An AI-assisted mathematics experiment exploring Crouzeix's conjecture through proof attempts, numerical searches, symbolic calculations and independent checks. The research used models including GPT-5.6 Sol and Claude Opus.
A matrix has an associated region in the complex plane called its numerical range. Crouzeix's conjecture says that applying a polynomial to the matrix can amplify vectors by at most twice the largest absolute value of that polynomial on this region.
The experiment's goal was to find a general proof or an exact counterexample, using AI agents to develop different approaches and check each other's work. The original task records the research setup.
- Revisit the existing proof machinery. We examined where the earlier bound of about 2.414 lost information and looked for ways to recover two.
- Search for counterexamples. Small nonnormal matrices and polynomial functions were optimized numerically, with known equality cases used as reference points.
- Solve structured families. Elliptic numerical ranges and weighted matrices made it possible to derive explicit formulas and check whole parameter ranges.
- Study perturbations of equality cases. We investigated how the bound changes when matrices already attaining two are slightly altered.
- Construct matrix certificates. Similarity transformations, positive kernels and semidefinite optimization provided alternative ways to establish bounds. Some of these approaches addressed the stronger version in which polynomial coefficients can themselves be matrices.
| Finding | What it describes | Notes |
|---|---|---|
| Exact elliptic-family constant | A closed formula for the optimal constant for the elliptic Crabb family in every matrix size, with an explicit transformation attaining it. | L117: elliptic Crabb axis |
| Weighted four-by-four family | A computer-assisted proof of the complete bound of two for a family with three independent positive weights and an elliptic numerical range. | L20/L59: similarity and certificates |
| Local second-order behaviour | Formulas expressing the change near a single Crabb equality block as negative quadratic terms, in arbitrary size. | L65: second variation |
| A three-dimensional neighbourhood | A local argument covering small complex perturbations of the three-by-three Crabb block, including directions where the quadratic term vanishes. | L73: local theorem |
| Equality and residual structure | Connections between equality cases, Toeplitz matrices and explicit square identities that simplify the local calculations. | L187: equality geometry, L355: residual square |
The literature ledger compares these derivations with earlier work, including Kenan Li's elliptic-family calculations and classical results on disk numerical ranges. The approach ledger also records failed shortcuts and the examples that identified their limits.
After the July research, the repository added a finite-matrix reconstruction of the general proof by Emiel Lorist and Felix Schwenninger. Their argument combines identities for all powers of an operator. Shanmu Jin's separate proof uses a positive-kernel sampling construction.
proof/— derivations, special-case results and proof audits.experiments/— Python programs, exact checks and recorded experiment data.LEMMA_LEDGER.md— statement IDs, scope and supporting evidence.LITERATURE_LEDGER.md— sources and comparisons with existing work.APPROACH_LEDGER.md— research decisions and retained unsuccessful approaches.COUNTEREXAMPLE_SEARCH.md— search methods, results and coverage.archive/— the full July lemma and approach ledgers.
The recorded environment uses Python 3.14. Dependencies are pinned in
requirements.txt. Run commands from the repository root:
python3 -m venv .venv
.venv/bin/python -m pip install -r requirements.txt
# Reconstruct the exact three-dimensional second-order identity.
.venv/bin/python experiments/p3_second_order_identity.py
# Check the elliptic-family formulas at selected sizes and parameters.
.venv/bin/python experiments/crabb_elliptic_axis_theorem.py \
--minimum-size 3 --maximum-size 8 \
--ellipse-parameters 0.15 0.6 --precision 100 \
--output tmp/elliptic-axis.jsonl
# Check the algebra used in the later all-powers proof.
.venv/bin/python experiments/global_power_lemma_exact_audit_20260905.py \
--output tmp/all-powers-exact.jsonIndividual proof notes give the corresponding reproduction commands and describe whether each check uses exact arithmetic or numerical approximation.