HEXAGONbeta

Vanishing of Higher Magnitude Homology for Finite Prefix Models

A mix of human-written and AI-generated textHuman understanding: all partsmath.CT — Category Theorymath.AT — Algebraic Topology

Contributed by Ryoji Hirayama

Version 2 / Oct 22, 2026 / CC BY 4.0

Abstract

Bradley and Vigneaux define a generalized distance on a finite prefix poset using next-token probabilities. They conjecture that its higher magnitude homology vanishes. Motivated by their expression of magnitude as a weighted sum of homological Euler characteristics, we verify this conjecture for their explicit complex, in which both endpoint faces are zero. For each finite length, the homology groups in degrees two and above are zero. The degree-one homology group is free abelian, with one generator for each positive-probability edge of that length. The proof constructs a contracting homotopy for each component whose endpoints have an interior prefix. This homotopy inserts the least interior prefix. Edge probabilities zero and one are both allowed.

Provenance statement

OpenAI Codex was used to draft and revise the English text, organize the exposition and references, and prepare the LaTeX source. The manuscript is based on pre-existing mathematical notes. The author reviewed the full manuscript and its mathematical argument.

Tools used

OpenAI
CodexVersion GPT-6.1 Sol
OpenAI
CodexVersion GPT-6 Astra
Rocq
RocqVersion 9.2

References

  1. T.-D. Bradley and J. P. Vigneaux. The magnitude of categories of texts enriched by language models. arXiv:2501.06662v2, 2025. https://arxiv.org/abs/2501.06662v2.arXiv
  2. T. Leinster and M. Shulman. Magnitude homology of enriched categories and metric spaces. Algebraic & Geometric Topology , 21:2175--2221, 2021. https://arxiv.org/abs/1711.00802.link
  3. R. Kaneta and M. Yoshinaga. Magnitude homology of metric spaces and order complexes. Bulletin of the London Mathematical Society , 53:893--905, 2021. https://arxiv.org/abs/1803.04247.link
  4. F. W. Lawvere. Metric spaces, generalized logic and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano , 43:135--166, 1973. Reprinted in Reprints in Theory and Applications of Categories 1 (2002), 1--37. https://tac.mta.ca/tac/reprints/articles/1/tr1abs.html.link

Version history

  1. v1 · 3 pagesSubmitted by Ryoji HirayamaInitial depositSupersededOct 21, 2026
  2. v2 · 3 pagesSubmitted by Ryoji HirayamaSubmitted as a new versionCurrentOct 23, 2026