\(s (83)\)$s(83)$ — 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 \(s(83) \le 9.63475764863109\)$s(83)\le 9.63475764863109$, the verified upper bound (T-101). Until then the verified upper bound was \(9.63475764863195\)$9.63475764863195$, the rounded-up side of an exact certificate of a binary64 parse of the catalogue’s picture (T-089).

Exact certificate, 2026-10-05. Allen Chang’s packing was certified here the same day, from a third party’s binary64 parse of the catalogue’s picture: an exact rational packing decided over \(\mathbb{Q}\)$\mathbb{Q}$ by two checkers that share no geometry or verification code proves \(s(83) \le 9.63475764863195\)$s(83)\le 9.63475764863195$, the verified upper bound until 6 October 2026, \(8.7 \times 10^{-13}\)$8.7\times {10}^{-13}$ above the printed side (T-089). Until then the verified upper bound was the \(10 \times 10\)$10\times 10$ grid.

Catalogue intake, 2026-10-05. The Kingbird catalogue’s capture of 30 September 2026 prints the side \(9.63475764863108\)$9.63475764863108$ at \(n = 83\)$n=83$ for Allen Chang’s improvement of the Hajba and Cantrell packing, optimized by David Ellsworth, below the side this case reported by about \(6.8 \times 10^{-5}\)$6.8\times {10}^{-5}$ (T-089). It is the reported upper bound here from this date.

External intake, 2026-10-02 (afternoon). wand125’s mixed-certificate source reports \(s(83) \ge 937/100 = 9.37\)$s(83)\ge 937/100=9.37$ (T-075), from a density of 728 rectangles of total mass \(8299999/100000 < 83\)$8299999/100000<83$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the linear certificate of the intake below by \(0.02\)$0.02$, and above Green’s reported \(9.2667335\ldots\)$9.2667335\dots$, which this case held by monotonicity before that intake, by \(0.10326646\ldots\)$0.10326646\dots$; the source’s \(0.1033\)$0.1033$ is measured from a value below Green’s. Its complete replay here on 2 and 3 October 2026 returned the certificate’s own record at all 201 directions, so it is also the verified lower bound. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-02. wand125’s linear-certificate source reported a direct \(187/20 = 9.35\)$187/20=9.35$ certificate for this case (T-073), whose reported bound the afternoon intake above raises, from a measure of points, segments and rectangles of total mass \(8299999/100000 < 83\)$8299999/100000<83$, accepted there at every net angle by its linear verifier, a checker the 2 October review read with no blocking defect. It is above Green’s reported \(9.2667335\ldots\)$9.2667335\dots$, which this case held by monotonicity, by \(0.083266\ldots\)$0.083266\dots$; the source’s “more than \(0.0833\)$0.0833$” is measured from a value below Green’s. Its complete replay here on 3 October 2026 returned the certificate’s own record at all 201 directions; the replayed mixed \(937/100\)$937/100$ above stays the verified lower bound. wand125’s README says parts of the work were produced with AI assistance under human direction.

Open. The best known packing gives \(s(83) \le 9.63475765\)$s(83)\le 9.63475765$, and the verified lower bound is \(s(83) \ge 937/100 = 9.37\)$s(83)\ge 937/100=9.37$, from wand125’s mixed rectangle-density certificate, leaving a gap of \(0.2648\)$0.2648$.

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 83 squares, each a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$, in a square of side \(9.6347576486310820294\ldots\)$9.6347576486310820294\dots$, \(2.0 \times 10^{-15}\)$2.0\times {10}^{-15}$ above the side the Kingbird catalogue prints, \(9.63475764863108\)$9.63475764863108$. 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 \(s(83) \le 9.63475764863109\)$s(83)\le 9.63475764863109$, 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

Allen Chang’s packing, optimized by David Ellsworth, as the Kingbird catalogue’s capture of 30 September 2026 prints it (T-089). In the catalogue’s words: “Improved by Allen Chang in September 2026, with GPT-5.6 Sol and GPT-6 Astra.” The catalogue adds “Optimized by David Ellsworth in September 2026” and “Polynomial root solution found by Allen Chang in September 2026 (see SVG source)”: its side is a root of degree 672 whose polynomial is printed only in the SVG.

The catalogue names no checker, and the SVG, which holds the pose and the polynomial, could not be fetched when this was recorded. The retained witness is therefore Evan Daniel’s binary64 parse of that SVG, whose side the printed decimal truncates. The SVG was fetched later on 5 October and read again by this repository’s own adapter: the parse agrees with it at binary64 but for one angle, which the picture gives as 0 and the parse as \(3.4 \times 10^{-49}\)$3.4\times {10}^{-49}$ degrees, so the certificate below is of the pictured packing. The SVG holds the degree-672 polynomial as a Mathematica Root object.

This repository certifies it exactly from that parse. The retained pose rounds to an exact rational packing of side \(9.6347576486319454\ldots\)$9.6347576486319454\dots$, \(8.65 \times 10^{-13}\)$8.65\times {10}^{-13}$ above the printed side, once dilated about its centre by \(1 + 10^{-13}\)$1+{10}^{-13}$ (the promotion’s step before it, \(1 + 10^{-15}\)$1+{10}^{-15}$, left one pair overlapping), 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 geometry or verification code with it (receipt). That proves \(s(83) \le 9.63475764863195\)$s(83)\le 9.63475764863195$, which was the verified upper bound until Evan Daniel’s exact certificate replaced it; it says nothing about optimality. The printed side is not certified: Evan Daniel’s exact certificate of the same packing (T-101) lies above it as well.

Found by Károly Hajba in 2024, via extension of a smaller record. Improved by David W. Cantrell in November 2024. Its side length is algebraic of degree 672 over \(\mathbb{Q}\)$\mathbb{Q}$.

The previous best known packing

Before this intake the best known packing was Hajba’s as Cantrell improved it in November 2024, of side \(9.63482562092335\)$9.63482562092335$, algebraic of degree 24, as the catalogue’s capture of 22 August 2026 printed it.

The lower bound

Before 2 October the selected external report was [Friedman DS7], which gives the lower-bound expression \(94 \sqrt{2} / 41 + 247 / 41\)$94\sqrt{2}/41+247/41$ for \(s(83)\)$s(83)$ (approximately \(9.266733533246\)$9.266733533246$). Friedman’s DS7 survey, Theorem 9, k=9, reports this bound at n=82; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at \(k = 9\)$k=9$ that point pattern leaves a unit square empty (review). Inherited at n=83 by monotonicity. That changed the reported source field only; the verified field now holds wand125’s replayed mixed certificate. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified field rests on a complete replay here (E-n083-wand125-mixed-937-source-replay, V3/C3) of wand125’s mixed certificate: the source’s unchanged checker and per-angle functions on the pinned bundle, all 201 directions on 2 and 3 October 2026, each returning the certificate’s own record. It runs the source’s own algorithm, so it confirms the source’s run rather than deciding coverage a second way. sqverify-fast, this repository’s clean-room measure verifier, decided the same certificate at all 201 net directions in its census of 3 October 2026, and two mutants scaled below coverage one were refused on 6 October (E-n083-wand125-mixed-937-sqverify-fast-replay): independently re-implemented, the same method decided a second way beside the producer’s code.

Until 2 October 2026 the verified lower bound was Nagamochi’s general closed form, now a reported bound (see the correction below), and from then until this replay it was 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.

See the Frontier corpus summary for the current aggregate count; source-reported bounds are recorded separately.

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.