\(s (211)\)$s(211)$ — open

Open. The best known published packing, Joost de Winter’s at Evan Daniel’s exact optimum (T-098), gives \(s(211) \le 14.9979607049673615652\ldots\)$s(211)\le 14.9979607049673615652\dots$, and the strongest lower bound independently verified here is \(14.544571\)$14.544571$ from Karakuş’s general theorem, leaving a gap of \(0.4534\)$0.4534$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The exact optimum

Evan Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing at its exact optimum (T-098): the same 211 squares in a square of side \(14.9979607049673615652\ldots\)$14.9979607049673615652\dots$, \(3.5 \times 10^{-13}\)$3.5\times {10}^{-13}$ below the side Joost de Winter prints. Square for square, its pose lies within \(5.2 \times 10^{-10}\)$5.2\times {10}^{-10}$ of the binary64 pose the atlas pictures for this count. The known-best witness this record lists is that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Evan Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over \(\mathbb{Q}\)$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves \(s(211) \le 14.9979607049673615652\ldots\)$s(211)\le 14.9979607049673615652\dots$, the verified upper bound; it says nothing about optimality. The source reports the exact point as a certified bound only: its numerical checks do not show it to be a local minimum of the side. On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The packing

Joost de Winter’s square-packing-211 reports 211 unit squares in a square of side \(14.99796070496771500150\)$14.99796070496771500150$, dated 16 September 2026 (T-057): the first packing on record that beats the \(15 \times 15\)$15\times 15$ grid. Its record names a previous side of \(14.99879247655475100150\)$14.99879247655475100150$ and describes its method as “Full-packing adaptive search followed by grouped-angle local refinement and interval-verified decimal export.” It reports an outward interval check at 80 digits but publishes no boxes or checker, and it says nothing about AI assistance. With the catalogue’s packings at \(n = 241, 273, 307\)$n=241,273,307$, found by Arslanov, Mustafin and Shangitbayev in 2019, it shows \(s(k^2 - k + 1) < k\)$s(k^2-k+1)<k$ for \(k = 15\)$k=15$ as well as \(16\)$16$, \(17\)$17$ and \(18\)$18$. The records at \(n = 31, 43, \ldots, 183\)$n=31,43,\dots ,183$ (\(k = 6\ldots14\)$k=6\dots 14$) still hold the grid, so on the catalogue and this record the smallest \(k\)$k$ shown to satisfy it moves from 16 to 15.

This repository certifies it exactly. The retained decimal pose rounds to an exact rational packing at centre dilation 1, of side \(14.9979607049676940014\)$14.9979607049676940014$, no larger than 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 \(s(211) \le 14.99796070496771500150\)$s(211)\le 14.99796070496771500150$, which was the verified upper bound until Evan Daniel’s exact optimum above replaced it; 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 bound (interval route).

The previous best known packing

Before this intake the best known packing was the trivial \(15 \times 15\)$15\times 15$ grid: the Kingbird catalogue as retained here does not picture \(n = 211\)$n=211$, and its live page of 29 September 2026 still does not.

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.