STAR-PolyaMath

Strategies for handling code execution issues during mathematical problem solving. Covers timeout resolution (reduce scale, optimize algorithm, change approach, parallelize), runtime error diagnosis, wrong output debugging, retry escalation, and Python optimization tips. WHEN: code timeout, runtime error, wrong output, code optimization, execution failed, MemoryError, RecursionError, code too slow, brute force timeout, when to abandon code verification.

Active falsification: try to refute a proposed conjecture, lemma, or intermediate claim by finding examples that satisfy its assumptions but violate its claimed conclusion. Used by the Reasoner to stress-test fragile conjectures before promoting them, and by the Verifier (via a sub-agent task) to challenge sharpness/optimality claims before tagging them `[Lemma]` in Verified Results. WHEN: a claim is conjectural rather than fully proved; a sharpness/extremal claim is on the verge of being recorded; a verifier sees an "answer-bearing" step claim with only one-sided evidence.

Reasoner's protocol for the pre-plan Exploration phase. Defines the goal, deliverable structure, time budget, and assessment categories. The Reasoner produces an `exploration.md` deliverable with five required sections; the orchestrator parses the Assessment section to decide the next phase. WHEN: Reasoner is in `exploration` phase (the very first phase before any plan is created). Used by the Reasoner agent.

Reference catalog of proof methods, problem-solving tactics, and domain techniques for competition math (induction, contradiction, extremal principle, invariants, etc.). WHEN: only load when strategic guidance is needed, such as pre-planning analysis of a new problem, selecting applicable methods, deadlock mediation, calibration failure diagnosis, choosing a fundamentally different approach after repeated failure. DO NOT USE WHEN: reasoning is executing normally, writing proofs, or running code.

Unified intervention protocol for the Meta-Strategist agent. Defines output formats, critical rules, and scenario-specific guidance for diagnosing failures and providing actionable tasks to the Reasoner. WHEN: Meta-Strategist is diagnosing a failure (calibration mismatch, timeout, code error, stuck), mediating a deadlock, advising on code issues, or providing pre-planning analysis. DO NOT USE WHEN: Reasoner or Verifier is executing normally.

Unified verification tag definitions ([verified], [easy-verify], [hard-verify]). Defines tag semantics, Reasoner usage rules, Verifier review checklists, and Orchestrator routing logic. WHEN: writing step reports with verification tags, reviewing tagged claims, routing reports based on tag types, understanding verification tag semantics.

Verifier's review process for a single step report or a challenge response. Defines the Two-gate evaluation (Goal Gate + Logic Gate), required output format, ACCEPT/CHALLENGE/TRACE_BACK verdict criteria, and the Verified Results summary block. WHEN: when reviewing a Reasoner step report, evaluating a Reasoner response in the challenge loop, or producing a verdict. Used by the Verifier agent.