arXiv:2610.33042v1 Announce Type: new Abstract: Let $D(n)$ be the maximum number of unit-square diagonals in an $n\times n$ square array such that no two selected diagonals share an endpoint and no unit square contains both diagonals. We prove that, for every odd positive integer $n$, $D(n)=n(n+1)/2+\lfloor(n+1)/6\rfloor$. The matching lower bound was established by Boyland, Roth, Pint\'er, Lauk\'o, Schoenfield, and Wasielewski. We obtain the upper bound by transforming any admissible selection $M$ in a $(2k-1)\times(2k-1)$ square array into an isometric path cover of the $k\times k$ vertex grid with at most $2k^2-|M|$ paths. An auxiliary matching decomposes the original vertices into alternating paths. On each such path, all segments between blocks are either of type $da$ or of type $bc$. Projecting these paths to the block grid gives coordinate-monotone sequences, which become geodesics when diagonal steps are expanded. The Fisher--Fitzpatrick theorem then yields the sharp upper bound. We also describe a Lean verification of the upper-bound reduction, with the grid-cover theorem supplied as an explicit hypothesis.
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