\(s (93)\)$s(93)$ — open

Verified from \(n = 91\)$n=91$, 6 October 2026. wand125’s \(s(91) \ge 781/80 = 9.7625\)$s(91)\ge 781/80=9.7625$ (T-091), decided here by sqverify-fast at all 201 net directions, is above this count’s verified \(39/4\)$39/4$, and a packing of 93 squares contains one of 91, so it bounded this case as well: the verified lower bound until this count’s own certificate below was decided here later that day.

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(93) \ge 247/25 = 9.88\)$s(93)\ge 247/25=9.88$ (T-090), from a density of 501 rectangles of total mass \(9299999/100000 < 93\)$9299999/100000<93$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(493/50\)$493/50$ this case reported, by \(0.02\)$0.02$. It supersedes the source’s mixed_n93_L986. 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, 2026-10-03. wand125’s mixed-certificate source reports the mixed certificate mixed_n93_L986 at side \(493/50 = 9.86\)$493/50=9.86$ (T-082), since superseded (T-090), from a density of 443 rectangles of total mass \(9299999/100000 < 93\)$9299999/100000<93$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(39/4\)$39/4$ this case reported, by \(0.11\)$0.11$. 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.

Reported from \(n = 92\)$n=92$, 2026-10-02 (afternoon). wand125’s mixed-certificate source reports \(s(92) \ge 39/4 = 9.75\)$s(92)\ge 39/4=9.75$ (T-075), from a density of total mass \(9199999/100000 < 93\)$9199999/100000<93$. It is above this count’s own certificates, and a packing of 93 squares contains one of 92, so it gave \(s(93) \ge 39/4 = 9.75\)$s(93)\ge 39/4=9.75$ by monotonicity, the reported lower bound until the intakes above. 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 here until 6 October. 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 a direct \(3889/400 = 9.7225\)$3889/400=9.7225$ certificate for this case, whose bound the 2026-10-02 afternoon intake above raises, with total mass \(9299/100 = 92.99 < 93\)$9299/100=92.99<93$, 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 was also the verified lower bound until the 2026-10-02 afternoon intake above. wand125’s README says parts of the work were produced with AI assistance under human direction.

Open. The best known packing gives \(s(93) \le 10\)$s(93)\le 10$, and the verified lower bound is \(s(93) \ge 247/25 = 9.88\)$s(93)\ge 247/25=9.88$, from wand125’s mixed rectangle-density certificate of 4 October for this count (T-090), leaving a gap of \(0.12\)$0.12$.

The packing

The record catalogue does not picture \(n = 93\)$n=93$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{93}\rceil = 10\)$\lceil \sqrt{93}\rceil =10$ 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-n093-wand125-mixed-988-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 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. 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-n091-wand125-mixed-97625-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October for 91 squares, whose mass, \(9099999/100000\)$9099999/100000$, is below 93, 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 the complete replay here of wand125’s mixed certificate for 92 squares (E-n092-wand125-mixed-975-source-replay, V3/C3): all 201 directions on 2 October 2026, each returning the certificate’s own record. Its mass, \(9199999/100000\)$9199999/100000$, is below 93, so a packing of 93 at a side below \(39/4\)$39/4$ would capture at least 93 from it.

From 2 October 2026 until that replay the verified lower bound was \(3889/400\)$3889/400$, by the replayed rectangle certificate of 1 October (E-wand125-rectangle-2026-10-01-source-replay).

Nagamochi’s general closed form, the verified lower bound before both certificates, is now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below both certificates 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.