\(s (77)\)$s(77)$ — solved

\(s(77) = 9\)$s(77)=9$. Established by wand125 (2026) with a mixed cover in Evan Daniel’s format, built from Daniel’s \(s(60) = 8\)$s(60)=8$ cover, and decided by his interval checker zmx2, whose check was replayed here in full (T-067); 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 78 too, so it gives \(s(78) = 9\)$s(78)=9$ as well.

External intake, 2026-10-01. wand125’s rectangle-density source reports a direct \(3573/400 = 8.9325\)$3573/400=8.9325$ certificate for this case, with total mass \(7699/100 = 76.99 < 77\)$7699/100=76.99<77$, accepted by Tokoharu’s unchanged interval checker. The replayed n75 certificate carried \(889/100 = 8.89\)$889/100=8.89$ here by monotonicity, 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. wand125’s exact-cover source reports \(s(77) = 9\)$s(77)=9$ (T-067), from a mixed cover of 28,273 weighted points and 6,420 segments of total mass \(76.99998460 < 77\)$76.99998460<77$, built from Evan Daniel’s \(s(60)\)$s(60)$ cover and checked there by his 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 \(891/100 = 8.91\)$891/100=8.91$ certificate for this case, with total mass \(7699/100 = 76.99 < 77\)$7699/100=76.99<77$, accepted by Tokoharu’s unchanged interval checker. Its n76 certificate, of mass \(7599/100 < 77\)$7599/100<77$, gives the stronger \(223/25 = 8.92\)$223/25=8.92$ by monotonicity, which the 2026-10-01 intake above raises. 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 \(222/25 = 8.88\)$222/25=8.88$ certificate for this case, with total mass \(7699/100 = 76.99 < 77\)$7699/100=76.99<77$, accepted by Tokoharu’s unchanged interval checker. Its n76 certificate, of mass \(7599/100 < 77\)$7599/100<77$, gave \(89/10 = 8.9\)$89/10=8.9$ by monotonicity, the reported lower bound until the 2026-09-28 intake above. 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 \(9 \times 9\)$9\times 9$ grid holds 81 unit squares, so it holds 77 with four cells empty, and \(s(77) \le 9\)$s(77)\le 9$. The record catalogue does not picture \(n = 77\)$n=77$, and no arrangement below side 9 has ever been found. What is new is that none exists.

The lower bound

The certificate is a mixed cover in Daniel’s format: 28,273 rationally weighted points of \([0,9]^2\)$[0,9{]}^2$ plus mass spread uniformly along 6,420 segments on the interior grid lines \(x, y \in \lbrace1, \ldots, 8\rbrace\)$x,y\in \{1,\dots ,8\}$, total \(43347137744028965/2^{49} = 76.99998460 < 77\)$43347137744028965/2^{49}=76.99998460<77$, exactly invariant under the square’s eight symmetries, such that every closed unit square inside \([0,9]^2\)$[0,9{]}^2$, at every centre and every angle, captures mass at least \(1\)$1$. If 77 unit squares fit in a side \(s < 9\)$s<9$, scaling by \(9/s\)$9/s$ gives 77 pairwise disjoint closed unit squares in \([0,9]^2\)$[0,9{]}^2$, which would capture at least 77 from a total of \(76.99998\)$76.99998$. It is Daniel’s \(s(60) = 8\)$s(60)=8$ cover cut at 4 in both coordinates with a central band inserted, plus a symmetric band measure of mass about \(16.43\)$16.43$; 84 points that lay on loaded grid lines were replaced by segments of length \(2/1000\)$2/1000$ on the same lines.

The source reports the cover accepted by zmx2 with --pair-points, its binary64 enclosures widened outward, over all 8,100 roots of the D4 fundamental region and, assuming no symmetry, all 64,800 roots of the whole pose space, and by Daniel’s exact-rational zm_mixed.py over 129,600 roots. Nothing about this cover is in Lean.

The verified field rests on a complete replay here of zmx2 (E-n077-wand125-mixed-cover-zmx2-replay, V3/C3). Built from the retained source the source’s verify.sh requires, it certified on 2 October all 8,100 D4 roots (5,810,824 boxes) in one run and all 64,800 unreduced roots (46,583,600 boxes) in two runs over halves of the root columns, which this repository’s audit read as one record, with no uncertified box and the box totals equal to the source’s; it 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, and the source’s own verify.sh cannot pass as published, because its checksum file lists run logs the source does not publish. 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 n75 rectangle-density certificate, replayed here, gave \(s (77) \ge 889 / 100 = 8.89\)$s(77)\ge 889/100=8.89$ by monotonicity, 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.