\(s (96)\)$s(96)$ — open

Verified from \(n = 95\)$n=95$, 6 October 2026. wand125’s \(s(95) \ge 1993/200 = 9.965\)$s(95)\ge 1993/200=9.965$ (T-090), decided here by sqverify-fast at all 201 net directions, is above this count’s verified \(249/25\)$249/25$, and a packing of 96 squares contains one of 95, so it bounded this case as well: the verified lower bound until this count’s own \(997/100\)$997/100$ (T-082) was decided here later that day. The reported field keeps the \(s(96) = 10\)$s(96)=10$ reported below (T-081), which has not been replayed here.

External intake, 2026-10-03. Evan Daniel reports \(s(96) = 10\)$s(96)=10$ as an instance of \(s(k^2 - 4) = k\)$s(k^2-4)=k$ at \(k = 10\)$k=10$ (jlevy/squares#316). The source’s Lean proves the general reduction conditional on ValidTilt9; the exact Python checker for that premise has not been replayed here, so the verified lower bound is unchanged. The source’s CREDITS.md says this work was produced by Claude (Anthropic) in a single session under human direction.

External intake, 2026-10-06. A second, separately written exact checker, wand125’s, reports ValidTilt9 on the same cover (6 October packet, E-k2m4-wand125-validtilt9-report). Its record check and a sample of its roots ran here; the full run has not been repeated, so the verified lower bound is unchanged.

External intake, 2026-10-03. wand125’s mixed-certificate source reports the certificate mixed_n96_L997 at side \(997/100 = 9.97\)$997/100=9.97$ (T-082), from a density of 376 rectangles of total mass \(9599999/100000 < 96\)$9599999/100000<96$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is below the \(s(96) = 10\)$s(96)=10$ reported above (T-081), and above the verified \(249/25\)$249/25$ of that day by \(0.01\)$0.01$, superseding the source’s mixed_n96_L996, and above the \(1993/200\)$1993/200$ carried from \(n = 95\)$n=95$ since 6 October by \(0.005\)$0.005$. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions. Its first control failed closed: the mutant scaled to \(99/100\)$99/100$ verified too, at the one direction that control runs it, which showed the certificate’s slack there and not a fault (think-0uia). The repaired control, which runs that mutant at every net direction, refused it at 187 of the 201, 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, 2026-10-02 (afternoon). wand125’s mixed-certificate source reports \(s(96) \ge 249/25 = 9.96\)$s(96)\ge 249/25=9.96$ (T-075), since superseded (T-082), from a density of 279 rectangles of total mass \(9599999/100000 < 96\)$9599999/100000<96$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above Nagamochi’s \(1 + \sqrt{79} = 9.8882\ldots\)$1+\sqrt{79}=9.8882\dots$, the verified lower bound here before it, by \(0.0718\)$0.0718$, the first certificate in this record above that value. The source calls Nagamochi’s value unproven, citing jlevy/squares#295; this certificate does not rest on it. That value has been a reported bound since Karakuş’s finding, and the verified floor it left, Karakuş’s general bound, is lower still (see the correction below). Its complete replay here on 2 October 2026 returned the certificate’s own record at all 201 directions, so it was also the verified lower bound until 6 October. wand125’s README says parts of the work were produced with AI assistance under human direction.

Open. The best known packing gives \(s(96) \le 10\)$s(96)\le 10$, and the verified lower bound is \(s(96) \ge 997/100 = 9.97\)$s(96)\ge 997/100=9.97$, from wand125’s mixed rectangle-density certificate of 3 October (T-082), leaving a gap of \(0.03\)$0.03$.

The packing

The record catalogue does not picture \(n = 96\)$n=96$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{96}\rceil = 10\)$\lceil \sqrt{96}\rceil =10$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

The verified field rests on a complete decision here (E-n096-wand125-mixed-997-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 3 October for this count 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, the one scaled to \(99/100\)$99/100$ at 187 of the 201 directions. 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.

Earlier on 6 October 2026 the verified field rested on a complete decision here (E-n095-wand125-mixed-9965-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October for 95 squares, whose mass, \(9499999/100000\)$9499999/100000$, is below 96, 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.

Before that the verified field rested on a complete replay here (E-n096-wand125-mixed-996-source-replay, V3/C3) of wand125’s mixed certificate for this count: 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; sqverify-fast decided the same certificate a second way on 3 October (E-n096-wand125-mixed-996-sqverify-fast-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.