\(s (67)\)$s(67)$ — open

External intake, 2026-10-05 (certificate of 5 October). wand125’s mixed-certificate source reports \(s(67) \ge 212/25 = 8.48\)$s(67)\ge 212/25=8.48$ (T-094), from a density of 536 rectangles of total mass \(6699999/100000 < 67\)$6699999/100000<67$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the source’s \(339/40\)$339/40$ of the day before, by \(0.005\)$0.005$, and the verified \(1691/200\)$1691/200$ by \(0.025\)$0.025$. It supersedes the source’s mixed_n67_L8475 (T-090). sqverify-fast, this repository’s clean-room verifier, decided it here on 5 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, 2026-10-05 (certificate of 4 October). wand125’s mixed-certificate source reports the mixed certificate mixed_n67_L8475 at side \(339/40 = 8.475\)$339/40=8.475$ (T-090), since superseded (T-094), from a density of 485 rectangles of total mass \(6699999/100000 < 67\)$6699999/100000<67$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(1691/200\)$1691/200$ 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: confirmed, independently re-implemented. The verified lower bound is unchanged, the certificate above being higher. 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 wand125’s rectangle-density source reports \(s (67) \ge 1691 / 200 = 8.455\)$s(67)\ge 1691/200=8.455$, since superseded (T-094), with total mass \(6699/100 = 66.99 < 67\)$6699/100=66.99<67$, 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 wand125’s rectangle-density source reports \(s (67) \ge 1691 / 200 = 8.455\)$s(67)\ge 1691/200=8.455$, since superseded (T-094), with total mass \(6699/100 = 66.99 < 67\)$6699/100=66.99<67$, 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 \(211/25 = 8.44\)$211/25=8.44$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(6699/100 = 66.99 < 67\)$6699/100=66.99<67$, 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 \(s(67) \le 8.70710679\)$s(67)\le 8.70710679$, and the verified lower bound is \(s(67) \ge 212/25 = 8.48\)$s(67)\ge 212/25=8.48$, from wand125’s mixed rectangle-density certificate, leaving a gap of \(0.2271\)$0.2271$.

The packing

Found by Frits Göbel in 1979, via a diagonal-strip construction — the strip family at \(a = 7\)$a=7$, whose side is \(8 + \sqrt{2} / 2\)$8+\sqrt{2}/2$ 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 earlier external report, [Friedman DS7], gives the lower-bound expression \(2 \sqrt{2} + 71 / 13\)$2\sqrt{2}+71/13$ for \(s(67)\)$s(67)$ (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=67 by monotonicity. wand125’s rectangle-density certificate above has since replaced it in the reported field, and its replay here on 2026-10-02 made it the verified lower bound until 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-n067-wand125-mixed-848-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 5 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 2 October 2026 until this decision the verified lower bound was \(1691/200\)$1691/200$, wand125’s rectangle-density certificate (E-wand125-rectangle-2026-09-28-source-replay).

Nagamochi’s general closed form, the verified lower bound before the rectangle 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.