Counterexamples to Bapat's q-permanent monotonicity conjecture

Abstract

Archival deposit of already public mathematical research by Yongxian Zhang. Original source bytes and release attachments are preserved, with exact Git and SHA-256 provenance. No external human peer review or worldwide priority is asserted. Existing licenses are preserved; previously unlicensed material remains all rights reserved. AI assistance and actual proof-verification scope are disclosed in the source materials. The GitHub releases below retain the original public disclosure times. The Zenodo deposit is a later archival publication; it does not retroactively create an earlier Zenodo timestamp. Historical proof states are not upgraded by later work. This 14-page manuscript gives two counterexamples to Bapat's original q-permanent monotonicity conjecture on [-1,1]: a specified 200 by 200 complex Hermitian positive-definite matrix with rational real and imaginary parts and P(q0)>P(1) for rational q0 in (0,1), and the existence of a real symmetric positive-definite integer matrix with P(q0)>P(q1) for rational 0<q0<q1<1. The latter proof provides no numerical matrix or explicit dimension bound. Both exact endpoints have complete Lean formalizations and public proof-closure verification records. The revision clarifies earlier special cases, distinguishes the Bapat–Sunder conjecture, explains the nonconstructive finite witness, adds reproduction entry points, and separates AI research and writing disclosures. The archive includes the audited arXiv source package, full formalization sources and existing verification records; this packaging is not a fresh Lean execution. arXiv submission awaits endorsement; the manuscript is not peer reviewed. The full source archive is split into byte-exact .partNN download files to accommodate the upload transport. Download every part, CHUNKS.json and reconstruct.py together, then run python3 reconstruct.py. All original source and certificate bytes are preserved.