evomath-tao

EvoMath (Tao-style)

EvoMath is a lightweight proof workflow for contest-style mathematical reasoning. Its job is to produce a rigorous proof, a verified counterexample, a useful partial result, or a clear handoff. Keep the process small; do not run a heavy audit pipeline by default.

Methodology Anchor — Terence Tao's Research-Math Practice

This skill operationalizes the way Terence Tao approaches research mathematics:

1. Compute small cases first (Kepler before Newton) — build intuition from data before reaching for theory.
2. Try the standard toolbox broadly before going deep — most hard problems crack to a standard technique; the few that don't only reveal which after several have failed.
3. Hold rigor and intuition together (post-rigorous mathematics) — trust intuition, but verify every step. "It feels right" is a hypothesis, not a proof.
4. Atomize when stuck — decompose into independently checkable sub-claims. A clean map of proved / conjectured / open beats a polished but shaky narrative.
5. Stay honest about what isn't proved — distinguish PROVED / VERIFIED_NUMERICALLY / CONJECTURED / HANDED_OFF. When blocked, name the precise gap.
6. Distill each result into reusable insight — after every problem, extract what worked into a strategy and what failed into a named pattern. Mathematical maturity is accumulated meta-insight.

Every phase below is a concrete operationalization of one or more of these principles.

Operating Rules

• Use Markdown notes for handoff between steps. Do not require JSON/YAML unless a script explicitly asks for it.
• Keep only compact state: plan, verified claims, failed attempts, final audit. Do not pass long failed derivations into later prompts.
• Prefer a few independent proof attempts over one long derivation.
• Numerical verification is NOT a proof step (math-olympiad rule). Checking a claim on n=1..100 and finding no counterexample does NOT make it PROVED; the strongest label such evidence can earn is VERIFIED_NUMERICALLY.
• Exact arithmetic can refute; approximate numerics only suggest.
• A proof is final only after an adversarial check of the clean proof.
• Calibrated abstention over bluffing: when verification fails repeatedly, admit it. Return partial results and mark unfixed gaps explicitly (math- olympiad rule). Final status HANDED_OFF with a structured wall report is always preferable to PROVED with hand-waved gaps.
• Every final answer must include a visible final-status: ... line.
• Use TodoWrite to drive the workflow. Each step is one todo; you cannot mark a todo completed unless the corresponding .md file passes its validator.