mathematical-reasoning

name	mathematical-reasoning
description	Rigorous mathematical reasoning, verification, proof checking, and human-guided problem solving. Use when Codex needs to verify provided mathematical material, inspect theorem/proposition statements, check derivations, identify missing assumptions or counterexamples, develop proofs, solve symbolic or abstract math problems, or guide a user through mathematical derivations step by step.

Mathematical Reasoning And Problem Solving

Use this skill for proof-heavy mathematical work where correctness, assumptions, and derivation clarity matter.

Core Posture

• Separate definitions, assumptions, claims, derivations, and conclusions.
• Preserve the user's notation when it is clear; explicitly define any new notation.
• Treat every nontrivial step as something that needs justification.
• Distinguish proven facts, plausible conjectures, examples, counterexamples, and open gaps.
• Ask for user input when a derivation has multiple meaningful routes or when notation or goals are ambiguous.
• Prefer rigorous concise reasoning over broad exposition.

Part 1: Verify Provided Material

When the user provides mathematical text, equations, proofs, derivations, or claims:

1. Restate the claim or goal precisely.
2. List the explicit assumptions and infer any implicit assumptions.
3. Check notation consistency, domains, dimensions, quantifiers, and boundary cases.
4. Verify each transformation or proof step.
5. Identify missing hypotheses, circular reasoning, unjustified existence or uniqueness claims, convergence gaps, regularity assumptions, sign errors, index errors, and invalid algebra.
6. Test fragile claims with simple examples, limiting cases, or counterexamples when useful.
7. Report findings in priority order:
   • Valid as written
   • Valid with minor clarification
   • Requires additional assumptions
   • Incorrect, with reason
   • Ambiguous or unverifiable from the provided material
8. Provide a corrected version only when the repair is clear; otherwise state what information is needed.

Part 2: Human-Guided Mathematical Derivations

When deriving a result with the user:

1. Clarify the target statement, known givens, allowed tools, and desired rigor level.
2. Choose a derivation route and explain why it is appropriate.
3. Work in small steps, showing the mathematical move and the reason it is valid.
4. Pause at meaningful decision points when different proof strategies are possible.
5. Keep a running list of assumptions introduced during the derivation.
6. Mark any step that depends on a theorem, lemma, approximation, limiting argument, or unproven condition.
7. Where helpful, offer a hint-first path before giving the full step.
8. End with a clean derivation or proof, followed by remaining caveats and possible generalizations.

Standards For Final Math Answers

Include, as appropriate:

• A precise statement of the result.
• Definitions and assumptions.
• The derivation or proof.
• Verification of edge cases or examples.
• Any gaps, caveats, or required additional hypotheses.
• A concise summary of what has been established.

Do not overclaim. If a result is only shown under special assumptions, say so directly.