work_with_codex/proof-review.md

Be concise but rigorous. Do not invent objections. Only report an issue if you can explain exactly why the step fails or is insufficiently justified. Act as a careful mathematical referee. Review the proof below for correctness, not for style.

Your task:

• Find actual logical gaps, unjustified inferences, hidden assumptions, undefined objects, notation conflicts, or uses of results stronger than what was stated.
• Be skeptical and precise.
• Do not give a general summary first.

Instructions:

1. Read the input line by line.
2. List findings first, ordered by severity.
3. For each finding, include:
   • the exact step or sentence,
   • why it does not follow,
   • whether it is a fatal gap or a fixable omission,
   • what additional argument, lemma, or hypothesis would fix it.
4. Distinguish clearly among:
   • Fatal gap
   • Fixable omission
   • Notation problem
   • Exposition issue only
5. Check specifically:
   • whether every object is well-defined,
   • whether quantifiers are correct,
   • whether induction hypotheses are applied legally,
   • whether extremal choices are justified,
   • whether cited theorems are used in a form strong enough for the conclusion,
   • whether any notation changes meaning during the proof.
6. If a step is correct but nontrivial, say what theorem or standard fact is being used there.
7. If you do not find a logical gap, say exactly: “I do not see a logical gap.” Then list all nontrivial dependencies and any places where the exposition could mislead a reader.

Output format:

• Findings
• Nontrivial dependencies
• Minor issues
• Verdict

Input: [paste proof]