Can the planar Sandier-Serfaty proof architecture reach three-dimensional bcc jellium? A feasibility study with a negative outcome

The Sandier–Serfaty conjecture (the triangular lattice minimizes the renormalized Coulomb energy of a planar point set in a uniform background) has a claimed proof in the openai/math release, which is AI-generated and not peer reviewed; our independent reproduction and audit of it is doi:10.5281/zenodo.23653827. The three-dimensional analogue, Wigner's conjecture that the bcc lattice minimizes the jellium energy among all point sets, is open (bcc is known to be optimal only among Bravais lattices, Muratov–Novaga, arXiv:2610.29732). This preprint asks whether the planar proof architecture can be transcribed to bcc. It is a feasibility study with a negative outcome; it proves nothing about bcc. The planar proof is decomposed into 13 steps and each is classified as dimension-agnostic, transferable or genuinely planar. The incidence count is not the obstruction: a per-cell Euler identity, sum(6 − nF) = 12 for simple Voronoi cells, transfers (Tier C). The obstruction is the exact calibration at the optimum. In 2D the Green function is a function of two variables in every sector (cos 6mφ = T6m(d/t)), so the trial-potential square term vanishes; in 3D the bcc Green function on a face circle of fixed distance t carries a cos 6ψ mode on hexagonal faces and a cos 4ψ mode on square faces (Tier X). A projection argument (Tier C, a human-style demonstration, not machine-checked, evaluated in floating point; not a rigorous enclosure) turns this into a ceiling: every certified constant from a radial, axial or hierarchical trial class whose data are invariant under the 60-degree rotation of the hexagonal pyramids (the number of parameters is not a hypothesis) is at most Wbcc − δ, with δ ≥ 8.75e-5 (hierarchical, 0.98 times the fcc−bcc gap 8.969e-5), ≥ 1.06e-4 (axial, 1.18 times the gap) and ≥ 3.09e-3 (radial). These slack values are floating-point quasi-Monte Carlo estimates (about 300 standard errors from zero, two seeds), not enclosures. The outcome is negative for the literal transcription (radial, axial and piece-intrinsic trial classes) and only heuristically negative for richer classes: for classes whose data are not invariant under that rotation (in particular classes that carry neighbour data) the negative rests only on a counting heuristic (about 12 conditions for 3 multipliers, against 3 for 3 in 2D), which is Tier C and is not a theorem. The paper also gives a weaker, non-sharp residual target (a computer-assisted improvement of the Lieb–Narnhofer constant −1.4508 towards −1.44423) with a falsification test that has not been run, and a certificate-size estimate under explicit assumptions (about 3e13 mesh nodes, of the order of 200 core-years, uniformly in the best case; not the binding constraint). Every claim is tagged and filed in an Elenchus ledger: Tier L (4 theorem-like quotations accepted by the citation gate), Tier C (hand arguments and verbatim literature remarks), Tier X (floating-point numerics with positive and negative controls). There is no Tier A: nothing is formalized in Lean. The internal study note's “A” (elementary inequality with proof sketch) is re-filed at C because kernel-checked is the ledger's meaning of A. The ledger gate reports 30 claims, no findings, exit code 0 (bookkeeping only). 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 has been done. One independent AI audit (a separate Claude agent instance; verdict accept with corrections, corrections applied) has been done; producer and writer are the same pipeline. Limitations include the unresolved cutoff-dependence of the renormalized energy for d ≥ 3 (Rougerie–Serfaty, Remark B.2), second-hand literature (Lieb–Narnhofer, Weaire–Phelan, Gersho; Hales–McLaughlin read at abstract level only), and a study null control that does not test the rotation convention of the pair-average bound (the independent auditor ran that control: convention correct).