\(s (130)\)$s(130)$ — open

Open. The best known published packing, Francisco Couzo’s (T-056), gives \(s(130) \le 11.911187706548755\)$s(130)\le 11.911187706548755$, and the strongest lower bound independently verified here is \(11.420164\)$11.420164$ from Karakuş’s general theorem, leaving a gap of \(0.491\)$0.491$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The packing

Francisco Couzo’s square-packing reports a packing of side \(11.911187706548755\)$11.911187706548755$ for this count, dated 26 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). The repository names no method and no tolerance, and itself states no AI assistance; its author said on issue #227 that he found the 102 and 103 packings “with the help of Claude”.

This repository certifies it exactly. The retained decimal pose rounds to an exact rational packing at centre dilation 1, of side \(11.9111877065487555592\ldots\)$11.9111877065487555592\dots$, \(5.592 \times 10^{-16}\)$5.592\times {10}^{-16}$ above the printed side, and every pair and every wall is decided over \(\mathbb{Q}\)$\mathbb{Q}$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That proves \(s(130) \le 11.911187706548756\)$s(130)\le 11.911187706548756$, the verified upper bound; it says nothing about optimality. Interval arithmetic on the printed pose itself, with each angle’s true cosine and sine and no rational rounding, decides every pair and wall again and gives the same verified upper bound (interval route).

The previous best known packing

Before this intake the best known packing was the Kingbird catalogue’s, of side \(11.91119052015898\)$11.91119052015898$.

Found by David Ellsworth in 2024. Improved by David W. Cantrell in November 2024. Improved by David Ellsworth in November 2024. The recorded construction method is an extension of a smaller record. Its side length is algebraic of degree 8 over \(\mathbb{Q}\)$\mathbb{Q}$.

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.

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.