HEXAGONbeta

The Mirror Clause of the Geometric Selection Conjecture Is Empty

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

The Geometric Selection Conjecture asks for a functional on the heterotic landscape, invariant under mirror symmetry, whose unique minimum is a phenomenologically viable vacuum. It was shown to be equivalent to the existence of a viable vacuum fixed by the mirror involution wherever that involution is defined. We prove that no viable vacuum is ever fixed: a fixed vacuum is isomorphic to its mirror, so its threefold has Euler number zero, the third Chern class of its bundle vanishes on its component, and by Hirzebruch-Riemann-Roch it has no net families. The conjecture is therefore equivalent to the existence of a viable vacuum on a component of the landscape carrying no standard embedding, and no vacuum built on the tangent bundle of a Calabi-Yau threefold is ever selected.

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 Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 23, 2026