Daily Papers

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.

Tool calling allows large language models (LLMs) to invoke external computation during problem solving, a useful capability in various fields including AI for mathematics. We study this setting through weighted sum-of-squares (SOS) decomposition, a machine-checkable route to proving polynomial nonnegativity and hence polynomial inequalities. A candidate decomposition can be checked exactly, but finding one requires choosing among non-unique regroupings and coordinating multiple symbolic transformations. We develop an agent that combines algebraic task training, symbolic tools, and verifier-grounded optimization for this task. Rather than training only on the composite SOS task, we construct 1.35 million synthetic examples covering eight supporting polynomial tasks together with weighted-SOS decomposition. We first apply supervised fine-tuning (SFT) to direct algebra problems and simulated symbolic traces, and then use Group Relative Policy Optimization (GRPO) with task-specific symbolic rewards. The SFT corpus contains no native tool-calling messages; at evaluation, the agent uses native SymPy calls for expansion, collection, reordering, and factorization. Every final SOS answer is checked by exact expansion and coefficient comparison. On held-out, same-generator synthetic problems, the full SFT+GRPO+tools system is the strongest of four evaluated configurations, reaching 78.96% verified success on weighted SOS, compared with 44.73% for the base model with the same tools, and 91.75% macro accuracy across nine polynomial tasks. Within this controlled setting, our work provides a case study of combining domain-specific skill training, executable tools, and verifier feedback, and may inform the design of tool-calling agents in other domains with exactly checkable outputs.

Classification accuracy and true-positive and false-positive rates alone do not determine whether restarting an AI agent improves success. Using an exact alarm-to-action identity for an independent clean restart with zero monitoring overhead, we show analytically that a monitor with fixed 90% accuracy can swing from a five-percentage-point loss to a 35-point gain in success rate, depending solely on the replacement attempt's residual-budget success probability. We introduce two techniques for evaluating monitors and progress signals in decision terms. First, a decision-sufficiency criterion, a binary-action specialization of Blackwell's comparison of experiments, identifies which progress representations or context compressions preserve the information needed to choose optimally between the specified actions, independent of trace reconstruction fidelity. Second, a finite-sample, model-free decision audit uses simultaneous binomial intervals over existing agent traces, without additional model calls, to certify a preference for "restart" or "continue" under stated sampling assumptions, or return "abstain" when evidence is insufficient. In a study of 194 externally evaluated model executions on 19 selected SymPy issues from SWE-bench Verified, with a retrospective audit of 130 of these traces, we find a 2.29-point apparent restart gain traced to a single surviving failed attempt, a 42.29-point loss on a second model, no positive replication of the initial gain on prospective task cohorts, and zero of 103 live comparisons meeting the specified certification bar. These results motivate evaluating agent monitors through recovery opportunity, decision value, and statistical support, rather than classification performance alone.