Order-Invariant Answers, Order-Sensitive Representations in Mathematical Reasoning

A paraphrase-quality audit of MathCheck (ICLR 2025) detected 4 semantically incorrect paraphrases in 129 groups (3.1%); removing them drops GPT-4o from rank 2 to rank 4 and elevates Claude Haiku and DeepSeek V3 above it; these ranking changes are invisible to any single-model evaluation. Cross-model unanimity found these errors automatically (>= 3/4 models for MathCheck; >= 6/9 for our primary evaluation) for under $10; in our own dataset the same protocol found that 47% of auto-generated connective-variation paraphrases were semantically incorrect. That flaw compounds a deeper measurement gap: Claude Haiku 4.5 achieves 86% accuracy yet SCR=50%, meaning half its theorems are answered differently under semantically equivalent restatements, while aggregate accuracy across 9 models spans only 86-96% yet Semantic Consistency Rates (SCR) span 50-82% -- a 32-point gap invisible to standard benchmarks. Formally, for any target ranking over 9 frontier models there exists a weighting over paraphrase families that realizes it (No-Free-Benchmark corollary), because no model Pareto-dominates all families -- so benchmark designers who select families are implicitly choosing which model wins. FormInv supplies the audit protocol (replicated on external benchmarks at 100% recall), SCR and per-theorem Cochran's Q as primary invariance measures evaluated on 9 models across 366-811 items (on Lean4-verified theorems), and FormInvSelector for regime-aware model selection.

Large language models now achieve high final-answer accuracy on mathematical reasoning benchmarks, but accuracy alone does not capture reasoning flexibility. We introduce a strategy-level evaluation framework instantiated on 80 AMC 10/12 and AIME problems with 217 AoPS-derived reference strategy families. Model outputs are annotated for strategy identity, validity, and correctness using dual-AI coding with human adjudication. Across four frontier models, we find a pronounced decoupling between answer accuracy and strategy diversity. Under a single-solution prompt, all models achieve high accuracy (95%-100%), but under a multiple-strategy prompt they recover substantially fewer strategies than the human reference set. Gemini, DeepSeek, GPT, and Claude generate 184, 152, 151, and 110 distinct valid strategies, respectively, with the largest gaps in Geometry and Number Theory. The models collectively produce 50 benchmark-novel valid strategies, indicating both incomplete coverage of human strategies and some capacity for alternative reasoning. A repeated-run robustness check on 20 problems shows diminishing gains in discovered strategies, with the strongest model recovering only 39 of 55 AoPS-reference strategies (71%) after three runs. These findings position strategy diversity as a complementary dimension for evaluating mathematical reasoning beyond answer correctness.

Modern reasoning language models generate dense, sequential chain-of-thought traces implicitly assuming that every token contributes and that steps must be consumed in order. We challenge both assumptions through a systematic intervention pipeline--removal, masking, shuffling, and noise injection--applied to model-generated reasoning chains across three models and three benchmarks. Our findings are counterintuitive on three dimensions. Order: Does the sequential order of a reasoning chain matter for answer extraction? No--line-level shuffling reduces accuracy by less than 0.5 pp; word-level shuffling retains 62%-89% accuracy; only token-level shuffling collapses to near zero. Pretrained-only and instruction-tuned variants exhibit near-identical tolerance (78.67% vs. 78.00% under line shuffling), indicating order-independence originates from pretraining rather than reasoning-specific fine-tuning. Dense: Is all the information in a reasoning chain important for answer extraction? No--masking numeric digits collapses accuracy to exactly 0%, while masking alphabetic prose improves accuracy by 4.7 pp. Robustness: Is a reasoning chain that is both order-shuffling and non-dense still robust? Yes--the most aggressively reduced representation (all natural language removed, lines arbitrarily shuffled) still achieves 83% accuracy, and injecting false answers at 3x true-answer frequency leaves accuracy unchanged (83.3%->83.3%), falsifying a frequency-based extraction account. These results establish that answer extraction operates on a sparse, order-insensitive, and structurally robust informational substrate, opening paths toward parallelized and token-efficient reasoning generation.