\(s (52)\)$s(52)$ — open

External intake, 2026-10-03. wand125’s mixed-certificate source reports \(s(52) \ge 151/20 = 7.55\)$s(52)\ge 151/20=7.55$ (T-082), from a density of 455 rectangles of total mass \(5199999/100000 < 52\)$5199999/100000<52$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(1507/200\)$1507/200$ this case reported, by \(0.015\)$0.015$. 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 wand125’s rectangle-density source reports \(s (52) \ge 1507 / 200 = 7.535\)$s(52)\ge 1507/200=7.535$, since superseded (T-082), with total mass \(5199/100 = 51.99 < 52\)$5199/100=51.99<52$, 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 wand125’s rectangle-density source reports \(s (52) \ge 1507 / 200 = 7.535\)$s(52)\ge 1507/200=7.535$, since superseded (T-082), with total mass \(5199/100 = 51.99 < 52\)$5199/100=51.99<52$, 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. 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 \(1501/200 = 7.505\)$1501/200=7.505$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(5199/100 = 51.99 < 52\)$5199/100=51.99<52$, 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. The complete exact replay of wand125’s n53 file also proves \(s (52) \ge 369 / 50 = 7.38\)$s(52)\ge 369/50=7.38$, the verified lower bound until 2026-10-02: its mass is \(51.48321764 < 52\)$51.48321764<52$, and the count occurs nowhere else in the covering theorem. This is a local verified deduction from the retained certificate; wand125’s README states the n53 claim rather than an n52 claim.

Open. The verified interval has endpoints \(7.55\)$7.55$ and approximately \(7.707106781\)$7.707106781$, exactly \(151/20\)$151/20$ and \(7 + \sqrt{2} / 2\)$7+\sqrt{2}/2$, leaving a gap of about \(0.1571\)$0.1571$. Its lower end is wand125’s mixed rectangle-density certificate of 3 October (T-082, decided here on 6 October 2026), which superseded its replayed rectangle-density certificate at \(7.535\)$7.535$ (T-070, 2026-10-02).

The packing

Found by Frits Göbel in 1979, via a diagonal-strip construction — the strip family at \(a = 6\)$a=6$, whose side is \(7 + \sqrt{2} / 2\)$7+\sqrt{2}/2$ exactly. cases/gobel_strip builds it and verifies it exactly over Q(sqrt 2), which is what moved verified_upper_bound from the grid ceiling onto the exact side. The retained witness declares that side rounded down at its own digits — about \(4.85 \times 10^{-30}\)$4.85\times {10}^{-30}$ below the exact value — and a different slack choice for the strip, so the certificate carries the construction’s coordinates rather than the witness’s (D-398).

The lower bound

The verified field rests on a complete decision here (E-n052-wand125-mixed-755-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 3 October 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. Until this decision the verified lower bound was \(1507/200\)$1507/200$ (E-wand125-rectangle-2026-09-28-source-replay).

The literal external report, [Friedman DS7], gives the lower-bound expression \(3 \sqrt{14} / 25 + 2 \sqrt{2} + 101 / 25\)$3\sqrt{14}/25+2\sqrt{2}+101/25$ for \(s(52)\)$s(52)$ (approximately \(7.317426011159\)$7.317426011159$). Friedman’s DS7 survey, Theorem 9, k=7, reports this bound at n=50; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at \(k = 7\)$k=7$ that point pattern leaves a unit square empty (review). Inherited at n=52 by monotonicity. It remains the reported_lower_bound because the stronger n52 number is this repository’s deduction, not a claim made by wand125 for n52. The source audit compares Green’s exact expression separately from opaque table decimals.

Until 2026-10-02 the verified lower bound was \(369/50\)$369/50$, by this deduction: the replay verifies all five point-certificate conditions, and the exact mass budget proves that the same geometry excludes 52 squares. wand125’s replayed rectangle-density certificate, \(1507/200\)$1507/200$ (T-070), superseded it. 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.