The harmonic triangular-to-oblique instability of the quantum Gaussian-core crystal: first order, near the Lindemann melting estimate, and absent in the self-consistent harmonic approximation (a preregistered negative result)

Negative result, preregistered. For the Gaussian pair potential V = ε exp(−π a r2/l2) at one particle per site, the harmonic energy of a planar Bravais lattice of shape τ is E = Est(τ) + ℏeff Ezp(τ), with ℏeff = ℏ/√(mεl2). We asked (Proposal 2, item B1; preregistration file results/proposal2/2026-10-26/preregistration.json) whether the triangular lattice then undergoes a quantum structural transition to a rhombic/oblique lattice, whether it comes before melting, and whether it survives beyond the harmonic approximation. The preregistered kill criterion is: “if the transition lies beyond melting or disappears beyond harmonic, report that (negative result)”. The criterion is met. Tier X (floating point; nothing here is a proof). Within the harmonic theory the triangular lattice stops being a local minimum at the spinodal ℏ*(a) = −cst/czp for a in (0.37668, 1.58148): minimum 0.0877 at a = 0.5, 0.1238 at a = 1. The transition is first order in the sense that a finite-distortion, dynamically stable oblique lattice reaches the triangular energy earlier, at Λ1 = 0.75–0.78 ℏ* for a = 0.45–1.2 (0.0942 at a = 1; the competitor sits on the harmonic stability edge and is not an interior local minimum). The order-3 stabiliser of the triangular point allows a cubic invariant (Re w3 in the disk coordinate w), which predicts equal and opposite slopes of the tie coupling on the two inequivalent mirror rays; the measured slope ratio is −1.0 within 2% for a = 0.5–1. The Λ1 winners lie on the harmonic stability edge, so harmonic Bravais theory does not identify the phase beyond. The harmonic Lindemann ratio at the spinodal is at least 0.2798 for every a tested, at or beyond the top of the quantum-Monte-Carlo band γc = 0.23–0.28 (but below the harmonic value 0.30 for a = 0.5–0.9); at Λ1 it is 0.245–0.273 for a = 0.5–1 (inside the band: the Lindemann test alone does not exclude Λ1 there, and the Lindemann coupling scales as γc2) and larger elsewhere. Where Λ1 lies inside the Lindemann band (a = 0.5–1 for γc ≤ 0.28, a = 0.45–1.1 for γc = 0.30) the negative verdict rests on SCHA alone. In the T = 0 self-consistent harmonic approximation (SCHA; two implementations, Python and Rust, of the same Duffy–Gauss zone rule, agree to 8.9e-16; an independent midpoint-grid recomputation by the auditor reproduces the a = 1 SCHA verdict) the triangular lattice stays a local minimum and the lowest stable sampled lattice up to ℏeff of about 0.27–0.33 (the SCHA spinodal, a-dependent), where the SCHA Lindemann ratio is 0.60–0.87; a continuation-seeded run over 150 shapes per case (103–114 of them beyond the harmonic stability edge) finds none below triangular. The transition disappears within SCHA. Caveats. SCHA is a variational Gaussian theory: a Rayleigh–Ritz upper bound for each shape that omits the second-order cubic O(ℏ2) term and exchange, and the SCHA stiffening is itself of order ℏ2, so the result holds within SCHA and is not a statement of physical absence. Path-integral Monte Carlo was not run (blocked: fixed commensurate box imposes the shape, the finite-size shift of the energy gap is about 70% of the gap at 64 particles and 24% at 100; numbers in the paper). Only Bravais lattices with one particle per cell were compared. The Lindemann constant is a heuristic; the QMC anchors are true-⟨u2⟩ values whereas the tabulated ratio is harmonic. Literature (Tier C, quotations verified verbatim against fetched text; no Tier L). The Elenchus citation gate admits only theorem-like quotations and refuses these; we therefore filed them as Tier C. Four sources were re-read (Astrakharchik et al. arXiv:cond-mat/0608246; Prestipino et al. arXiv:1809.07515; Henkel et al. arXiv:1001.3250; Zeiher et al. arXiv:1602.06313). Rydberg dressing realises a soft-shoulder potential with a negative Fourier component (we confirm it numerically), not a Gaussian; a pure Gaussian has a positive Fourier transform and is called the marginal non-clustering case by Prestipino et al. (that ordinary quantum melting preempts the harmonic transition is our Tier X inference from the Lindemann numbers and SCHA, not a literature statement). Statements read only by a sub-agent are marked [sub] and are in no ledger row. Improvement: a new second-implementation SCHA module in rusty-SUNDIALS (crates/qf-yukawa-crystal/src/scha.rs, commit 9e85e15b23be0c74a665b6a5b74ec02b0bbffc5f, not pushed at the time of writing); the crate tests were rerun at paper stage: 63 passed, 0 failed, 1 ignored. The earlier result that the triangular lattice is a local maximum of the harmonic zero-point energy (doi:10.5281/zenodo.23653833) stands as a statement about the harmonic functional only. Provenance and audit. Every claim is filed in an Elenchus claim ledger (39 rows: Tier C arguments and quotation-checked literature, Tier X numerics; no Tier A or L row). The ledger gate reports exit code 0 with no findings; that is bookkeeping only. The work was produced by AI agents (Claude) under the author's direction; the author is responsible for the content. Audit: an independent AI audit by a separate Claude agent (AUDIT.md in the paper directory; verdict ACCEPT_WITH_CORRECTIONS; it recomputed the headline numbers with its own code and a different quadrature) found four blocking and eleven nonblocking items; all blocking items are corrected in this version (REVISION.md). No human has audited the work. Producer, writer and paper-stage reruns belong to the same pipeline.