\(s (61)\)$s(61)$ — solved

\(s(61) = 8\)$s(61)=8$, by monotonicity from \(s(60) = 8\)$s(60)=8$ (T-063): removing a square from a packing leaves a packing, so \(s(61)\)$s(61)$ is at least \(s(60)\)$s(60)$, and the \(8 \times 8\)$8\times 8$ grid holds 61. The premise is Evan Daniel’s mixed cover (T-062), whose interval check was replayed here in full; its total is below 61 too, so the same certificate decides this case. wand125’s \(s(59) = 8\)$s(59)=8$ (T-066), replayed here with the same checker, gives the value a second route. The source’s CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.

External intake, 2026-10-02. wand125’s point-only cover, 15,193 \(D_4\)$D_4$-invariant points of total \(8584985072679551/2^{47} < 61\)$8584985072679551/2^{47}<61$, gives the value by points alone, on a cover built by someone else. Evan Daniel’s unmodified zmx2, the checker the source’s own verify.sh runs, was replayed on it here on 2 October: VERIFIED-D4 over all 6,400 roots, none uncertified, and two mutated covers refused (E-n061-wand125-point-cover-zmx2-replay). It is the same checker as T-062’s and the one the source pins, so it is a second route by the source’s pinned checker on T-063 and not a second method. 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 \(199/25 = 7.96\)$199/25=7.96$ certificate for this case, with total mass \(6099/100 = 60.99 < 61\)$6099/100=60.99<61$, 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 the verified lower bound until 2 October 2026. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. Evan Daniel’s source reports \(s(61) = 8\)$s(61)=8$ twice: as a corollary of its reported \(s(60) = 8\)$s(60)=8$ by monotonicity (T-063), and as the case \(k = 8\)$k=8$ of its family \(s(k^2 - 3) = k\)$s(k^2-3)=k$ (T-064). The \(s(60)\)$s(60)$ cover’s zmx2 sweeps were replayed here on 2 October 2026, which closed the case; the family’s certificate had not been replayed when this was written, and was replayed here on 2 and 3 October. wand125 reports the same value by a separate point-only route, taken in on 2 October below. The source’s CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(199/25 = 7.96\)$199/25=7.96$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(6099/100 = 60.99 < 61\)$6099/100=60.99<61$, 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.

The packing

The upper bound is trivial: the \(8 \times 8\)$8\times 8$ grid holds 64 unit squares, so it holds 61 with three cells empty, and \(s(61) \le 8\)$s(61)\le 8$. The checked record catalogues do not picture \(n = 61\)$n=61$, and no arrangement below side 8 has ever been found. What is new is that none exists.

The lower bound

The verified field rests on the complete replay here of zmx2 on Daniel’s \(s(60)\)$s(60)$ cover (E-n060-evand-mixed-cover-zmx2-replay, V3/C3): all 6,400 D4 roots and all 51,200 unreduced roots certified on 2 October with no uncertified box, every root carrying the census of the source’s record. The cover’s total, \(748233441/12500000 = 59.85867528\)$748233441/12500000=59.85867528$, is below 61, so a packing of 61 at a side below 8, scaled to side 8, would capture at least 61 from it. The case record for 60 squares describes the certificate and the review that found its premises holding, the corollary included.

Earlier lower bounds

Bašić and Slivková (2018, Theorem 10) applied their piercing-number framework specifically to this case and proved

That is a genuine case-specific result, but it is weaker than Nagamochi’s 2005 general closed form, which was the lower bound before 2026 and applies to every \(N \ge 4\)$N\ge 4$:

Correction, 2 October 2026. Nagamochi’s closed form no longer gives the standing verified lower bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false, so it is now a reported bound (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516. The verified lower bound here was then Karakuş’s general bound (T-083), about 7.8654 (superseded 3 October 2026, below), which is weaker than Bašić and Slivková’s 7.8906; their bound is not registered as evidence in this record, so it does not hold the verified field. The paragraph above is kept as written.

The Bašić–Slivková paper nevertheless matters methodologically: it explicitly connects the piercing number of the continuous family of unit-square poses to \(s(n)\)$s(n)$. It cannot be cited as an improvement over Nagamochi at \(n = 61\)$n=61$.

Update, 3 October 2026. Bašić and Slivková’s bound is now registered (T-087), \(7\sqrt3/2 + 2\sqrt2 - 1 \approx 7.890604\)$7\sqrt{3}/2+2\sqrt{2}-1\approx 7.890604$, above Karakuş’s \(7.865459\)$7.865459$; the paragraphs above are kept as written. Their Theorem 10 was read on the rendered pages and its proof re-derived, and check_piercing_lower_bounds replays its arithmetic exactly; the geometry is read, not machine-checked. Its proof uses nothing of Nagamochi 2005, which it cites only for values it reproduces. It was the verified lower bound here on its own line of the register until that line merged, the same day, with the replayed cover above, which proves \(s(61) = 8\)$s(61)=8$ (T-063) and superseded it.

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.