{
  "abstract": "For a pair potential V(r) = g(r2) and a planar Bravais lattice of covolume one, the harmonic zero-point energy is Ezp = ½⟨∑j ωj(k)⟩ over the Brillouin zone. Planar universal optimality (the Cohn–Kumar conjecture; a theorem for Gaussians by Montgomery; its general proof in openai/math is AI-generated and not peer reviewed) concerns the static energy. The natural quantum analogue, conjecture QU-1, says the triangular lattice also minimises Ezp for every completely monotone g. We tested it before trying to prove it, and it fails numerically (Tier X); we do not claim a proof that it is false. Every claim is filed in an Elenchus claim ledger with tiers A (kernel-checked Lean), L (cited literature), C (human argument, not machine-checked) and X (floating-point exploration, which never supports a claim by itself). Tier X: for the Gaussian exp(−π r2) the rhombic lattice τ = 0.5 + 0.966025 i is dynamically stable and has ΔEzp = −1.3074e-2 relative to the triangular lattice (Ezp = 1.4902783183 against 1.5033523522). Two formulas in four implementations agree (real-space folded-grid Rust code and an independent numpy re-implementation; a reciprocal-space Rust code with a spectrally convergent zone rule, quadrature error about 4e-15; a short third numpy implementation). The conjecture is numerically refuted for 8 of 12 completely monotone potentials tested (Gaussian a = 0.5 and 1, Yukawa, dipolar-exciton, Riesz s = 3 and 4); the riesz:4 refutation rests on a single competitor with a small long-wave stability margin (0.08 % of the triangular value), its finite-k stability comes from grid computations only, and its triangular point is a local minimum (c > 0). Four potentials (Gaussian a = 0.25, 2, 4 and Riesz s = 6) are not refuted, on a 567-point sample plus two dense paths, which is not an exhaustive search. At the triangular point the curvature of Ezp is isotropic and, for exp(−π a r2), numerically negative for a in (0.37668, 1.58148) within 0.15 ≤ a ≤ 4 (a scan of 23 values of a between 0.1 and 8 followed by bisection of the sign changes; gaps between scan points are not excluded). A restricted conjecture QU-1′ for 0.15 ≤ a ≤ 4 is stated; it is supported only by these numbers and is not proved. A first-reported largest-margin point (0.295 + 0.9555 i) is only marginally stable and is corrected in the paper; the competitors are not stress-free and the harmonic approximation, Bravais lattices and two dimensions are assumed. Tier A (Lean 4.34.1, pinned toolchain and Mathlib; axioms within propext, Classical.choice, Quot.sound; planted sorry and planted-axiom controls caught): 19 finite and algebraic statements (derivatives and Laplacian identities of the lattice-vector length, the Gaussian curvature densities, the 120-degree rotation lemmas that force an isotropic Hessian at the triangular point, and an inversion step for the mean-square-frequency sum rule). They certify none of the statements about Ezp. Tier L: Montgomery's theorem (cited through Deng–Luo; original not opened) and two results of Luo–Wei, with the Elenchus citation gate accepting the three quotations. Tier C: the sum rule, the Einstein bound and a concavity decomposition of the curvature. The curvature scan, the mechanism and QU-1′ were run after the preregistered falsification gate refuted the conjecture and are post hoc. Conditionality: no result here rests on openai/math apart from the pinned toolchain; the refutation needs only that exp(−π a r2) is completely monotone in r2. Improvement along the way: a new rusty-SUNDIALS module (reciprocal-space Gaussian dynamical matrix and a Duffy–Gauss–Legendre zone rule), crate tests 59 passed, 0 failed, 1 ignored at the commit used for all reported numbers; after the audit a hardening commit (ezp_gauss_duffy_stable, which refuses to return a clipped energy for an unstable lattice) brings the crate to 60 passed, 0 failed, 1 ignored with clippy and fmt clean, changing no reported number. No LeanFlow change was made. The Elenchus ledger gate reports 0 blocking findings and 8 flags (exit code 1), one for each Tier A row whose statement adequacy no human has audited. The work was produced by AI agents (Claude) under the author's direction; the author is responsible for the content. No human audit of this preprint, its Lean module or its Rust code has been done. A separate AI agent (Claude) audited the preprint (AUDIT.md in the bundle, verdict accept with corrections; the required corrections are applied): it recompiled the Lean module with its own planted sorry and axiom controls, reproduced the gauss:1 and gauss:0.5 refutations with an independent real-space code, regenerated all numbers and tables and re-checked the three citations, and reran the crate tests; it did not independently recompute the dipolar, Riesz and Yukawa refutations, the long-wave elastic tensor, the Lin/Conc decomposition, Result 5, c(a) beyond a = 1 and 2, or phonon.rs. The paper-stage reruns by the producing pipeline are not a producer-verifier check.",
  "abstract_shown": true,
  "affiliations": [
    "SocrateAI Lab"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Xavier Callens"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqa2bvmdf2vyn.bib",
  "cited_by": 8,
  "doi": "10.5281/zenodo.23653832",
  "fulltext_url": null,
  "id": "bqa2bvmdf2vyn",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23653832",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
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  "openalex": "W7223664836",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/bqa2bvmdf2vyn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7223664836",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23653832"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2612|Mathematical Approximation and Integration",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Does the triangular lattice minimise the harmonic zero-point energy? A numerical counterexample for completely monotone potentials, with kernel-checked curvature identities",
  "title_text": "Does the triangular lattice minimise the harmonic zero-point energy? A numerical counterexample for completely monotone potentials, with kernel-checked curvature identities",
  "type": "preprint",
  "updated_at": "2026-11-05T13:20:40Z",
  "url": "https://research.draconitic.com/paper/bqa2bvmdf2vyn",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}