CrouzeixConjecture: a Lean 4 workspace for a candidate proof

What the CrouzeixConjecture repository actually contains

The README describes the repository as a research workspace rather than a library. The status list is explicit on three points: the candidate proof is written as a standalone LaTeX manuscript, several independent computational and adversarial audits found no specific mathematical error, and formal peer review is still pending. That third line is the one to keep in view. A clean audit is not a proof of correctness, and the project does not claim otherwise.

The audience is narrow. This is for people who already work on numerical range questions and want to inspect a candidate argument, or for people who want to see how a Lean 4 formalization of a spectral-set statement is laid out. The top-level entries are AnnMath/, LaTeX/, Lean/, README.md, conversation-019f7059-public/, crouzeix_conjecture_prompt.txt, and preprint/. Each of those is a different artifact of the same effort, not a product surface.

The three parallel artifacts and how they relate

There is no single source of truth here, and the README does not say which artifact is authoritative. The LaTeX/crouzeix_conjecture_proof.tex file holds the candidate proof manuscript, with a compiled PDF beside it. LaTeX/main_problem.tex states the problem. AnnMath/the_numerical_range_is_a_2_spectral_set.tex is the same result formatted for Annals of Mathematics submission, with aomart.cls as the document class. The Lean/ directory holds a Lean 4 formalization, a verification entry point, and an axiom audit.

The naming difference is worth noting: the manuscript title says the numerical range is a 2-spectral set, while the repository is named after the conjecture. Those are the same subject expressed at different levels, and the README does not explain the relationship between the two documents. If you intend to review the mathematics, read LaTeX/main_problem.tex first, then the proof manuscript, then decide whether the Lean development covers the same statement or a narrower one. The README does not state which.

crouzeix_conjecture_prompt.txt is described as project provenance and the original task specification. That file is the place to look if you want to know what was asked before anything was proved.

Building the manuscript from the AnnMath directory

The README gives the build command for the submission manuscript. It is run from inside the AnnMath directory, not from the repository root, and it uses latexmk with nonstop and halt-on-error flags. The document class aomart.cls sits in the same directory, so no separate class installation is described.

cd AnnMath
latexmk -pdf -interaction=nonstopmode -halt-on-error the_numerical_range_is_a_2_spectral_set.tex

The expected result is a compiled PDF named the_numerical_range_is_a_2_spectral_set.pdf in that directory, matching the PDF already committed. Because halt-on-error is set, any LaTeX error stops the run rather than producing a partial document. The README does not document a bibliography step, so if the manuscript carries citations, what latexmk does with them is not described.

Running the Lean 4 verification and its axiom audit

The Lean side has a single documented entry point. From the Lean directory, lake run builds the complete library and then runs the exported-theorem axiom audit.

cd Lean
lake run

According to the README, that one command does both jobs: the full library build and the audit over exported theorems. What you should look for in the output is the audit report itself, since that is what tells you which axioms the exported theorems depend on. The README does not list the expected axiom set, does not say what the audit prints on success, and does not document a separate flag for running the audit alone. It also does not state a Lean toolchain version beyond Lean 4, so the lakefile or toolchain file in the Lean directory is the place to check before you build. Treat a successful lake run as a statement about the formalized statements only, not about the LaTeX manuscript, because the README never claims the two are in correspondence.

The AI contribution and the public conversation record

The README states plainly that OpenAI ChatGPT contributed to proof development, manuscript preparation, and adversarial checking. That is unusual to see stated up front, and it is the right place for it. The conversation-019f7059-public/ directory holds a privacy-filtered record of the Codex proof-development conversation and all persisted subagent branches, including the formal reasoning summaries that OpenAI exposes for review.

The filtering is described in concrete terms: visible messages and formal reasoning summaries are included, while hidden reasoning blocks, platform instructions, credentials, and personal account identifiers are excluded. So the record is not the full transcript, and the README says so. If your interest in this repository is methodological, meaning how a proof was developed with a model in the loop, that directory is the primary evidence and the exclusion list is part of what you are evaluating. The README does not state how the branches were selected for persistence.

Where this repository is the wrong tool

If you need a result you can cite, this is not it yet. The README's own status section says formal peer review is still pending, and the audits it mentions are described as having found no specific mathematical error, which is a weaker statement than verified. Nothing in the README reports a journal acceptance, a referee report, or an arXiv identifier for the manuscript.

There is a second limitation that matters for anyone planning to build on the formalization. The README does not state that the Lean development proves the same statement as the LaTeX manuscript. It lists them as separate contents. A reader who assumes the Lean entry point discharges the paper's main theorem is making an assumption the documentation does not support. The same gap applies to the AnnMath version: it is a reformatted submission manuscript, and the README does not say whether its statement matches the LaTeX/crouzeix_conjecture_proof.tex statement exactly.

Finally, the repository has no releases and no documented versioning, so there is no tagged state to pin a citation to.

Compared with a conventional formalization project

The usual shape of a Lean formalization repository is a library with a stable API, a CI configuration, and a README that maps theorem names to paper statements. This repository inverts that. The paper comes first, the formalization is one directory among several, and the mapping between them is left to the reader. mathlib-style projects also tend to document their axiom dependencies in the README; here the audit exists as a command but its expected output is not written down.

The other clear alternative is simply waiting for the mathematical community to respond to the manuscript. That costs nothing and requires no build. What you lose is the ability to inspect the formal development and the persisted conversation record yourself, which is the specific thing this repository offers that a preprint alone would not.

Maintenance, licence, and what a fork inherits

The last push to the default branch was on 2026-08-16, which is recent enough that the repository is not dormant. There are no releases, so there is nothing to pin. The README does not describe a contribution process, a code of conduct, or an issue policy.

The licence is not stated in the repository's documentation, and no licence file appears among its top-level entries. That is a real problem for reuse. The README describes an Annals-formatted submission manuscript and a document class file, and it does not say under what terms any of it may be redistributed. Before you copy the LaTeX sources or the Lean development into your own work, check whether a licence file exists in the repository itself. The README is silent on this, and silence is not permission. This is a factual gap in the documentation, not a legal conclusion, and it is the kind of thing to resolve by reading the repository rather than by assuming.

Editorial conclusion

Adopt this repository if you want to read or audit a candidate proof of Crouzeix's conjecture that ships with its own Lean 4 formalization and an exported-theorem axiom audit, and if you accept that the README states formal peer review is still pending. Do not treat it as an established theorem: the README's own status list says the candidate proof is a standalone LaTeX manuscript and that review has not happened. Before citing anything, run lake run from the Lean directory and read the axiom audit output, then check whether the theorem names it reports match the statements you intend to rely on.

Frequently asked questions

Official sources

1. Issues
2. jinshanmu/CrouzeixConjecture on GitHub
3. README