HEXAGONbeta

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 1 / 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

Deep Bhattacharjee asked for a proof of the Geometric Selection Conjecture of his paper with Ushashi Bhattacharya, citing its Hexagon version 2610.00446v1. Claude (Anthropic), through Claude Code, found that the conjecture cannot be closed as stated, since conditions (C7) to (C9) are fixed by comparison with measurement, and instead proved that the mirror-fixed clause of its equivalent form (Theorem 22 of the strings paper) is never met by a viable vacuum; it drafted the note and wrote the check, a Python file that carries the lemma and the corollary on the discrete data of vacua (Hodge numbers, c_3, order of a free quotient) and tests them on self-mirror and three-family Calabi-Yau pairs with refuters (511 checks). The authors directed the work and are responsible for the mathematics. Code: release v1.0.0, https://github.com/creelie/geometric-selection-conjecture/releases/tag/v1.0.0.
AI provenance filesAI-PROVENANCE-HEXAGON.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