\(s (85)\)$s(85)$ — open

External intake, 2026-10-02 (afternoon). wand125’s mixed-certificate source reports \(s(85) \ge 473/50 = 9.46\)$s(85)\ge 473/50=9.46$ (T-075), from a density of 525 rectangles of total mass \(8499999/100000 < 85\)$8499999/100000<85$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It supersedes the source’s \(471/50\)$471/50$ certificate of the 2026-10-02 intake below, and its mass, below 86, carries it to \(n = 86\)$n=86$ as well. 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.

External intake, 2026-10-02. wand125’s mixed-certificate source reported a direct \(471/50 = 9.42\)$471/50=9.42$ certificate for this case (T-071), from a density of 587 rectangles of total mass \(8499999/100000 < 85\)$8499999/100000<85$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above Green’s reported \(9.2667335\ldots\)$9.2667335\dots$, which this case held by monotonicity, by more than \(0.1532\)$0.1532$. By monotonicity the value is also above the rectangle certificates reported at \(n = 86\)$n=86$ and 87, \(1873/200\)$1873/200$ and \(941/100\)$941/100$ (T-068). Its complete replay here on 2 October 2026 returned the certificate’s own record at all 201 directions, so it was the verified lower bound until the afternoon intake above, and its mass, below 86 and 87, carried the same bound to both. wand125’s README says parts of the work were produced with AI assistance under human direction.

Open. The best known packing gives \(s(85) \le 9.74264069\)$s(85)\le 9.74264069$, and the verified lower bound is \(s(85) \ge 473/50 = 9.46\)$s(85)\ge 473/50=9.46$, from wand125’s mixed rectangle-density certificate, leaving a gap of \(0.2826\)$0.2826$.

The packing

Found by Erich Friedman in 1997, via the off-centre variant of Göbel’s family — the rule of [Friedman DS7] section 3 at \((a, b) = (4, 6)\)$(a,b)=(4,6)$, a rectangle packing with the tilted block centred in the rectangle plus a column of \(2a + 1\)$2a+1$ unit squares against its tall side. cases/gobel_offcentre 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 up at its own digits, so the construction fits inside the witness’s declared value (D-398’s comfortable direction), but its layout is the witness’s own; the certificate carries the construction’s coordinates.

The lower bound

Before 2 October the selected external report was [Friedman DS7], which gives the lower-bound expression \(94 \sqrt{2} / 41 + 247 / 41\)$94\sqrt{2}/41+247/41$ for \(s(85)\)$s(85)$ (approximately \(9.266733533246\)$9.266733533246$). Friedman’s DS7 survey, Theorem 9, k=9, reports this bound at n=82; 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 \(k = 9\)$k=9$ that point pattern leaves a unit square empty (review). Inherited at n=85 by monotonicity. That changed 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-n085-wand125-mixed-946-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. sqverify-fast, this repository’s clean-room measure verifier, decided the same certificate at all 201 net directions in its census of 3 October 2026, and two mutants scaled below coverage one were refused on 6 October (E-n085-wand125-mixed-946-sqverify-fast-replay): independently re-implemented, the same method decided a second way beside the producer’s code.

From the morning of 2 October 2026 until this replay the verified lower bound was \(471/50\)$471/50$, by the replay of the source’s earlier certificate (E-n085-wand125-mixed-942-source-replay).

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.