T-066: $s(59)=8$$s(59)=8$, by a mixed cover of points and grid-line segments

S3 V3 C3 optimality confirmed

2026-10-01 published · wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · $n=59$$n=59$

$s(59)=8$$s(59)=8$: the lower half by wand125's mixed cover of 1 October 2026, the upper half by the $8\times 8$$8\times 8$ grid. The cover's total is below 60 and 61 too, so it gives $s(60)=8$$s(60)=8$ and $s(61)=8$$s(61)=8$ as well, a second route to T-062 and T-063.

The cover is 26,308 weighted points of [0,8]^2 plus mass spread uniformly along 5,240 segments of length 1/50 on the interior grid lines, total 1474762899/25000000 = 58.99051596 < 59, exactly D4-invariant, in Evan Daniel's mixed format. Every closed unit square in [0,8]^2 captures mass at least one, so a packing of 59 at side below 8, scaled up to side 8, would capture at least 59. It was made with Daniel's line-cover linear program from his $s(60)=8$$s(60)=8$ cover, and scaled to a margin the source puts at about 0.09% over the exact threshold.

The source reports the cover accepted by Daniel's zmx2, with outward-rounded binary64 enclosures, over 6,400 symmetry-reduced roots and 51,200 unreduced ones, none uncertified. Its exact-rational evidence is two zm_mixed.py runs whose joint coverage cannot be checked from what is published, so it is not recorded here. It has no Lean statement.

zmx2 was replayed here in full on 2 October 2026, over all 6,400 D4 roots and all 51,200 unreduced roots, none uncertified, with the box totals the source states; its run records are not published, so no root-for-root comparison was possible.

wand125, square-packing-bounds, building directly on Evan Daniel's method, format, checkers and $s(60)$$s(60)$ cover. Registration was requested in jlevy/squares#280. The source says parts of the work were produced with AI assistance under human direction.