HEXAGONbeta

Reflexive Descent to Minimal Seeds in the Kreuzer--Skarke List

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.AG — Algebraic Geometrymath-ph — Mathematical Physics

Contributed by Deep Bhattacharjee ↗, Ushashi Bhattacharya ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Kreuzer and Skarke built the list of reflexive polytopes by growing it from small seed polytopes. We ask how far a polytope can be from its seeds when it is shrunk through reflexive polytopes only, one at a time and without skipping any. Such a descent always exists and always ends at a seed. In four dimensions no descent takes more than 348 steps, which proves the conjecture of a uniform transition depth, and in two and three dimensions the exact maxima are five and eleven. In four dimensions two polytopes with thirty-seven lattice points need exactly twenty-two steps, so the maximum lies between twenty-two and 348.

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 long descents and for the certificate of depth 22 was organised on the BLLM engine Anaxia (creelie/eBLLM, release 3.0.5, https://github.com/creelie/eBLLM/releases/tag/v3.0.5) by its learn-from-failure protocol, the whole record was rerun on release 3.1.1 (https://github.com/creelie/eBLLM/releases/tag/v3.1.1), and it is kept in release 3.2.1 (https://github.com/creelie/eBLLM/releases/tag/v3.2.1). The bound in every dimension follows by hand from Blichfeldt's theorem and the volume bound of Balletti, Kasprzyk and Nill. The Kreuzer–Skarke list is due to M. Kreuzer and H. Skarke and is read from its public copy, which is cited. The theorems are proved in the paper from finite data computed from the list (the cover relation in dimensions two and three, the joint distribution of point counts in dimension four), which is archived; the certificate of depth 22 (5,592 polytopes with their complete lists of covers) and the descent of the 680-point polytope are archived and rechecked by independent programs (87 checks, 98 with the full run, including a check that every cut met in dimension three and at the certificate is reflexive and equals the cover). The exact value of D_4 is not claimed. Both authors approved the final version. Code, checks and Lean: release v1.2.0, https://github.com/creelie/seed/releases/tag/v1.2.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026