mst-ai/linalg-bench-llm

LinAlg-Bench: Where LLMs Stop Computing and Start Hallucinating

Ten frontier LLMs drop from near-perfect to near-zero on 5×5 eigenvalue problems. Complete computational collapse is dimension-gated: rare at 3×3, dominant at 4×4 and 5×5. Failures dissociate cleanly by task — eigenvalues fail by constraint-aware fabrication (invented eigenvalues that still match the matrix trace), determinants by sign-accumulation drift. Nearly a third of irrational-spectrum eigenvalue failures are "symbolic stops": the model derives a correct characteristic polynomial, then halts.

LinAlg-Bench is a diagnostic benchmark of 1,194 SymPy-verified linear-algebra problems (nine task types, 3×3–5×5 matrices) with 10,509 graded model outputs and 3,933 per-failure forensic error annotations under a fourteen-tag, validated taxonomy.

Quick start

Anonymized for blind review — browse the dataset at anonymous-hf.com/a/may6smri97a3.

Programmatic access via datasets.load_dataset(...) will be restored once the repository id is de-anonymized after review.

What's inside

Config	Rows (3×3 / 4×4 / 5×5)	Total	Contents
benchmark	317 / 442 / 435	1,194	Problems with LaTeX statements and ground-truth answers (SymPy-verified), keyed by Problem_ID and task subcategory
model_outputs	2,687 / 3,898 / 3,924	10,509	One row per (model, problem) for eight models. GPT-5.2 and OpenAI-o1 — both costly to run at scale — were each called only on the preceding model's failures: GPT-5.2 on GPT-4o's, OpenAI-o1 on GPT-5.2's. Their results/accuracy correct count carries the preceding model's correct answers forward as credit, not re-verified
error_annotations	622 / 1,277 / 2,034	3,933	Per-failure error tags: 14 top-level tags (SIGN_ERROR, HALLUCINATION, ARITHMETIC, ...) plus validated subtags (Alternating_Drift, Complete_Collapse, Teleological_Zeroing, ...)
format_sensitivity	— / 5,400 / 5,940	11,340	The same problems restated in three input formats (LaTeX, List, Tabular) and re-graded, isolating presentation effects from computation
human_feedback	372 / 797 / 1,351	2,520	Human re-adjudication of the automated judge — a full census, every row independently human-labelled: human tag/subtag and first-error location alongside the judge's. Manual_Review is the agreement flag (1 = agree, 0 = human overrode). Agreement 2,298/2,520 = 91.2% (87.9% / 89.6% / 93.0%)
forced_gaussian_ablation	— / — / 174	174	Five cofactor-dominant models re-run on their own 5×5 determinant failures under a prompt mandating Gaussian elimination, testing whether strategy choice causes the failures. Per-model pass/fail totals in forced_gaussian_ablation_summary (5 rows)
model_outputs_run2 / model_outputs_run3	1,790 / 1,862 / 1,909 and 1,790 / 1,859 / 1,904	5,561 + 5,553	Two temperature-0 repetition runs over a 660-problem-per-model subset (all 10 models, all nine task types), for run-to-run stability. Per-model run1/run2/run3 accuracy deltas in inter_run_accuracy (10 rows)
scaffold_gradient	— / — / 4,320	4,320	Scaffolding gradient L0–L4 on 5×5 irrational-spectrum eigenvalue pools (symmetric and non-symmetric), 30 problems × 3 repeats per cell, with per-response symbolic_stop flags. Cell-level means in scaffold_gradient_aggregated (64 rows)
opcount_control	883	883	Operation-count control: chained matrix-vector and related tasks at varying operation depth k, isolating raw operation count from matrix dimension across 9 models. Phase/model/k means in opcount_control_aggregated (78 rows)
ibp_depth_ladder	2,720	2,720	Cross-domain depth-ladder probe outside linear algebra: 340 integration-by-parts problems at increasing nesting depth n, 4 models × 2 repeats. Per-(model, depth, repeat) accuracy with CIs in ibp_depth_ladder_by_level (136 rows)
results	41 / 41 / 41	123	Pre-aggregated summary tables in four groups: accuracy, error_tags, sign_subtypes, hallucination_subtypes

Task types (nine): determinant, eigenvalue, rank, nullity, trace, transpose, multiplication, matrix_power, matrix_vector. Eigenvalue and determinant are deepened (120–175 problems per size); the other seven carry 20 problems per size.

Models covered: ten frontier LLMs, all at temperature 0 — Claude-4.5-Sonnet, DeepSeek-V3, Gemini-3.1-Pro, GPT-4o, GPT-5.2, Llama-3.3-70B, Mistral-Large, OpenAI-o1, Qwen2.5-72B, Qwen3-235B.

Headline findings

• The 5×5 cliff: near-ceiling accuracy on shallow tasks; near-zero on 5×5 eigenvalues for eight of ten models, with two exceptions (Gemini-3.1-Pro 56.7%, OpenAI-o1 30.2%).
• Spectrum-conditional: eigenvalue accuracy is over 40% on integer spectra vs under 15% on irrational — required numerical depth, beyond matrix size, drives failure.
• Dimension-gated collapse: Complete_Collapse hallucinations: 11 cases at 3×3 → 219 at 4×4 → 960 at 5×5.
• Two failure mechanisms: constraint-aware confabulation (eigenvalues) vs sign drift (determinants; over half of failures).
• Teleological zeroing: models suppress values they just computed to force premature termination (mainly in eigenvalue derivations and determinant's "det = 0" pattern, both over 10 of 35 cases).

Grading

Symbolic/numeric grading with published tolerance (per-value deviation ≤ 0.011, unordered-set matching for eigenvalues). No string matching. Full grading spec in the paper's Appendix B.

Intended uses

Evaluating exact-computation ability of LLMs; studying hallucination under computational load; training/evaluating process-reward or verification models on annotated failure traces; depth-of-state and working-memory research.

Limitations

1. Domain-specific: linear algebra only, aside from appendix_data/ibp_deliverables (a calculus probe). Generalization beyond these two domains is untested.

2. Snapshot in time: inferences were collected during the 2026 cycle (baseline May 2026, extension and consolidation through September 2026). Model updates or API changes after that are not reflected.

3. Scale limit: maximum 5×5 matrices. Behaviour at larger scales is untested.

Ethical Considerations & Responsible Use

Recommended Use Cases

✅ Research on symbolic reasoning in LLMs ✅ Analyzing systematic error patterns and failure modes ✅ Identifying mechanistic bottlenecks (working memory, strategy rigidity) ✅ Developing better post-training curricula for mathematical reasoning ✅ Training or evaluating process-reward and verification models on annotated failure traces

Caution / Not Recommended For

❌ Definitive model ranking — a single benchmark is insufficient for production decisions ❌ Knowledge claims — this measures execution under specific conditions (T=0, no tools) ❌ General reasoning ability assessment — linear algebra is a narrow domain ❌ Marketing claims — results are domain-specific diagnostics, not overall capability metrics

Known Biases

• Synthetic problems: small integer entries may not reflect real-world matrix distributions
• Format bias: LaTeX consistently helps at recursive levels (model-specific); this is measured directly in format_sensitivity rather than assumed
• Training data: newer models with larger training corpora may have seen more matrix problems
• Failure-conditioned annotations: error_annotations covers failures only, so tag frequencies describe the failure population, not all model behaviour
• Human-review selection: cases sent for human adjudication were not drawn uniformly at random

Citation

@article{linalgbench2027,
  title   = {LinAlg-Bench: When LLMs Stop Computing --- Structured Hallucination,
             Sign Drift, and Depth-Gated Collapse in Matrix Arithmetic},
  author  = {ANONYMOUS — replace after ICLR review},
  journal = {arXiv preprint arXiv:XXXX.XXXXX},
  year    = {2027}
}

License

CC BY 4.0.