\(s (65)\)$s(65)$ — open

External intake, 2026-10-01. wand125’s mixed-certificate source reports \(s(65) \ge 167/20 = 8.35\)$s(65)\ge 167/20=8.35$ (T-069), from a density of 787 rectangles of total mass \(6499999/100000 < 65\)$6499999/100000<65$, 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, so it is also the verified lower bound. wand125’s README says parts of the work were produced with AI assistance under human direction.

Open. The best known packing gives \(s(65) \le 8.53553391\)$s(65)\le 8.53553391$, and the verified lower bound is \(s(65) \ge 167/20 = 8.35\)$s(65)\ge 167/20=8.35$, from wand125’s mixed rectangle-density certificate, leaving a gap of \(0.1855\)$0.1855$.

The packing

Found by Frits Göbel in 1979, via a hand construction.

The lower bound

The selected external report, [Friedman DS7], gives the lower-bound expression \(2 \sqrt{2} + 71 / 13\)$2\sqrt{2}+71/13$ for \(s(65)\)$s(65)$ (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). This changes the reported source field only; the verified field now holds wand125’s replayed mixed certificate. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified field rests on a complete replay here (E-n065-wand125-mixed-835-source-replay, V3/C3) of wand125’s mixed certificate: the source’s unchanged checker and per-angle functions on the pinned bundle, all 201 directions on 2 October 2026, each returning the certificate’s own record. It runs the source’s own algorithm, so it confirms the source’s run rather than deciding coverage a second way.

Until 2 October 2026 the verified lower bound 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.