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arxiv: 2605.16675 · v1 · pith:B367BSOWnew · submitted 2026-05-15 · 💻 cs.AI

LinAlg-Bench: A Forensic Benchmark Revealing Structural Failure Modes in LLM Mathematical Reasoning

Shradha Agarwal , Deepak Rajbhar , Tariq J This is my paper

Pith reviewed 2026-05-20 17:39 UTC · model grok-4.3

classification 💻 cs.AI
keywords LLM mathematical reasoninglinear algebra benchmarkfailure mode analysisstructured hallucinationcomputational abandonmentmatrix dimension effectserror classification pipelineworking memory limits
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The pith

LLMs switch from calculation errors to abandoning math and fabricating answers once matrix size hits 4x4.

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

The paper introduces LinAlg-Bench to test frontier language models on linear algebra tasks using matrices sized 3x3, 4x4, and 5x5 across nine problem types. It applies an automated pipeline to label over a thousand failures and identifies a clear change in behavior tied to matrix dimension. On smaller matrices models attempt the work but make execution mistakes such as losing track of signs or drifting in arithmetic. Starting at 4x4 the same models largely stop trying to compute and instead produce invented responses that pretend to use tools or stay consistent with problem constraints. The pattern holds across model families and points to a capacity limit in handling longer sequences of exact steps rather than missing knowledge of the subject.

Core claim

The central claim is that LLM failure on structured linear algebra is not random but follows a sharp scale-dependent transition. Below 4x4 matrices, errors are execution-based, including sign tracking failures, arithmetic drift, and parity mistakes. At and above 4x4, models shift to computational abandonment, responding through tool roleplay, constraint-consistent confabulation, and structured hallucination instead of performing the required operations. This transition is accompanied by three error types that emerge only at the larger sizes and appears consistent across model tiers.

What carries the argument

The three-stage automated forensic pipeline that tags failures into ten primary error categories on 660 SymPy-verified problems spanning a strict 3x3 to 5x5 dimensional gradient.

If this is right

  • Solution strategy rigidity predicts 5x5 determinant accuracy with near-perfect correlation.
  • Constraint-aware confabulation emerges as a repeatable structured hallucination pattern distinct from random invention.
  • The execution-to-abandonment transition occurs in every tested model architecture and size tier.
  • Three scale-emergent error types appear only at 4x4 and 5x5 and are absent at 3x3.

Where Pith is reading between the lines

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

  • The observed threshold may mark a general bound on how many exact sequential operations LLMs can maintain before defaulting to surface-level response generation, testable in other multi-step symbolic domains.
  • Explicit tool-use instructions or external calculators might reduce outright abandonment but risk new forms of roleplay where the model describes tool calls without actually invoking them.
  • Fine-tuning on progressively larger matrix problems could shift the transition point outward, providing a direct test of whether the limit is fixed or trainable.

Load-bearing premise

The automated pipeline correctly sorts every failure into the ten error tags without systematic bias introduced by the judge model or prompt wording.

What would settle it

Re-running the 4x4 and 5x5 problems while forcing models to output every intermediate calculation step and checking whether abandonment and fabrication rates drop compared to the original free-response setting.

Figures

Figures reproduced from arXiv: 2605.16675 by Deepak Rajbhar, Shradha Agarwal, Tariq J.

Figure 4.1
Figure 4.1. Figure 4.1: Accuracy trajectories across matrix dimensions for the Recursive level (determinant, [PITH_FULL_IMAGE:figures/full_fig_p004_4_1.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Error tag distribution across matrix dimensions, expressed as percentage of total failures [PITH_FULL_IMAGE:figures/full_fig_p006_5_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Forced Gaussian elimination ablation. Panel A: 5 × 5 determinant accuracy under natural zero-shot prompting versus forced Gaussian elimination for five cofactor-dominant models. Enforcing the algorithmically efficient O(n 3 ) strategy yields no meaningful accuracy recovery. Mistral-Large shows partial improvement (26.9% on previously failed problems); Claude-4.5-Sonnet: 5.3% (n=19). Panel B: Percentage o… view at source ↗
read the original abstract

We introduce LinAlg-Bench, a diagnostic benchmark evaluating 10 frontier large language models on structured linear algebra computation across a strict dimensional gradient of 3x3, 4x4, and 5x5 matrices. Spanning 9 task types and 660 SymPy-certified problems, the benchmark exhaustively evaluates 6,600 model outputs. Beyond binary accuracy, LinAlg-Bench introduces a three-stage automated forensic pipeline classifying 1,156 failures into ten primary error tags with fine-grained subtypes, revealing that LLM mathematical failure is not random but structurally constrained by algorithm type and matrix dimension. Our central finding is a sharp behavioral threshold at 4x4 scale: below it, models fail through execution errors -- sign tracking failures, arithmetic drift, and parity errors; above it, failure transitions to computational abandonment, with models fabricating responses through tool roleplay, constraint-consistent confabulation, and structured hallucination rather than attempting computation. This fabrication-to-abandonment transition is near-universal across all model tiers and architectures, suggesting a working memory limit rather than a knowledge gap, supported by three scale-emergent error types absent at 3x3 but present at 4x4 and 5x5. We further show that solution strategy rigidity is a near-perfect predictor of 5x5 determinant accuracy, document constraint-aware confabulation as a novel structured hallucination failure mode, and release all data, model outputs, error labels, and judge pipeline publicly.

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 introduces LinAlg-Bench, a diagnostic benchmark for evaluating 10 frontier LLMs on 660 SymPy-certified linear algebra problems spanning 9 task types and matrix sizes 3x3, 4x4, and 5x5 (6,600 total outputs). It deploys a three-stage automated forensic pipeline to classify 1,156 failures into ten primary error tags with subtypes, documenting a near-universal shift at the 4x4 scale from execution errors (sign tracking, arithmetic drift, parity) to computational abandonment via tool roleplay, constraint-consistent confabulation, and structured hallucination. Additional claims include solution strategy rigidity as a near-perfect predictor of 5x5 determinant accuracy and the public release of all data, outputs, labels, and pipeline code.

Significance. If the classification pipeline proves reliable, the work supplies concrete empirical evidence that LLM mathematical failures are structurally tied to problem scale rather than random, with a reproducible transition suggestive of working-memory limits. The exhaustive scale (660 problems, 10 models), public data release, and identification of scale-emergent error types absent at 3x3 constitute clear strengths for the field of LLM evaluation and reasoning diagnostics.

major comments (3)
  1. [§4.2] §4.2 (Forensic Pipeline): the central claim of a dimension-driven transition rests on the automated three-stage classifier; the manuscript provides no inter-rater agreement metrics, human validation subset, or ablation on judge-model prompt sensitivity, leaving open the possibility of systematic bias in tagging execution errors versus abandonment.
  2. [§5.1] Results §5.1 and Figure 4: the 'sharp behavioral threshold at 4x4' is asserted as near-universal, yet no statistical test (e.g., McNemar or transition-probability analysis) or per-model breakdown is reported to distinguish an abrupt shift from a gradual increase in abandonment rate.
  3. [§5.3] §5.3 (Strategy Rigidity Predictor): the claim that rigidity is a 'near-perfect predictor' of 5x5 determinant accuracy requires the exact correlation value, number of observations, and controls for model scale or task type; without these the predictive strength cannot be assessed.
minor comments (2)
  1. [Table 2] Table 2: column headers for error subtypes are abbreviated without a legend; expand or footnote for readability.
  2. [Abstract] The abstract states 'near-universal across all model tiers' while the main text reports minor exceptions for two smaller models; reconcile the wording for precision.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense of the work while acknowledging where additional evidence or clarification is warranted. We have outlined specific revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (Forensic Pipeline): the central claim of a dimension-driven transition rests on the automated three-stage classifier; the manuscript provides no inter-rater agreement metrics, human validation subset, or ablation on judge-model prompt sensitivity, leaving open the possibility of systematic bias in tagging execution errors versus abandonment.

    Authors: We agree that the absence of explicit validation metrics for the automated classifier represents a limitation in the current manuscript. The three-stage pipeline combines SymPy-certified correctness checks with rule-based tagging for execution errors and LLM-assisted classification for abandonment modes, chosen to maximize reproducibility across the 6,600 outputs. However, to directly address concerns about systematic bias, the revised manuscript will add a human validation study on a random stratified subset of 150 failure cases, reporting Cohen's kappa between the automated labels and two independent human raters. We will also include an ablation on judge-model prompt sensitivity by testing two alternative prompt phrasings and reporting tag consistency rates. revision: yes

  2. Referee: [§5.1] Results §5.1 and Figure 4: the 'sharp behavioral threshold at 4x4' is asserted as near-universal, yet no statistical test (e.g., McNemar or transition-probability analysis) or per-model breakdown is reported to distinguish an abrupt shift from a gradual increase in abandonment rate.

    Authors: The manuscript documents the threshold via the emergence of three scale-specific error types (tool roleplay, constraint-consistent confabulation, and structured hallucination) that are absent at 3x3 but appear at 4x4 and 5x5 across all ten models. To strengthen the claim of abruptness rather than gradual change, the revised version will expand Figure 4 with per-model abandonment rate breakdowns and add a McNemar test comparing paired error-type shifts between 3x3 and 4x4 dimensions, along with transition probability matrices aggregated across models. revision: yes

  3. Referee: [§5.3] §5.3 (Strategy Rigidity Predictor): the claim that rigidity is a 'near-perfect predictor' of 5x5 determinant accuracy requires the exact correlation value, number of observations, and controls for model scale or task type; without these the predictive strength cannot be assessed.

    Authors: The current claim is based on the observation that models exhibiting rigid adherence to a single solution strategy (as classified in the forensic pipeline) consistently solve 5x5 determinant problems correctly, while those shifting to abandonment do not. We acknowledge that the manuscript presents this qualitatively. In the revision we will report the exact correlation (point-biserial r), the number of observations (all 660 determinant instances across the 10 models), and controls by stratifying results by model parameter count and task subtype to allow readers to evaluate the predictive strength directly. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical benchmark with observed thresholds

full rationale

The paper introduces LinAlg-Bench as an empirical diagnostic tool with 660 SymPy-certified problems and a three-stage failure classification pipeline. The central claim of a behavioral threshold at 4x4 (execution errors below, computational abandonment above) is presented as an observed pattern across 6,600 outputs from 10 models, not as a first-principles derivation or fitted prediction that reduces to its own inputs by construction. No equations, ansatzes, or self-citations are invoked to define or force the threshold; the result follows directly from the experimental data and error tagging. The work contains no load-bearing self-referential steps, uniqueness theorems, or renamings of known results that would create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the assumption that SymPy provides ground-truth labels and that the chosen 9 task types and matrix sizes capture the relevant failure modes; no free parameters are fitted to produce the threshold itself.

axioms (1)
  • domain assumption SymPy correctly computes all ground-truth linear algebra results for the 660 problems
    Invoked to certify the 660 problems as the evaluation basis.

pith-pipeline@v0.9.0 · 5805 in / 1360 out tokens · 88087 ms · 2026-05-20T17:39:23.466343+00:00 · methodology

discussion (0)

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