Integer-charge electron-number convexity counterexamples

Public review release — candidate proof, not peer reviewed. The exact finite computations pass reproducible rational-arithmetic certificates. The many-body spectral argument still requires independent expert verification.

This repository proposes an explicit fixed-nuclei, nonrelativistic Coulomb system with ten protons for which the exact fermionic ground-state energy is not convex in the integer electron number. For the stated rational geometry and all sufficiently large separation scales L, the central claim is

E_L(16) + E_L(18) - 2 E_L(17) < 0.

The proposed asymptotic coefficient is certified exactly:

D = C_6 + C_8 - 2 C_7
  = -0.033585733779687126075624519286... < -0.032.

The companion research note develops further candidate theorems: stability on an explicit open set, an infinite rational one-parameter family, a rational counterexample with exactly K protons for every K >= 10, and replicated systems with arbitrarily many violating electron counts.

Evidence boundary

What has been machine-checked:

• exact rational evaluation of a Chandrasekhar-type trial state proving E(H-) < E(H);
• exhaustive rational-interval enumeration of the ten-site subset problem;
• the negative coefficient and optimizer gaps;
• perturbation radii, the parametric interval, shadow-site inequalities, and the supporting-line inequalities used for replicated gadgets.

What has not yet been independently certified:

• the full operator-domain implementation of the many-particle IMS argument;
• every use of HVZ, exponential localization, and the maximum-ionization theorem in the assembled proof;
• literature priority beyond the searches described in the manuscripts;
• peer review or journal acceptance.

See STATUS.md for the release gate and claim-by-claim status.

Repository map

paper/
  core_proof.md                    Base ten-proton proof draft
  core_proof_review.pdf            Rendered review copy
  infinite_families.md             Uniform and infinite-family extension
  infinite_families_review.pdf     Rendered review copy
certificates/
  atomic_hydride_certificate.py
  base_geometry_certificate.py
  infinite_families_certificate.py
results/
  Recorded outputs for all certificates

Reproduce the certificates

Only Python's standard library is required.

make verify

Or run the files individually:

python3 certificates/atomic_hydride_certificate.py
python3 certificates/base_geometry_certificate.py
python3 certificates/infinite_families_certificate.py

The longest certificate performs exhaustive subset enumeration and may take a little time. A successful run ends with:

All exact-arithmetic assertions passed.

More detail is in REPRODUCIBILITY.md.

How to review this claim

The highest-value review target is Section 5 of paper/core_proof.md and Theorem 2.1 of paper/infinite_families.md. Please report a suspected gap with an exact section/equation reference and classify it as fatal, repairable, or expository. See CONTRIBUTING.md.

AI provenance

Generative-AI systems were used extensively for candidate discovery, proof exploration, drafting, adversarial revision, and certificate development. The named human author is responsible for the public claims and is actively seeking human expert verification. See AI_PROVENANCE.md.

Citation

Citation metadata are provided in CITATION.cff. Until an arXiv identifier or DOI exists, cite the tagged v0.1.0-review version by author, title, version, release date, and URL rather than the mutable default branch.

License

• Source code: MIT License, see LICENSE-CODE.
• Manuscripts and documentation: Creative Commons Attribution 4.0 International, see LICENSE-MANUSCRIPT.