\(s (31)\)$s(31)$ — open

External intake, 2026-10-01. wand125’s rectangle-density source reports \(s (31) \ge 2381 / 400 = 5.9525\)$s(31)\ge 2381/400=5.9525$, with total mass \(3099/100 = 30.99 < 31\)$3099/100=30.99<31$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it is also verified. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(1187/200 = 5.935\)$1187/200=5.935$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(3099/100 = 30.99 < 31\)$3099/100=30.99<31$, accepted by Tokoharu’s unchanged interval checker. Its complete coverage replay has not yet run here; the verified lower bound was \(148/25 = 5.92\)$148/25=5.92$ until the 2026-10-01 intake above, from the source’s certificate for this count in the 2026-09-27 intake, whose complete 201-direction replay passed here. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(148/25 = 5.92\)$148/25=5.92$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(3099/100 = 30.99 < 31\)$3099/100=30.99<31$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it was also the verified lower bound until the 2026-10-01 intake above.

External intake, 2026-09-22. Tokoharu’s retained rectangle-density source reports \(s (29) \ge 571 / 100 = 5.71\)$s(29)\ge 571/100=5.71$; monotonicity carries the reported bound to \(n = 31\)$n=31$. The complete source replay verifies the source premise, so the same monotonicity argument verified that bound at \(n = 31\)$n=31$ until the direct certificate above superseded it. Tokoharu’s README says parts of the work were produced with AI assistance under human direction.

Open. The verified interval is \(5.92 \le s(31) \le 6\)$5.92\le s(31)\le 6$, with width \(0.08\)$0.08$.

The packing

The record catalogue does not picture \(n = 31\)$n=31$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{31}\rceil = 6\)$\lceil \sqrt{31}\rceil =6$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

wand125’s n31 rectangle-density certificate, replayed here over all 201 directions, verifies \(148/25\)$148/25$ directly: 155 rectangle orbits, total mass \(3099/100\)$3099/100$, below 31 by \(1/100\)$1/100$. Before it, the complete n29 interval replay verified \(571/100\)$571/100$, which deleting two squares carried to n31. A method-distinct rectangle coverage implementation would add confidence and improve generalization, but it is not a missing premise of either bound.

Corrected 2 October 2026: Nagamochi’s Lemma 1 is false (Karakuş 2026), so his closed form below is now a reported bound whose published proof is incomplete (review). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s earlier general closed form remains part of the evidence history and applies to every \(N \ge 4\)$N\ge 4$:

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.