\(s (264)\)$s(264)$ — open

Exact certificate, 2026-10-06. Evan Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing, decided here over \(\mathbb{Q}\)$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves \(s(264) \le 16.77817459305203\)$s(264)\le 16.77817459305203$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound \(17\)$17$.

Open. The best known packing gives \(s(264) \le 16.7781746\)$s(264)\le 16.7781746$, and the strongest lower bound independently verified here is \(16.255951\)$16.255951$ from Karakuş’s general theorem, leaving a gap of \(0.5222\)$0.5222$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The verified upper bound is a ceiling

verified_upper_bound for this case is \(16.77817459305203\)$16.77817459305203$, proved by Evan Daniel’s exact rational certificate of this packing (E-evand-exact-ceilings-2026-10-05-exact-replay, T-101). It is larger than the best known \(16.77817459305202\)$16.77817459305202$ two fields above it, by \(1 \times 10^{-14}\)$1\times {10}^{-14}$.

It is not the value of \(s(264)\)$s(264)$ and not a different packing: it is the certificate’s own side, \(16.7781745930520227686\ldots\)$16.7781745930520227686\dots$, rounded up at the fourteen decimals the catalogue prints. The catalogue gives the packing’s side exactly, as \(9 + (11 / 2) \sqrt{2}\)$9+(11/2)\sqrt{2}$, and the certificate’s side lies \(1.7 \times 10^{-19}\)$1.7\times {10}^{-19}$ above that: its squares are rational and each is kept clear of its neighbours and the walls, so it bounds the exact side from above and does not reach it. That side is irrational, so no rational certificate reaches it. The mathematics blocker in the frontmatter records the difference. Read reported_upper_bound for the best known side length.

The exact certificate

Evan Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register lists as best known, solved from its own witnesses, and offered those that do not lower a printed side as a replay of the existing bounds, asking for nothing to be registered from them. The certificate for this count holds the same 264 squares, each a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$, in a square of side \(16.7781745930520227686\ldots\)$16.7781745930520227686\dots$, \(2.8 \times 10^{-15}\)$2.8\times {10}^{-15}$ above the side the Kingbird catalogue prints, \(16.77817459305202\)$16.77817459305202$. Rounded to binary64, square for square, its pose is the one the atlas pictures for this count. 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.

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 \(s(264) \le 16.77817459305203\)$s(264)\le 16.77817459305203$, the verified upper bound: the certificate’s side rounded up at the fourteen decimals the catalogue prints, as the record writes any certificate of a printed side. It says nothing about optimality. 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

Found by an unrecorded author, via a hand construction.

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.