T-097: Mixed rectangle-measure lower bound verified at $n=66$$n=66$

S3 V3 C3 lower bound confirmed

2026-10-05 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · $n=66$$n=66$

A rectangle density of wand125/square-packing-bounds, checked at coverage one and published on 5 October 2026, after those of T-094, proves $s(66)\ge 843/100$$s(66)\ge 843/100$ = 8.43.

The certificate is a density of 713 uniform rectangles in D4 orbits, and no point mass, of total mass 6599999/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, of the kind and checker of T-069 and later. The source reports it accepted at all 200 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, and at the axis by exact integer tables. It supersedes the source's mixed_n66_L842 (T-069).

The value is above T-069's 421/50 = 8.42, the reported and verified lower bound at $n=66$$n=66$ before it, by 0.01. A packing of 67 squares contains one of 66, but at $n=67$$n=67$ the record already holds more in both lanes (212/25), so it does not carry further.

The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions of the retained candidate, and two mutants scaled below coverage one were refused: confirmed, independently re-implemented. The verifier shares no code with the source's checker and runs the same net-and-shrink method, so it is a second implementation and not a second method. The source's own checker was replayed here at 12 of the 201 directions, each returning the certificate's own record, and not in full.

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 5 October 2026 on jlevy/squares#282. The source says parts of the work were produced with AI assistance under human direction.