first commit

.gitignore

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+_*/
+
+.ds_store
+.vscode/
+*.sage.py
+*.pdf
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bbl-SAVE-ERROR
+*.bcf
+*.bcf-SAVE-ERROR
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+*.rubbercache
+rubber.cache
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hypdoc
+*.hd
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib

AGENT.md

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+You will be solving an extremely challenging mathematics question. The answer may not be known to anyone.
+
+Feel free to think out loud as much as you want. You also have a MacOS environment and some tools available to help you.
+
+Tips for solving:
+* **Try your hardest to answer it.** Even if it seems impossible, spend some time thinking about it before giving up.
+* **Create a plan.** I strongly recommend that you start by making a high-level plan for how you will tackle the problem. Revise your plan along the way if necessary.
+
+Ideas to try if you get stuck:
+* Think about other, similar problems.
+* Try first solving a simpler version of the problem.
+* Pursue lines of investigation that might not seem like they will end up helping.
+* Brainstorm new approaches and try each of them.
+
+Tips for writing: in markdown use `#` for sections and `##` for subsections. Write title in yaml section.
+

math-macros.md

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+\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
+\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
+\newcommand{\set}[1]{\left\{ #1 \right\}}
+\newcommand{\norm}[1]{\left\| #1 \right\|}
+\newcommand{\rm}[1]{\operatorname{#1}}
+
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\C}{\mathbb{C}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\e}{\varepsilon}
+
+\newcommand{\mex}{\operatorname{mex}}
+\newcommand{\lcm}{\operatorname{lcm}}
+\newcommand{\dist}{\operatorname{dist}}
+\newcommand{\poly}{\operatorname{poly}}
+\newcommand{\polylog}{\operatorname{polylog}}
+\newcommand{\span}{\operatorname{span}}

paper-review.md

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+```
+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.
+```

proof-review.md

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+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]