T-133: $s(40)>335427/50000=6.70854$$s(40)>335427/50000=6.70854$, by a clipped-corner transfer on a 401-direction net

S3 V3 C3 lower bound confirmed superseded by T-145

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

$s(40)>335427/50000$$s(40)>335427/50000$ = 6.70854, by Guzhou0806, published on 10 October 2026 as release n40-670854-20261010 of Guzhou0806/n40-square-packing and reported on jlevy/squares#485. It raises the reported and verified lower bound at $n=40$$n=40$ by 427/50000 = 0.00854 over T-068's 67/10, from the same certificate. The release also gives the weaker $s(40)>67000$$s(40)>67000$√(6400006889)/798988091 = 6.7084890815…, which needs no density peak.

The certificate is wand125's rect_n40_L67 rectangle density of T-068, unchanged, of mass 3999/100 at core side 9977/10000. Every closed square of that side inside [0, 67/10]^2, at each of 401 half-angle tangents of step 83/80000, twice as fine as T-068's net of 201, captures mass at least 10001/10000.

A continuous-angle transfer carries this to every unit square scaled into that container: the net direction above a square's angle gives a reference square inside it when the angle is close enough, and otherwise the one below it does, less the density's essential supremum, at most 2818711359413/10^9, times a bound on the area of the reference's four clipped corners. Each square then captures at least 99979/100000, and forty of them exceed the mass by 1/625, so no packing fits at side 335427/50000; compactness makes the bound strict.

The source decided the nodal statement with this repository's sqverify-fast, vendored unchanged, at all 401 directions, and the finite steps with its own exact Python, in a local run and a GitHub Actions run.

Both parts were decided here on 10 October 2026. sqverify-fast, built from reviewed source, verified all 401 directions on an input regenerated from the retained density, every row equal to both release receipts apart from timing, and refused two mass mutants: confirmed, reproduced with the producer's code, since the release ran the same crate.

This repository's exact audit of the transfer, which shares no code with the release's finite checks, recomputed the density's mass, symmetry and essential supremum and every step of the chain, and refused four altered parameters: the finite steps were independently re-implemented. The 10 October review re-derived the transfer, the clipped-corner bound and the strict endpoint and found no blocking defect. No second method decided the nodal statement.

Guzhou0806 after wand125, Tokoharu and Levy, n40-square-packing. The source credits the density and the original verification method to wand125, Tokoharu and their cited predecessors, and says Guzhou0806's contribution, the finer-net verification, the strict core transfer and the clipped-corner composition, came from AI-assisted research; the issue says AI assisted the exploration, proof drafting, implementation, testing, packaging and submission under the author's direction.