\(s (221)\)$s(221)$ — open

External intake, 2026-10-03. Evan Daniel reports \(s(221) = 15\)$s(221)=15$ as an instance of \(s(k^2 - 4) = k\)$s(k^2-4)=k$ at \(k = 15\)$k=15$ (jlevy/squares#316). The source’s Lean proves the general reduction conditional on ValidTilt9; the exact Python checker for that premise has not been replayed here, so the verified lower bound is unchanged. The source’s CREDITS.md says this work was produced by Claude (Anthropic) in a single session under human direction.

External intake, 2026-10-06. A second, separately written exact checker, wand125’s, reports ValidTilt9 on the same cover (6 October packet, E-k2m4-wand125-validtilt9-report). Its record check and a sample of its roots ran here; the full run has not been repeated, so the verified lower bound is unchanged.

Open. The best known packing gives \(s(221) \le 15\)$s(221)\le 15$, and the strongest lower bound independently verified here is \(14.896180\)$14.896180$ from Karakuş’s general theorem, leaving a gap of \(0.1038\)$0.1038$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The packing

The record catalogue does not picture \(n = 221\)$n=221$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{221}\rceil = 15\)$\lceil \sqrt{221}\rceil =15$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

The strongest lower bound independently verified in this record is 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.

Update, 3 October 2026. The reported field now holds Evan Daniel’s reported \(s(221) = 15\)$s(221)=15$ (T-081, above), which is stronger than Nagamochi’s value; that value stays on record under T-007, and the verified lower bound is still Karakuş’s.

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.