TRACE-PERTURBED MINIMIZERS OF THE MIXED DISCRIMINANT
Scott Schubr
Version 1.0, October 20, 2026

CONTENTS

Trace_Perturbed_Mixed_Discriminants.pdf -- complete short note.
main.tex -- complete LaTeX source, with an embedded bibliography.
CITATION.bib -- citation for this note and its three references.
MANIFEST.sha256.json -- byte counts and SHA-256 hashes of the other files.

MATHEMATICAL SCOPE

For every fixed n, sufficiently small changes in the individual traces
preserve the unique scalar-tuple mixed-discriminant minimizer among
Hermitian PSD n-tuples summing to the identity. The proof answers
Question 1.7 of Gurvits--Leake, arXiv:2609.21083v2 (October 15, 2026),
using Gurvits's earlier global uniqueness theorem and standard local
stability. A supporting proposition identifies the exact region of
positive definiteness of the constrained Hessian at the scalar tuple.

No explicit or dimension-uniform global neighborhood, general
real-stable-polynomial theorem, or algorithm is asserted. The local
Hessian criterion is not a global minimizer criterion. The proof is
analytic and requires no computational certificates or data.

BUILDING

The source is self-contained apart from standard LaTeX packages.
With Tectonic installed, create a separate build directory and run:

  tectonic --untrusted --outdir build main.tex

Alternatively, run pdflatex twice in a separate build directory with
main.tex as the input. BibTeX is not required; CITATION.bib is supplied
for readers' reference managers. A rebuilt PDF may differ at the byte
level because compiler versions, fonts, or metadata may differ.

SOURCE VERSIONS AND ATTRIBUTION

Question and theorem numbering for Gurvits--Leake refers specifically
to arXiv:2609.21083v2. Numbering for the older Gurvits paper refers
to arXiv:math/0406420v1; its journal citation is also supplied.
The main global uniqueness input and the close older Corollary 4.2
perturbation argument are credited in the note. The simultaneously
diagonalizable subcase was already covered by permanent results.

The note includes a disclosure of AI assistance. It does not claim
external human refereeing, formal verification, or worldwide priority.
