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Explicit String Vacua and the Geometric Selection Conjecture

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.AG — Algebraic Geometry

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗

Version 2 / Oct 23, 2026 / CC BY 4.0

Abstract

We combine explicit spectral calculations for the Dwork and weighted Fermat pencils with an audit of the Geometric Selection Conjecture. The residue calculation determines the character labels, cubic couplings and invariant-direction mass maps, while an index argument excludes three standard-embedding families on free quotients of thirteen hypergeometric complete-intersection types. For a nonstandard construction we fix one tetraquadric equation, prove smoothness and a free degree-four action, and calculate equivariant bundle cohomology by an exact determinant certificate. Its projected charged spectrum contains three families and one Higgs doublet pair. At the specified split-bundle point, however, an additional Abelian gauge factor survives, the perturbative Yukawa matrices fail the required rank test, and the tree-level volume remains flat. We then show that the first of these failures is forced: on every Calabi--Yau threefold with at most four K\"ahler parameters, every polystable sum of five line bundles with vanishing first Chern class has a charge matrix of rank at most three and keeps at least one Abelian gauge boson massless at its split locus. On the tetraquadric the polystable K\"ahler classes of the fixed bundle form exactly the volume ray. For the second failure we decide the perturbative selection rules on eleven split data, the fixed bundle and ten further line-bundle sums with three families on the same manifold, by a flow criterion for the singlet insertions: six of the eleven lose a Yukawa matrix identically and five are not excluded. We then follow the extension branch: a single non-split extension never removes the forced gauge boson, while for the fixed bundle and five of the ten further sums a polystable sum of two rank-two extensions and a line bundle, with the same Chern character, exists on one side of a stability wall arbitrarily close to the split point and carries no massless Abelian gauge boson; for three further sums no such bundle exists at any K\"ahler class, and for one it does not exist on the general tetraquadric. We fix the acceptance windows for the stabilization, Yukawa and supersymmetry breaking conditions, so that the conjecture becomes a definite statement, and show that the failures found do not depend on them. No vacuum satisfying all nine conditions is constructed; the Geometric Selection Conjecture remains unproved.

Provenance statement

Human authors: Deep Bhattacharjee, Ushashi Bhattacharya. Submitting contributor: Deep Bhattacharjee, ORCID 0000-0003-0466-750X. Text classification: to be chosen by the submitting author on the form (version 1 of the record, hexagon:2610.00493v1, was submitted as partly checked and understood by a human; this version has been read in full by the submitting author before upload if the stronger declaration is chosen). Basis: the submitting author's own declaration on the Hexagon form, not a claim verified by the assistant. Claude Code (Anthropic) assisted the preparation of the manuscript, the merger of the two earlier Hexagon parts, the proofs and the writing of Section 16 (including the extension-branch results of Section 16.3 and the acceptance windows of Definition 9.1), the Python checks of the finite arithmetic and the submission package. The Anaxia engine (creelie/eBLLM, release 3.1.0, 10.5281/zenodo.23484160) supplied the ten tetraquadric line-bundle classes of Proposition 16.4 and the atlas record that conditions C7 to C9 of the conjecture are fixed by comparison with measurement; it is cited in the paper. Documented tools: Anthropic Claude / Claude Code; Python 3 (exact rational arithmetic, 511 + 45 + 61 checks in the repository, the last with SymPy); LaTeX / pdfLaTeX (amsart); Matplotlib for the figures. Exact historical model versions are not inferred from tool names. Verification scope: every statement of the paper is proved in the text by hand; the programs check finite arithmetic (the matrix certificates, the charge matrices, the Yukawa flow bounds and the slope spaces, extension groups and wall certificates of the extension branch) and no statement rests on them alone. No Lean formalisation is claimed for this paper. The authors retain responsibility for the final mathematics, citations and approval of the manuscript. AI systems are not added as human contributors. The package creation step prepared the source archive and the submission metadata; it did not submit the manuscript to Hexagon. No private prompts, chat history or hidden reasoning traces are included or represented as an exported transcript. Earlier Hexagon versions: https://hexagonmath.org/2610.00493v1 (mirror clause), https://hexagonmath.org/2610.00446v1 (spectral calculations) Research manuscript/source DOI: https://doi.org/10.5281/zenodo.23688897 (release v3.1.0; all versions https://doi.org/10.5281/zenodo.23567114) Repository: https://github.com/creelie/geometric-selection-conjecture
AI provenance filesAI_PROVENANCE.md · 3 KB

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 23, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 27, 2026