The S-matrix conjecture

Source abstract

Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the even-dimensional case for $n\ge1000$. We complete the remaining even-dimensional cases. Starting from the structural identities in Frankel--Urschel Lemma 2.1, we derive an exact global defect budget and combine binary rounding with Gram projection. A refined ten-row obstruction handles every even $n\ge66$; a finite exact calculation handles $4\le n\le64$, $n\ne6$; and a separate multi-column energy argument treats $n=6$. The order-two case follows from a direct calculation. The new even-dimensional proof has been formalized in Lean 4, with Frankel--Urschel Lemma 2.1 as its sole external mathematical input. Together with Cheng's odd-dimensional theorem, this proves the S-matrix conjecture in every dimension.