\(s (89)\)$s(89)$ — open

Verified from \(n = 88\)$n=88$, 5 October 2026. wand125’s \(s(88) \ge 481/50 = 9.62\)$s(88)\ge 481/50=9.62$ (T-091), decided here by sqverify-fast at all 201 net directions, is above this count’s verified \(1913/200\)$1913/200$, and a packing of 89 squares contains one of 88, so it bounded this case as well: the verified lower bound until this count’s own certificate below was decided here on 6 October 2026.

External intake, 2026-10-03. wand125’s mixed-certificate source reports \(s(89) \ge 193/20 = 9.65\)$s(89)\ge 193/20=9.65$ (T-082), from a density of 416 rectangles of total mass \(8899999/100000 < 89\)$8899999/100000<89$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(1913/200\)$1913/200$ this case reported, by \(0.085\)$0.085$. 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 wand125’s rectangle-density source reports \(s (89) \ge 1913 / 200 = 9.565\)$s(89)\ge 1913/200=9.565$, since superseded (T-091), with total mass \(8899/100 = 88.99 < 89\)$8899/100=88.99<89$, 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 \(191/20 = 9.55\)$191/20=9.55$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(8899/100 = 88.99 < 89\)$8899/100=88.99<89$, 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.

Open. The best known packing gives \(s(89) \le 9.94974747\)$s(89)\le 9.94974747$, and the verified lower bound is \(s(89) \ge 193/20 = 9.65\)$s(89)\ge 193/20=9.65$, from wand125’s mixed rectangle-density certificate of 3 October (T-082), leaving a gap of \(0.2997\)$0.2997$.

The packing

Found by Evert Stenlund in 1980, via a hand construction.

The lower bound

The verified field rests on a complete decision here (E-n089-wand125-mixed-965-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.

From 5 October 2026 until this decision the verified field rested on a complete decision here (E-n088-wand125-mixed-962-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October for 88 squares, whose mass, \(8799999/100000\)$8799999/100000$, is below 89, by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 5 October 2026, with two mutants scaled below coverage one refused. Before that the verified lower bound was \(1913/200\)$1913/200$ (E-wand125-rectangle-2026-10-01-source-replay).

Nagamochi’s general closed form, the verified lower bound before this certificate, is now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this \(n\)$n$, applies to every nonsquare \(N \ge 8\)$N\ge 8$:

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, \(s(N) \ge \min(\lceil\sqrt{N}\rceil, \sqrt{N - 2\lfloor\sqrt{N}\rfloor + 1} + 1)\)$s(N)\ge min(\lceil \sqrt{N}\rceil ,\sqrt{N-2\lfloor \sqrt{N}\rfloor +1}+1)$, which is stronger at this \(n\)$n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

See the Frontier corpus summary for the current aggregate count; source-reported bounds are recorded separately.

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.