\(s (69)\)$s(69)$ — 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 \(s(69) \le 8.82719465572974\)$s(69)\le 8.82719465572974$, the verified upper bound (T-101). Until then the verified upper bound was \(8.82719465572975\)$8.82719465572975$, the rounded-up side of an exact certificate of a binary64 parse of the catalogue’s picture (T-088).

Exact certificate, 2026-10-05. David Ellsworth’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 \(s(69) \le 8.82719465572975\)$s(69)\le 8.82719465572975$, the verified upper bound until 6 October 2026, \(2 \times 10^{-14}\)$2\times {10}^{-14}$ above the printed side (T-088). Until then the verified upper bound was the \(9 \times 9\)$9\times 9$ grid.

Catalogue intake, 2026-10-05. The Kingbird catalogue’s capture of 30 September 2026 prints the side \(8.82719465572973\)$8.82719465572973$ at \(n = 69\)$n=69$ for David Ellsworth’s optimization of the packing the UnitSquare release reported in July 2026, below that release’s side by about \(1.1 \times 10^{-5}\)$1.1\times {10}^{-5}$ (T-088). It is the reported upper bound here from this date.

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(69) \ge 431/50 = 8.62\)$s(69)\ge 431/50=8.62$ (T-090), from a density of 547 rectangles of total mass \(6899999/100000 < 69\)$6899999/100000<69$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(2153/250\)$2153/250$ this case reported, by \(0.008\)$0.008$. It supersedes the source’s mixed_n69_L8612. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions, so it is also the verified lower bound: confirmed, independently re-implemented. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-03. wand125’s mixed-certificate source reports the mixed certificate mixed_n69_L8612 at side \(2153/250 = 8.612\)$2153/250=8.612$ (T-082), since superseded (T-090), from a density of 556 rectangles of total mass \(6899999/100000 < 69\)$6899999/100000<69$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(1717/200\)$1717/200$ this case reported, by \(0.027\)$0.027$. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions: confirmed, independently re-implemented. The verified lower bound is unchanged, the certificate above being higher. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. wand125’s rectangle-density source reports \(s (69) \ge 1717 / 200 = 8.585\)$s(69)\ge 1717/200=8.585$, since superseded (T-090), with total mass \(6899/100 = 68.99 < 69\)$6899/100=68.99<69$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it is also verified, and it was the verified lower bound until 6 October. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(343/40 = 8.575\)$343/40=8.575$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(6899/100 = 68.99 < 69\)$6899/100=68.99<69$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(1709/200 = 8.545\)$1709/200=8.545$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(6899/100 = 68.99 < 69\)$6899/100=68.99<69$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged.

External intake, 2026-09-22. wand125’s retained point-certificate source reports \(s (69) \ge 841 / 100 = 8.41\)$s(69)\ge 841/100=8.41$, which was the verified lower bound until 2026-10-02. The complete replay through the native exact verifier passed all 201 directions and all five certificate conditions.

Open. The best known packing, David Ellsworth’s (T-088), gives \(s(69) \le 8.82719466\)$s(69)\le 8.82719466$, and the strongest verified lower bound is \(431/50 = 8.62\)$431/50=8.62$, from wand125’s mixed rectangle-density certificate of 4 October (T-090). The exact optimum remains open.

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 69 squares, each a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$, in a square of side \(8.8271946557297389151\ldots\)$8.8271946557297389151\dots$, \(8.9 \times 10^{-15}\)$8.9\times {10}^{-15}$ above the side the Kingbird catalogue prints, \(8.82719465572973\)$8.82719465572973$. 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. 1 square moves by more than \(1 \times 10^{-8}\)$1\times {10}^{-8}$, one the source lists as carrying no force; every other square’s centre rounds to the witness’s. 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 \(s(69) \le 8.82719465572974\)$s(69)\le 8.82719465572974$, 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

David Ellsworth’s packing, as the Kingbird catalogue’s capture of 30 September 2026 prints it (T-088). In the catalogue’s words it was “Refound by David Ellsworth in September 2026, using his modified version of Thomas Schadt’s simulated annealing program, starting from \(s(69)\)$s(69)$ found by Maurizio Morandi in June 2010”, and “Optimized by David Ellsworth in September 2026”. Its side is the root near \(8.82719465572973\)$8.82719465572973$ of the degree-38 polynomial the catalogue prints, \(8.8271946557297389149\ldots\)$8.8271946557297389149\dots$, which the catalogue prints cut short.

It is the packing the UnitSquare Project’s release of 29 July 2026 reported from hmbelvedere.com, which the catalogue credits to hmbelvedere. In the catalogue’s words: “Improved by hmbelvedere in July 2026, using an undisclosed iterative method (probably with AI).” The optimization lowers the release’s side by \(0.0000108520861817\ldots\)$0.0000108520861817\dots$.

The catalogue names no checker, and its SVG, which holds the pose, could not be fetched when this was recorded. The retained witness is therefore Evan Daniel’s binary64 parse of that SVG, whose side is the polynomial’s root to 30 digits. The SVG was fetched later on 5 October and read again by this repository’s own adapter: every coordinate of the parse is the picture’s rounded to binary64, so the certificate below is of the pictured packing.

This repository certifies it exactly from that parse. The retained pose rounds to an exact rational packing of side \(8.8271946557297478\ldots\)$8.8271946557297478\dots$, \(1.78 \times 10^{-14}\)$1.78\times {10}^{-14}$ above the printed side, once dilated about its centre by \(1 + 10^{-15}\)$1+{10}^{-15}$ (undilated, 29 pairs overlap), 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 \(s(69) \le 8.82719465572975\)$s(69)\le 8.82719465572975$, 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 Maurizio Morandi in 2010. Improved by David W. Cantrell in August 2023. The recorded construction method is simulated annealing. Its side length is algebraic of degree 38 over \(\mathbb{Q}\)$\mathbb{Q}$.

The previous best known packing

Before this intake the best known packing was the UnitSquare Project’s, of side \(8.827205507815920656880680735618736931225314831\)$8.827205507815920656880680735618736931225314831$, from its release of 29 July 2026, which improved the public Morandi-Cantrell parent by \(0.000006548113079343119\)$0.000006548113079343119$. The old parent’s degree-82 polynomial does not describe that numerical construction and was removed from the current-bound fields then. The release says it used interval arithmetic, a 300-digit recomputation, and an independent checker. It does not publish the interval boxes, governed receipt, or replayable checker needed to inspect the formal claim, so this repository recorded the value as reported.

The lower bound

The verified field rests on a complete decision here (E-n069-wand125-mixed-862-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 6 October 2026, with two mutants scaled below coverage one refused. It shares no code with the source’s checker, so it decides coverage a second way with the same method; the source’s own checker was not run here. Until this decision the verified lower bound was \(1717/200\)$1717/200$, wand125’s rectangle-density certificate of 1 October (E-wand125-rectangle-2026-10-01-source-replay), and until 2 October the point certificate below.

The retained cert_n69_L841.json has exact nonnegative mass \(846701027/12500000 < 69\)$846701027/12500000<69$. The native verifier checks D4 symmetry, the complete rational direction net, strict inner-square containment, and exact minimum coverage over every reachable translation cell. Its least covered mass is \(100001017/100000000 > 1\)$100001017/100000000>1$, at direction 118. The replay log and mathematical review retain the complete decision and the theorem-to-code review.

This is replayed external exact evidence, not a first-party discovery or a second coverage method. Its mass also gives the locally derived n68 bound recorded in n-068.md. Nagamochi’s general bound \(1 + \sqrt{54}\)$1+\sqrt{54}$ remains valid historical evidence and is superseded as this case’s strongest verified floor. Corrected 2 October 2026: its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false, so it is now a reported bound (review). This register had recorded that proof as verified, its own error, logged as defect D-516.

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.