\(s (321)\)$s(321)$ — solved

\(s(321) = 18\)$s(321)=18$, as the case \(k = 18\)$k=18$ of Evan Daniel’s \(s(k^2 - 3) = k\)$s(k^2-3)=k$ for every integer \(k \ge 6\)$k\ge 6$ (T-064). The lower half rests on one finite statement, Valid7, and on a Lean reduction from it to every \(k \ge 6\)$k\ge 6$. Daniel’s exact checker of Valid7 was replayed here in full on the retained cover, every root’s leaves equal to the published run’s, and the reduction was built here with the pinned toolchain and depends only on the standard axioms. The \(18 \times 18\)$18\times 18$ grid holds 321 squares. The source’s CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.

External intake, 2026-10-01. Evan Daniel’s source reports \(s(321) = 18\)$s(321)=18$ as the case \(k = 18\)$k=18$ of \(s(k^2 - 3) = k\)$s(k^2-3)=k$ for every integer \(k \ge 6\)$k\ge 6$ (T-064). By the source’s report, its Lean development derives the family from one finite covering statement, Valid7, which a single exact Python checker at the source decides. Neither had been run here when this was written, so the verified lower bound was unchanged and the case stayed open; both were rebuilt on 2 and 3 October. The source’s CREDITS.md says the work was produced by Claude (Anthropic) in a single session under human direction.

The packing

The record catalogue does not picture \(n = 321\)$n=321$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{321}\rceil = 18\)$\lceil \sqrt{321}\rceil =18$ grid, so the grid is still the best known packing. That was a statement about what had been searched, not a proof; T-064 now proves that none exists.

The lower bound

Nagamochi’s general closed form was the verified lower bound until 2 October 2026 and is now a reported bound (see the correction below); from then until T-064’s replay it was Karakuş’s general bound, which 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.

Source-reported bounds are recorded separately from this independently verified theorem.

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.