Protected corners and a trichotomy for Han's conjecture

Marco Armenta

arXiv:2607.20849v1 Primary: math.RT Secondary: math.KT, math.RA

Abstract

Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. In the tau-Hochschild framework of Cibils, Lanzilotta, Marcos and Solotar, it splits into persistence (Gap A) and survival (Gap B), and a Gap-A failure is already a counterexample. We prove a protected corner theorem bounding Ext at a surviving vertex, settling the Liu-Morin extension conjecture beyond the monomial and special biserial cases. We then establish a trichotomy for Gap-A failures on three strongly connected vertices: the all-infinite case is impossible, and the two-infinite case is completely classified as a mutual dumbbell.

Disclosure

“Disclosure During the preparation of this work, the author used Claude, Anthropic’s Fable 5 model, for deep research, formulation of theorems, and drafts of their proofs. The author reviewed and edited the output as needed and takes full responsibility for the content of the published article. References [”
PDF page 29

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T19:03:42.608050Z

Structural counts

Pages 31 pdf

Theorems 12 source

Lemmas 8 source

Propositions 12 source

Corollaries 11 source

Definitions 2 source

Displayed equations 34 source

Bibliography entries 38 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main_5_.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.