\(s (66)\)$s(66)$ — open

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(66) \ge 843/100 = 8.43\)$s(66)\ge 843/100=8.43$ (T-097), from a density of 713 rectangles of total mass \(6599999/100000 < 66\)$6599999/100000<66$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the source’s \(421/50\)$421/50$ of 1 October, which was also the verified lower bound, by \(0.01\)$0.01$, and supersedes its mixed_n66_L842 (T-069). 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 wand125’s mixed-certificate source reports \(s(66) \ge 421/50 = 8.42\)$s(66)\ge 421/50=8.42$ (T-069), since superseded (T-097), from a density of 631 rectangles of total mass \(6599999/100000 < 66\)$6599999/100000<66$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. Its complete replay here on 2 October 2026 returned the certificate’s own record at all 201 directions, which made it the verified lower bound until 6 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 a direct \(67/8 = 8.375\)$67/8=8.375$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(6599/100 = 65.99 < 66\)$6599/100=65.99<66$, 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 \(1669/200 = 8.345\)$1669/200=8.345$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(6599/100 = 65.99 < 66\)$6599/100=65.99<66$, 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.

Open. The best known packing gives \(s(66) \le 8.65685425\)$s(66)\le 8.65685425$, and the verified lower bound is \(s(66) \ge 843/100 = 8.43\)$s(66)\ge 843/100=8.43$, from wand125’s mixed rectangle-density certificate of 5 October, leaving a gap of \(0.2269\)$0.2269$.

The packing

Stenlund’s diagonal strip of width three (1980), for which the survey states no generating rule — only the figure. cases/lifted_q2 lifts the retained witness’s every coordinate into Q(sqrt 2) at small height and verifies the lifted pose exactly, which is what moved verified_upper_bound from the grid ceiling onto the published exact side. The lift is a candidate generator and the exact verifier is the proof; the certificate is the witness’s own geometry, made exact (D-398, and the operation D-402 does not foreclose).

The lower bound

The earlier external report, [Friedman DS7], gives the lower-bound expression \(2 \sqrt{2} + 71 / 13\)$2\sqrt{2}+71/13$ for \(s(66)\)$s(66)$ (approximately \(8.289965586285\)$8.289965586285$). Friedman’s DS7 survey, Theorem 9, k=8, reports this bound at n=65; 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 \(k = 8\)$k=8$ that point pattern leaves a unit square empty (review). Inherited at n=66 by monotonicity. wand125’s rectangle-density certificate above has since replaced it in the reported field; the verified field now holds wand125’s mixed certificate of 5 October. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified field rests on a complete decision here (E-n066-wand125-mixed-843-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 5 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 run here at 12 of the 201 directions, each returning the certificate’s record, and not in full.

From 2 October 2026 until this decision the verified lower bound was \(421/50\)$421/50$, from a complete replay here (E-n066-wand125-mixed-842-source-replay) of wand125’s mixed certificate of 1 October: the source’s unchanged checker and per-angle functions on the pinned bundle, all 201 directions, each returning the certificate’s own record. Before that it was Nagamochi’s general closed form, 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.