HEXAGONbeta

Which String Vibrations Become Which Generations on the Dwork Quintic

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 22, 2026 / CC BY 4.0

Abstract

We determine, for the heterotic string on the one-parameter family of quintic threefolds, which vibration modes become which families of charged particles. The hundred and one families are labelled by the characters of a symmetry group of order one hundred and twenty-five. Every Yukawa coupling that this symmetry allows is a power of the modulus, and every other coupling vanishes. The invariant deformation pairs each family with exactly one partner, so that fifty families are massless at the Fermat point and none anywhere else, and at the Fermat point the masses are given in closed form by the Gamma function. On the three other one-parameter families of the same type, in weighted projective spaces, the symmetry no longer tells the families apart and some families stay massless at every member. We also prove that the Geometric Selection Conjecture is equivalent to the existence of one viable vacuum fixed by mirror symmetry, and that a free quotient of a Calabi–Yau complete intersection of any of the thirteen one-parameter hypergeometric types never carries three families with the standard embedding; on seven of these types, among them the sextic and dectic pencils, no finite group acts freely at all.

Provenance statement

The proofs, the text, the figures and the programs were carried out by Deep Bhattacharjee with AI assistance (Claude, Anthropic) in the Claude Code environment. The search for constructions was organised on the BLLM engine Anaxia (creelie/eBLLM; record made on release 3.1.0, current release 3.2.1, https://github.com/creelie/eBLLM/releases/tag/v3.2.1) by its learn-from-failure protocol. Every statement is proved in the paper; the computations are archived in creelie/strings, release 1.2.0, and rechecked by 215 independent checks and 14 Lean 4 theorems. The Geometric Selection Conjecture is not claimed. Both authors approved the final version. Code, checks and Lean: release v1.2.0, https://github.com/creelie/strings/releases/tag/v1.2.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 21, 2026