\(s (59)\)$s(59)$ — solved

\(s(59) = 8\)$s(59)=8$. Established by wand125 (2026) with a mixed cover in Evan Daniel’s format, made by Daniel’s line-cover linear program from his \(s(60) = 8\)$s(60)=8$ cover and decided by his interval checker zmx2, whose check was replayed here in full (T-066); the method goes back to Sam Burns’s and Gustavo Massaccesi’s weighted exact-rational covers. wand125’s README says parts of the work were produced with AI assistance under human direction. Its total is below 60 and 61 too, so it gives \(s(60) = s(61) = 8\)$s(60)=s(61)=8$ a second route, beside Daniel’s own cover (T-062, T-063).

External intake, 2026-10-01. wand125’s rectangle-density source reports a direct \(3173/400 = 7.9325\)$3173/400=7.9325$ certificate for this case, with total mass \(5899/100 = 58.99 < 59\)$5899/100=58.99<59$, accepted by Tokoharu’s unchanged interval checker. Its complete coverage replay has not run here; the verified lower bound was \(s (59) \ge 198 / 25 = 7.92\)$s(59)\ge 198/25=7.92$, from the source’s certificate for this count in the 2026-09-28 intake, whose complete 201-direction replay passed here, 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. wand125’s exact-cover source reports \(s(59) = 8\)$s(59)=8$ (T-066), from a mixed cover of 26,308 weighted points and 5,240 grid-line segments of total mass \(58.99051596 < 59\)$58.99051596<59$, checked there by Evan Daniel’s zmx2 and zm_mixed.py. Its zmx2 sweeps were replayed here on 2 October 2026, which closed the case; the lower bound below says how. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(198/25 = 7.92\)$198/25=7.92$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(5899/100 = 58.99 < 59\)$5899/100=58.99<59$, 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 \(1581/200 = 7.905\)$1581/200=7.905$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(5899/100 = 58.99 < 59\)$5899/100=58.99<59$, 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 59 with five cells empty, and \(s(59) \le 8\)$s(59)\le 8$. The record catalogue does not picture \(n = 59\)$n=59$, 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 Daniel’s format: 26,308 rationally weighted points of \([0,8]^2\)$[0,8{]}^2$ plus mass spread uniformly along 5,240 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 \(1474762899/25000000 = 58.99051596 < 59\)$1474762899/25000000=58.99051596<59$, 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 \(1\)$1$. If 59 unit squares fit in a side \(s < 8\)$s<8$, scaling by \(8/s\)$8/s$ gives 59 pairwise disjoint closed unit squares in \([0,8]^2\)$[0,8{]}^2$, which would capture at least 59 from a total of \(58.99\)$58.99$. The source puts the margin at about 0.09% over the exact threshold.

The source reports the cover accepted by zmx2, with its binary64 enclosures widened outward, over all 6,400 roots of the D4 fundamental region and, assuming no symmetry, all 51,200 roots of the whole pose space. Its exact-rational evidence is two zm_mixed.py runs: the complete sweep leaves six boxes uncertified in one root, and a deeper run of that root certifies it. Whether the six boxes lie in that root is not decided by anything published, so that route is not recorded here. Nothing about this cover is in Lean.

The verified field rests on a complete replay here of zmx2 (E-n059-wand125-mixed-cover-zmx2-replay, V3/C3). Built from the retained source the source’s verify.sh requires, it certified all 6,400 D4 roots (9,844,124 boxes) and all 51,200 unreduced roots (79,108,328 boxes) on 2 October with no uncertified box, the box totals equal to the source’s, and refused two mutated copies of the cover; the receipts are in the packet. The source’s run records are not published, so no root-for-root comparison was possible. That replay is interval-certified, so it assumes correctly rounded binary64 arithmetic on the host, and it runs Daniel’s own checker. The review of 2 October found no mathematical defect in the claim.

Earlier lower bounds

wand125’s rectangle-density certificate of 28 September, replayed here, gave \(s (59) \ge 198 / 25 = 7.92\)$s(59)\ge 198/25=7.92$, 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.