T-076: Linear-measure lower bound replayed at $n=82$$n=82$

S3 V3 C3 lower bound confirmed

2026-10-02 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · $n=82$$n=82$

A measure of points, segments and rectangles that wand125/square-packing-bounds published on 2 October 2026 proves $s(82)\ge 233/25$$s(82)\ge 233/25$ = 9.32. The measure is of the kind and checker of T-073: point masses, segments of uniform density and rectangles of uniform density, in D4 orbits with exact rational geometry, of total mass 82 - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, with 86 point, 222 segment and 774 rectangle orbits. They are those of T-073's $n=83$$n=83$ certificate, every coordinate scaled by exactly 932/935, with the masses solved again.

The source reports it accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp, byte for byte the verifier of T-073, in 153,579,479 nodes, and says the full replay was run again from the published tarball by the same implementation. The value exceeds Green's reported 2 sqrt(2) + (247 + 12 sqrt(2))/41 = 9.2667335..., the value this record reported at $n=82$$n=82$, by 0.0532664..., and does not rest on Green's argument, which jlevy/squares#308 reports the published material does not establish for k >= 4. Its mass is below 83, where T-075's 937/100 is higher.

It passed a complete replay here on 3 October 2026 of the source's unchanged checker and replay functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replay runs the source's own algorithm and is not an independent decision of coverage.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 2 October 2026 on jlevy/squares#294. The source says parts of the work were produced with AI assistance under human direction.