\(s (129)\)$s(129)$ — 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 \(s(129) \le 11.88130621809001\)$s(129)\le 11.88130621809001$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound \(12\)$12$.

Open. The best known packing gives \(s(129) \le 11.88130622\)$s(129)\le 11.88130622$, and the strongest lower bound independently verified here is \(11.374281\)$11.374281$ from Karakuş’s general theorem, leaving a gap of \(0.507\)$0.507$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

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 129 squares, each a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$, in a square of side \(11.8813062180900030634\ldots\)$11.8813062180900030634\dots$, \(3.1 \times 10^{-15}\)$3.1\times {10}^{-15}$ above the side the Kingbird catalogue prints, \(11.88130621809000\)$11.88130621809000$. Square for square, its pose lies within \(1.5 \times 10^{-4}\)$1.5\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 27 squares move by more than \(1 \times 10^{-8}\)$1\times {10}^{-8}$, 4 of them squares the source lists as carrying no force and the other 23 by at most \(6.1 \times 10^{-8}\)$6.1\times {10}^{-8}$; every other square moves by at most \(4.3 \times 10^{-9}\)$4.3\times {10}^{-9}$. 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. At this count it started from the witness after the source’s own local squeeze, which it has not published; that bears on reproducing the solve and not on checking the certificate.

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 \(s(129) \le 11.88130621809001\)$s(129)\le 11.88130621809001$, 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 verified upper bound and the printed side now agree to one unit of the printed side’s last place, the precision at which the record compares them. The printed side itself is not certified here: the certificate’s side lies above it.

The packing

Found by David Ellsworth in 2024. Improved by David Ellsworth in December 2025, via simulated annealing. Optimized by David Ellsworth in January 2026. Its side length is algebraic of degree 20 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.