\(s (126)\)$s(126)$ — open

Open. The best known published packing, Joost de Winter’s at Evan Daniel’s exact optimum (T-098), gives \(s(126) \le 11.7747351323878328426\ldots\)$s(126)\le 11.7747351323878328426\dots$, and the strongest lower bound independently verified here is \(11.235455\)$11.235455$ from Karakuş’s general theorem, leaving a gap of \(0.5393\)$0.5393$. 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 126 squares in a square of side \(11.7747351323878328426\ldots\)$11.7747351323878328426\dots$, \(1.9 \times 10^{-11}\)$1.9\times {10}^{-11}$ below the side the Kingbird catalogue prints for Joost de Winter’s packing. No certificate of the packing was on record before, so the verified ceiling here was the trivial grid bound \(12\)$12$ until this one. Square for square, its pose lies within \(5.5 \times 10^{-4}\)$5.5\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 2 squares move by more than \(1 \times 10^{-8}\)$1\times {10}^{-8}$, all of them squares the source lists as carrying no force; every other square moves by at most \(6.5 \times 10^{-11}\)$6.5\times {10}^{-11}$. 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 \(s(126) \le 11.7747351323878328426\ldots\)$s(126)\le 11.7747351323878328426\dots$, the verified upper bound; it says nothing about optimality. The source reports the exact point as a KKT local minimum: multipliers that keep it in equilibrium, a reduced Hessian positive definite once its exact flat motions are set aside (its per-count report), and no first-order descent across corner-to-corner contacts, each computed numerically at that point; none of that is verified here, and it bears on this packing alone, not on \(s(126)\)$s(126)$. 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 Joost de Winter in 2026, via simulated annealing. In the catalogue’s words: “Found by Joost de Winter in August 2026, working with unspecified AI, using an evolutionary beam search with simulated annealing, starting from the latest s(105) as of December 2025.”

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.