T-145: $s(40)>1340000/199529=6.7158157\dots$$s(40)>1340000/199529=6.7158157\dots$, by a continuous-pose interval kernel

S3 V3 C3 lower bound confirmed

2026-10-10 published · Guzhou0806 after wand125, Tokoharu, Levy, Stromquist, Burns, Massaccesi · $n=40$$n=40$

$s(40)>1340000/199529$$s(40)>1340000/199529$ = 6.715815746082023…, by Guzhou0806, published on 10 October 2026 as release n40-continuous-1340000-199529-20261010 of Guzhou0806/n40-square-packing and reported on jlevy/squares#485. It raises T-133's 335427/50000 by 72586117/9976450000 = 0.0072757…, from the same certificate.

The certificate is wand125's rect_n40_L67 rectangle density of T-068, unchanged, of mass 3999/100. Every closed square of side 199529/200000 inside [0, 67/10]^2, at every angle and every legal centre, captures mass at least 4999/5000, and 40 × 4999/5000 − 3999/100 = 1/500, so no packing fits at side 1340000/199529; compactness makes the bound strict.

A new interval kernel covers the continuous pose domain, centre and half-angle tangent together, in 43 closed slabs of t = tan(θ/2) over [10^-6, 83/200], by outward-rounded binary64 branch and bound with a concave-section integral bound and translation-derivative bounds; an exact integer certificate for horizontal squares of side 997643/1000000, which lie inside every side-B square at t ≤ 10^-6, covers the rest. No net, no transfer and no density peak are needed.

The source decided both parts with its two C++ kernels, in a local run and a GitHub Actions run.

Both were decided here on 10 October 2026: the two kernels, built from the retained source into the CI run's binaries, ran the axis certificate and all 43 slabs, every output equal to both release receipts apart from timing, and refused every synthetic density whose least capture is below the threshold: confirmed, reproduced with the producer's code.

This repository's exact audit, which shares no code with the release's, regenerated both kernel inputs and decided every finite step, and sqverify-fast decided the near-axis statement a second way on the original density. The 10 October review re-derived the kernel's lemmas and found no blocking defect. No second method decided the continuous cover.

Guzhou0806 after wand125, Tokoharu and Levy, n40-square-packing. The source credits the density to wand125, the rectangle-density method and the translation-derivative structure to Tokoharu, says the concave-section bound is informed by this repository's sqverify_fast, and gives Guzhou0806's contribution, the continuous-pose extension, the smaller-square coverage, the near-axis patch and the composition, as AI-assisted research; the comment says AI assisted the exploration, proof drafting, implementation, tests, packaging and the update under the author's direction.