\(s (60)\)$s(60)$ — solved

\(s(60) = 8\)$s(60)=8$. Established by a mixed cover with two exhaustive checks of its covering condition, Evan Daniel (2026), building on Sam Burns’s and Gustavo Massaccesi’s weighted exact-rational covering method; its interval check was replayed here in full (T-062). The source’s CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction. wand125’s \(s(59) = 8\)$s(59)=8$ (T-066), from a cover built on this one and replayed here with the same checker, gives the value a second route by monotonicity.

External intake, 2026-10-01. wand125’s rectangle-density source reports a direct \(397/50 = 7.94\)$397/50=7.94$ certificate for this case, with total mass \(5999/100 = 59.99 < 60\)$5999/100=59.99<60$, accepted by Tokoharu’s unchanged interval checker. The replayed n59 certificate carried \(s (60) \ge 198 / 25 = 7.92\)$s(60)\ge 198/25=7.92$ here by monotonicity, which 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 mixed-cover source reports \(s(60) = 8\)$s(60)=8$ (T-062), from 23,744 weighted points and 5,216 grid-line segments of total mass \(59.85867528 < 60\)$59.85867528<60$, checked there by zm_mixed.py and zmx2. Its zmx2 sweeps were replayed here on 2 October 2026, which closed the case; the lower bound below says how. The source’s CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(397/50 = 7.94\)$397/50=7.94$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(5999/100 = 59.99 < 60\)$5999/100=59.99<60$, 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.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(198/25 = 7.92\)$198/25=7.92$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(5999/100 = 59.99 < 60\)$5999/100=59.99<60$, 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.

The packing

The upper bound is trivial: the \(8 \times 8\)$8\times 8$ grid holds 64 unit squares, so it holds 60 with four cells empty, and \(s(60) \le 8\)$s(60)\le 8$. The record catalogue does not picture \(n = 60\)$n=60$, and no arrangement below side 8 has ever been found. What is new is that none exists.

The lower bound

The certificate is a mixed cover in the format of \(s(21)\)$s(21)$’s: 23,744 rationally weighted points of \([0,8]^2\)$[0,8{]}^2$ plus mass spread uniformly along 5,216 segments of length \(1/50\)$1/50$ on the interior grid lines \(x, y \in \lbrace1, \ldots, 7\rbrace\)$x,y\in \{1,\dots ,7\}$, total \(748233441/12500000 = 59.85867528 < 60\)$748233441/12500000=59.85867528<60$, exactly invariant under the square’s eight symmetries, such that every closed unit square inside \([0,8]^2\)$[0,8{]}^2$, at every centre and every angle, captures mass at least \(1\)$1$, a point on its boundary counting and a segment along one of its edges counting in full. If 60 unit squares fit in a side \(s < 8\)$s<8$, scaling by \(8/s\)$8/s$ gives 60 pairwise disjoint closed unit squares in \([0,8]^2\)$[0,8{]}^2$, which would capture at least 60 from a total of \(59.86\)$59.86$. The total is below 61 too, so the same cover decides \(s(61) = 8\)$s(61)=8$.

The source certifies it with two checkers. zm_mixed.py, in exact rational arithmetic, certifies all 102,400 root boxes of the cover’s D4 fundamental region, and the Rust zmx2, with its binary64 enclosures widened outward, all 6,400 roots of the same region and, assuming no symmetry, all 51,200 roots of the whole pose space. Nothing about this cover is in Lean: the reduction is the \(m = 8\)$m=8$ case of lemmas the source proves in Lean for every side, but there is no data file and no top theorem.

The verified field rests on a complete replay here of zmx2 (E-n060-evand-mixed-cover-zmx2-replay, V3/C3). Built from the retained source that made the source’s records, it certified all 6,400 D4 roots (2,617,534 boxes) and all 51,200 unreduced roots (21,036,120 boxes) on 2 October with no uncertified box, every root carrying the census of the source’s record of the same root, and refused two mutated copies of the cover; the receipts are in the packet. That replay is interval-certified, so it assumes correctly rounded binary64 arithmetic on the host, and it runs the source author’s own checker. The exact checker was not run here; a complete re-sweep, recorded as a second, exact-algebraic replay, would give this value a second machine method beside its C3. The review of the geometric premises of 2 October found every premise holding and no blocking defect.

Earlier lower bounds

wand125’s rectangle-density certificates, replayed here, gave \(s (60) \ge 198 / 25 = 7.92\)$s(60)\ge 198/25=7.92$ by monotonicity from its n59 certificate, which was the verified lower bound until 2 October 2026. Before 2026 it was Nagamochi’s general closed form, now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below the exact value proved here, 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.