T-214: Mixed rectangle-measure lower bound $s(106)\ge 10503/1000$$s(106)\ge 10503/1000$ reported, on the 201-direction net

S3 V0 C0 Lower bound recorded

2026-10-16 published · Hosono after Tokoharu and Levy · $n=106$$n=106$

A rectangle measure of wand125/square-packing, published on 16 October 2026 and reported in jlevy/squares#446, is reported to prove $s(106)\ge 10503/1000$$s(106)\ge 10503/1000$ = 10.503.

The certificate, mixed_n106_L10503, is a density of 735 uniform rectangles in D4 orbits, and no point mass, of total mass 10599999/100000, on the net its candidate declares, the standard net of 201 half-angle tangents of step 83/40000 at core side 9977/10000. Since 9977/10000 (1 + 83/40000) = 399908091/400000000 < 1, and the last tangent 83/200 is past tan(pi/8), every unit square contains a concentric core at a net angle strictly in its interior. The search that found it started from a published certificate at a smaller count, mapped to this side and re-solved by linear programming (lifted, the source's provenance/search-notes.md); the certificate is the measure alone.

The source reports every such core capturing mass at least one at all 201 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, in two complete runs, and a control with every mass scaled by 197/200 refused at 32 sampled directions.

The value is above the reported lower bound at $n=106$$n=106$, T-180's 10.403976649170..., Hosono's closed form (jlevy/squares#600), by 0.0990, and above the verified one, 10.3107084351... (Karakuş's strip bound, T-083 and T-084), by 0.1923; the bundle's own README compares it with the latter.

Here the maintained readers accepted its exact mass, declared-net and check2 premises from the retained release asset. No complete replay has run here, so the value stands as reported.

wand125 after Tokoharu and Levy, square-packing. The source says parts of the work were produced with AI assistance under human direction.