifies key areas where additional rigor will strengthen the manuscript. We address each major comment below and will incorporate revisions to improve experimental controls, statistical reporting, and quantitative analyses.

read point-by-point responses

  1. Referee: [Experimental Setup] The experimental setup provides no controls or matching for prompt length, syntactic complexity, or lexical diversity between answerable and unanswerable examples within each form. Because the reported separation is form-conditional and absent for factual prompts, the observed geometric deviation could be driven by these surface properties rather than answerability per se; this directly affects the interpretation of the ROC-AUC 0.78-0.84 results on mathematical prompts.

    Authors: We agree this is a valid concern that could affect causal interpretation. While the form-conditional pattern (strong effects for math, absent for facts) offers indirect evidence against a purely surface-driven account, as surface confounds would be expected to appear across forms, we did not perform explicit matching. In the revised manuscript we will construct length-, syntax-, and lexical-diversity-matched subsets within each form, recompute the deviation metric and ROC-AUC on these subsets, and report correlations between deviation scores and the surface metrics to quantify any residual confounding. revision: yes

  2. Referee: [Results] Section 4 reports concrete ROC-AUC values and baseline comparisons but supplies no statistical significance tests, confidence intervals, or details on reference-set size and sampling procedure. Without these, it is impossible to judge whether the separation is robust or sensitive to the particular choice of answerable centroid.

    Authors: We acknowledge the omission of these statistical details. The reference sets were constructed from 200 randomly sampled answerable prompts per form, with centroids as the mean of the corresponding hidden-state vectors. The revised version will add 95% bootstrap confidence intervals around all reported ROC-AUC values, permutation tests for significance versus the refusal baseline and self-consistency, and a sensitivity analysis showing how AUC varies with reference-set size (50–300 examples). revision: yes

  3. Referee: [Layer-wise Analysis] The layer-wise analysis states that the signal arises early and attenuates, yet no quantitative comparison (e.g., layer-by-layer AUC curves or statistical tests across models) is given to support the claim that answerability geometry is established before final generation stages.

    Authors: The manuscript currently describes the early-layer origin and attenuation qualitatively from the per-layer deviation trajectories. To provide the requested quantitative support, we will include layer-by-layer ROC-AUC curves for every model and prompt form, together with repeated-measures ANOVA and post-hoc tests across layers and models to statistically confirm the early peak and subsequent attenuation pattern. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct geometric computation on independent reference set

full rationale

The paper computes deviation of hidden-state representations from a pre-chosen answerable reference centroid. This is a fixed, non-parametric geometric operation (distance to mean) with no fitted parameters, no self-referential definitions, and no load-bearing self-citations. The reference set is selected independently of the test prompts, and the reported ROC-AUC values are direct empirical measurements rather than predictions derived from the same data by construction. The form-conditional nature of the signal is acknowledged but does not introduce circularity in the derivation. No steps reduce to tautology or fitted-input renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the empirical observation that answerability manifests as measurable geometric deviation in hidden-state space for certain task forms. No new entities are postulated and no parameters are fitted beyond the unsupervised construction of the reference centroid.

axioms (1)
  • domain assumption Hidden-state geometry in LLMs encodes task-form-specific information about query answerability that can be captured by centroid deviation.
    Invoked when the authors interpret deviation as an answerability signal and when they restrict the claim to mathematical prompts.

pith-pipeline@v0.9.0 · 5563 in / 1330 out tokens · 49022 ms · 2026-05-08T17:52:19.072123+00:00 · methodology

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Reference graph

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