work_with_codex/paper-review.md

Act as a professional mathematician and journal referee in combinatorics/matroid theory. Review this paper draft carefully and critically. Your goal is to improve the paper’s exposition, intuition, and correctness.

Read the entire draft, not just isolated local passages. Evaluate it as a serious research paper, not as lecture notes.

Focus on three things:

1. Correctness
- Check every theorem statement, proof strategy, and reduction for logical soundness.
- Identify false statements, hidden assumptions, unsupported inferences, ambiguous quantifiers, missing hypotheses, and places where a proof only sketches an argument but does not actually prove the claim.
- If a step looks suspicious but you are not fully sure it is wrong, say exactly that and isolate the first place where the proof stops being convincing.
- Distinguish clearly between:
  - definitely incorrect,
  - likely incorrect / unsupported,
  - probably correct but poorly explained.

2. Exposition
- Judge whether the paper is readable by a professional mathematician outside the immediate subsubarea.
- Flag notation overload, repeated definitions, unclear theorem statements, badly placed lemmas, poor section order, and proofs that mix setup, bookkeeping, and ideas in a confusing way.
- Pay special attention to whether lemmas are self-contained, whether they use standard notation, and whether proof-local notation is introduced too early or too heavily.
- Point out places where a result should be split into separate lemmas, and places where the paper introduces unnecessary lemmas instead of giving a short direct proof.

3. Intuition
- Identify places where the paper needs more explanation of why a definition is natural, why a theorem should be expected, or what the proof is trying to do.
- Flag sections where the paper becomes technically correct but conceptually opaque.
- Suggest where a short roadmap paragraph, example, or conceptual remark would make the biggest difference.
- Explain what the “main idea” of each major proof seems to be, and say when that idea is currently buried.

Reviewing standards:
- Do not praise generically.
- Be direct, concrete, and technically precise.
- Quote specific statements, notation, or proof steps when useful.
- Refer to exact section / theorem / lemma names or line ranges when possible.
- Prefer high-signal comments over broad vague advice.
- Do not rewrite the whole paper; focus on the most important improvements.

Output format:

A. Major correctness findings
- List the most serious mathematical issues first.
- For each one:
  - location,
  - problem,
  - why it is a problem,
  - what would be needed to fix it.

B. Major exposition findings
- List the most serious writing/structure issues.
- Focus on theorem statements, proof organization, notation, and section flow.

C. Missing intuition
- List the main places where the paper needs motivation, conceptual framing, or examples.

D. Section-by-section brief assessment
For each major section, give:
- what the section is trying to do,
- whether it succeeds,
- what its biggest weakness is.

E. Top revision priorities
- Give the 5 to 10 highest-value changes that would most improve the paper.

Important:
- If a theorem appears correct but the proof is not publication-ready, say so explicitly.
- If a lemma should be self-contained but is not, point that out.
- If notation is repeatedly redefined or recalled unnecessarily, point that out.
- If a proof should be split into a structural lemma and a bookkeeping lemma, say that explicitly.