Bang--bang representation of $3\times 3$ embeddable stochastic matrices

Leonel Robert

Abstract

We prove that every embeddable $3\times3$ row-stochastic matrix is a product of at most seven elementary row-stochastic matrices. Frydman's example shows that this bound is sharp. The proof combines Frydman's six-factor criterion and spiral theorem with a reduction to a three-parameter critical family and an analysis of the membership certificates that persist along it.

Disclosure

“odel and Codex through a ChatGPT Plus account. I supplied ChatGPT with an initial report on the problem which discussed relevant ideas and identities and the overall strategy of studying a first exit from R7 . Starting from this framework, ChatGPT assisted in solving the algebraic conditions arising at first exit and in finding a convenient normalization of the matrices satisfying them. This led ChatGPT to the explicit three-parameter critical family U (x, r, y) and to more tractabl”
PDF page 20

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T05:01:22.508548Z

Structural counts

Pages 23 pdf

Theorems 6 source

Lemmas 11 source

Propositions 6 source

Corollaries 5 source

Definitions 2 source

Displayed equations 203 source

Bibliography entries 12 source

Appendix pages 4 estimated

Count notes

Source counts use the expanded primary TeX file 7moves10_xry.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.