Fixed-Witness Complementary-Minor Inequalities for Strictly Sign-Regular Matrices

Let A be a real n×n matrix that is strictly sign-regular with signature ε: every minor of order k is nonzero and has sign ε_k. We prove that for every nontrivial rank 1 ≤ r < n and every r-element column set J, there is an r-element row set I(ε,J), selected using only the signature and J, such that

(−1)^(ΣI+ΣJ) · det A[I,J] · det A[I^c,J^c] / det A > 1

for every matrix A in that signature class. In particular, |det A[I,J] det A[I^c,J^c]| > |det A|. The same term in the generalized Laplace expansion therefore has the sign of det A and exceeds the determinant in magnitude. Equivalently, every column of the combined matrix of the r-th multiplicative compound has an entry greater than one at a position fixed throughout the strictly sign-regular chamber; this entry is also the determinant of a block relative gain.

The rank-one selector has the explicit form 1 + |j − h(ε)|. The proof combines corner Schur complements, an arbitrary-core Sylvester determinant identity using increasing-order minors, and a row-position lifting lemma. The resulting exact factorization reduces the all-rank theorem to the rank-one case. The totally positive specialization is the classical strict Fischer inequality. The strict bound need not have a uniform margin above one.

The deposit comprises the 16-page mathematical preprint, complete LaTeX source and a reproducible exact-arithmetic Python verification archive. Its finite checks and mutation tests are regression evidence, not a substitute for the mathematical proof. The work was developed with substantial assistance from OpenAI ChatGPT and Anthropic Claude, including proof development, code and manuscript drafting. AI-based checks are not human peer review; this is an unrefereed author preprint.

Copyright © 2026 Henric Larsson. PDF, LaTeX, documentation and data: Creative Commons Attribution 4.0 International (CC BY 4.0). Python verification code: MIT License.

DOI: https://doi.org/10.5281/zenodo.23675800