\(s (72)\)$s(72)$ — open

External intake, 2026-10-02. wand125’s rectangle-density source reports the rectangle certificate rect_n72_L874 at side \(437/50 = 8.74\)$437/50=8.74$, since superseded (T-090), with total mass \(7199/100 = 71.99 < 72\)$7199/100=71.99<72$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again on 3 October, 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 the verified lower bound from its entry on 6 October (T-070) until T-090’s certificate was decided here later that day. wand125’s README says parts of the work were produced with AI assistance under human direction.

Verified from \(n = 71\)$n=71$, 5 October 2026. wand125’s \(s(71) \ge 8721/1000 = 8.721\)$s(71)\ge 8721/1000=8.721$ (T-091), decided here by sqverify-fast at all 201 net directions, is above this count’s verified \(1737/200\)$1737/200$, and a packing of 72 squares contains one of 71, so it bounded this case as well: the verified lower bound until this count’s own certificates were entered here on 6 October, the replayed rect_n72_L874 (T-070, above) and later that day mixed_n72_L876 (T-090, below).

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(72) \ge 219/25 = 8.76\)$s(72)\ge 219/25=8.76$ (T-090), from a density of 488 rectangles of total mass \(7199999/100000 < 72\)$7199999/100000<72$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(437/50\)$437/50$ this case reported, by \(0.02\)$0.02$. It is the source’s first mixed certificate at this count. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions, so it is also the verified lower bound: confirmed, independently re-implemented. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. wand125’s rectangle-density source reports the rectangle certificate rect_n72_L874 at side \(437/50 = 8.74\)$437/50=8.74$, since superseded (T-090), with total mass \(7199/100 = 71.99 < 72\)$7199/100=71.99<72$, accepted by Tokoharu’s unchanged interval checker. The replayed n71 certificate carries \(s (72) \ge 1737 / 200 = 8.685\)$s(72)\ge 1737/200=8.685$ here by monotonicity, which was the verified lower bound until 5 October. 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 the rectangle certificate rect_n72_L874 at side \(437/50 = 8.74\)$437/50=8.74$, since superseded (T-090), with total mass \(7199/100 = 71.99 < 72\)$7199/100=71.99<72$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay had not yet run here, so the verified lower bound was unchanged until that replay was entered on 6 October (T-070, above). 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 \(1741/200 = 8.705\)$1741/200=8.705$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(7199/100 = 71.99 < 72\)$7199/100=71.99<72$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged.

External intake, 2026-09-22. wand125’s retained point-certificate source reports \(s (72) \ge 861 / 100 = 8.61\)$s(72)\ge 861/100=8.61$, which was the verified lower bound until 2026-10-02. The complete exact replay verifies the retained certificate.

Open. The verified interval is \(8.76 \le s(72) \le 9\)$8.76\le s(72)\le 9$, with width \(0.24\)$0.24$. Its lower end is wand125’s mixed rectangle-density certificate of 4 October for this count (T-090, decided here on 2026-10-06). Earlier that day it was wand125’s rectangle-density certificate of 28 September for this count, rect_n72_L874, \(8.74\)$8.74$ (T-070, replayed here on 2026-10-03 and entered on 2026-10-06); before that the n71 mixed certificate’s \(8.721\)$8.721$ by monotonicity (T-091, 2026-10-05), and before that the n71 rectangle-density certificate’s \(8.685\)$8.685$ (T-070, 2026-10-02).

The packing

The record catalogue does not picture \(n = 72\)$n=72$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{72}\rceil = 9\)$\lceil \sqrt{72}\rceil =9$ 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

The verified field rests on a complete decision here (E-n072-wand125-mixed-876-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October for this count by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 6 October 2026, with two mutants scaled below coverage one refused. It shares no code with the source’s checker, so it decides coverage a second way with the same method; the source’s own checker was not run here.

Earlier on 6 October the verified field rested on the complete 201-direction replay here (E-wand125-rectangle-2026-09-28-source-replay, V3/C3) of wand125’s rect_n72_L874, of mass \(7199/100 < 72\)$7199/100<72$, by Tokoharu’s unchanged verify.cpp, which reproduced the upstream accepting run at every direction: confirmed, reproduced with the producer’s code. It ran in a cloud batch on 3 October 2026 and was entered on 6 October. From 5 October until then it rested on a complete decision here (E-n071-wand125-mixed-8721-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October for 71 squares, whose mass, \(7099999/100000\)$7099999/100000$, is below 72, by sqverify-fast: all 201 net directions of the retained candidate on 5 October 2026, with two mutants scaled below coverage one refused. Before 5 October the verified lower bound was \(1737/200\)$1737/200$, from the replayed n71 rectangle certificate (the same 28 September replay entry).

The exact replay verifies all five covering-certificate conditions over the complete rational direction net. A second interval implementation or generalized native importer would add confirmation and reuse, but it is not a missing premise of this fixed 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.