$n=83$$n=83$

$s(83)$$s(83)$ — open

Exact regularized certificate, 2026-10-12. This project’s regularization pass, Daniel’s exact solver and regularizer run here on this register’s witness, produced an exact rational packing of this arrangement, decided here over $\mathbb{Q}$$\mathbb{Q}$ by two exact checkers: it proves $s(83)\le 9.634757648631082029356431370282$$s(83)\le 9.634757648631082029356431370282$, a bound Daniel’s exact certificate of this packing already proves at the same side (T-101). The verified upper bound is now that side exactly, resting on both certificates (T-101, T-155), and the record still prints it as $9.63475764863109$$9.63475764863109$. Until then the verified upper bound was exactly $9.63475764863109$$9.63475764863109$, the side of Daniel’s exact certificate of this packing rounded up at the printed precision (T-101).

Exact certificate, 2026-10-06. 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 from 6 to 12 October 2026 (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. 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, 8.7×10−13$8.7\times {10}^{-13}$ 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 Chang’s improvement of the Hajba and Cantrell packing, optimized by Ellsworth, below the side this case reported by about 6.8×10−5$6.8\times {10}^{-5}$ (T-089). It is the reported upper bound here from this date.

External intake, 2026-10-02 (afternoon). Hosono’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$$0.02$, and above Green’s reported $9.2667335\dots$$9.2667335\dots$, which this case held by monotonicity before that intake, by $0.10326646\dots$$0.10326646\dots$; the source’s $0.1033$$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. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-02. Hosono’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\dots$$9.2667335\dots$, which this case held by monotonicity, by $0.083266\dots$$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. Hosono’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 Hosono’s mixed rectangle-density certificate, leaving a gap of $0.2648$$0.2648$.

The regularized exact certificate

This project’s regularization pass of 11 and 12 October 2026 ran Daniel’s exact solver and regularizer (evand/square-packing, MIT, vendored here) on this register’s witness for this count, with the conservative angle policy and coincide placement, which wrote its certificate at the side the solver reached, the solved point scaled by 1+10−20$1+{10}^{-20}$ and the side rounded up. The result is an exact rational packing of the same 83 squares in a square of side $9.634757648631082029356431370282$$9.634757648631082029356431370282$, 8.0×10−15$8.0\times {10}^{-15}$ below the verified upper bound the record held before, which was that same side rounded up at the printed precision: it is the side of Daniel’s certificate of this packing (T-101). Measured against the witness under the box’s eight symmetries, its squares pair one to one with the witness’s; 14 move by more than 1×10−3$1\times {10}^{-3}$, the largest by 3.1×10−2$3.1\times {10}^{-2}$, and none turns by more than 1×10−6$1\times {10}^{-6}$ as a sine. It redraws the packing visibly, and is drawn only once the regularized layer admits it under the regularization policy.

Each pair and each wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(83)\le 9.634757648631082029356431370282$$s(83)\le 9.634757648631082029356431370282$. Daniel’s certificate of this packing already proves the same bound at the same side (T-101), and its replay here, by two checkers that share no code with each other or with the source, is independent of the code that made this view. That side is the exact form of the case’s verified_upper_bound, which cites both certificates (T-101, T-155) and prints it as $9.63475764863109$$9.63475764863109$, the decimal it printed before and an upper display of it. It says nothing about optimality, and the packing stays its finders’.

The earlier exact certificate

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)$$t=\tan (\theta /2)$, in a square of side $9.6347576486310820294\dots$$9.6347576486310820294\dots$, 2.0×10−15$2.0\times {10}^{-15}$ 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. 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 until 12 October 2026: 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.”

Its rounded-up side and the printed side 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

Chang’s packing, optimized by Ellsworth, as the Kingbird catalogue’s capture of 30 September 2026 prints it (T-089). In the catalogue’s words: “Improved by Chang in September 2026, with GPT-5.6 Sol and GPT-6 Astra.” The catalogue adds “Optimized by Ellsworth in September 2026” and “Polynomial root solution found by 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 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 3.4×10−49$3.4\times {10}^{-49}$ 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\dots$$9.6347576486319454\dots$, 8.65×10−13$8.65\times {10}^{-13}$ above the printed side, once dilated about its centre by 1+10−13$1+{10}^{-13}$ (the promotion’s step before it, 1+10−15$1+{10}^{-15}$, left one pair overlapping), and every pair and every wall is decided over $\mathbb{Q}$$\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 Daniel’s exact certificate replaced it; it says nothing about optimality. The printed side is not certified: Daniel’s exact certificate of the same packing (T-101) lies above it as well.

Found by Hajba in 2024, via extension of a smaller record. Improved by 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 942​/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 Hosono’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 Hosono’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)≥min(⌈N​⌉,N−2⌊N​⌋+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.