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A Counterexample to the Subtraction-Free Conjecture for Bounded Ratios of Minors

Primarily AI-generated textHuman understanding: some partsmath.RA — Rings and Algebrasmath.CO — Combinatorics

Contributed by Qiyue Tang ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

We disprove the subtraction-free conjecture of Boocher and Froehle for bounded ratios of minors of totally positive matrices. We construct products (P,M,Q) of ordinary minors of a matrix of order (3207) such that [ 0<\frac{P}{MQ}\leq\frac1{64} ] throughout the totally positive locus, whereas (MQ-P) has a coefficient equal to (-1) in the standard bidiagonal parameters.

Provenance statement

GPT-6-Pro was used for approximately five hours of documented research, including the development of the counterexample, mathematical analysis, and computational verification. GPT-6-Pro was also used for detailed proof checking, independent mathematical review, manuscript revision, and language polishing. A further independent audit using Rethlas-OSV is planned. Its results will be incorporated into a subsequent version of the manuscript once the review is completed. Uploading this paper to this platform does not mean that I claim ownership or copyright over it. My sole intention is to share this AI-generated solution and make it available for independent review. Once its correctness has been independently verified, the paper may be cited directly. I welcome anyone who would like to organize this work; if you do so, the credit will be yours. I also welcome further research building on this work, including extensions, new applications, and related questions. For additional references or detailed research materials, please contact me by email. Email: tangqy24@mails.tsinghua.edu.cn

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Version history

  1. v1 · 16 pagesSubmitted by Qiyue TangInitial depositCurrentNov 10, 2026