Small Matrices with Large Inverses: Unimodular $4 \times 4$ Cases

Steven Finch

arXiv:2607.07688v4 Primary: math.CO Secondary: cs.DM, math.NT

Abstract

How close to singularity can an $n \times n$ unimodular matrix be? For ternary cases as $n$ increases, exact expressions are unlikely, but upon fixing $n=4$ and assessing $(2k+1)$-ary cases as $k$ increases, we make significant progress; similarly for $(k+1)$-ary cases of $4\times 4$ nonnegative unimodular matrices.

Disclosure

“were allowed to be zero). It seemed important to restate our conjecture in full generality, removing the zerofree condition. Also, the case k = 1 was empty in [10] but it is meaningful here. 6. Acknowledgements Microsoft Copilot assisted in writing CUDA programs, which opened the door for me to run intensive experimental jobs on Nvidia GPUs. Anthropic Claude Opus 4.8 played a crucial role in turning my conjectures into theorems. The creators of Mathematica, as wel”
PDF page 17

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-24T00:34:15.228Z

Structural counts

Pages 21 pdf

Theorems 0 source

Lemmas 0 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 102 source

Bibliography entries 10 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file mtrx3.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.