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]