\(s (54)\)$s(54)$ — 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 \(s(54) \le 7.84666719284349\)$s(54)\le 7.84666719284349$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound \(8\)$8$.

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(54) \ge 1537/200 = 7.685\)$s(54)\ge 1537/200=7.685$ (T-091), from a density of 364 rectangles of total mass \(5399999/100000 < 54\)$5399999/100000<54$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(3069/400\)$3069/400$ this case reported, by \(0.0125\)$0.0125$. It is the source’s first mixed certificate at this count. sqverify-fast, this repository’s clean-room verifier, decided it here on 5 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, 2026-10-01. wand125’s rectangle-density source reports \(s (54) \ge 3069 / 400 = 7.6725\)$s(54)\ge 3069/400=7.6725$, since superseded (T-091), with total mass \(5399/100 = 53.99 < 54\)$5399/100=53.99<54$, 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. 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 \(3067/400 = 7.6675\)$3067/400=7.6675$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(5399/100 = 53.99 < 54\)$5399/100=53.99<54$, 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 \(1533/200 = 7.665\)$1533/200=7.665$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(5399/100 = 53.99 < 54\)$5399/100=53.99<54$, 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.

Open. The best known packing gives \(s(54) \le 7.84666720\)$s(54)\le 7.84666720$, and the verified lower bound is \(s(54) \ge 1537/200 = 7.685\)$s(54)\ge 1537/200=7.685$, from wand125’s mixed rectangle-density certificate of 4 October, leaving a gap of about \(0.1617\)$0.1617$.

The verified upper bound is a ceiling

verified_upper_bound for this case is \(7.84666719284349\)$7.84666719284349$, proved by Evan Daniel’s exact rational certificate of this packing (E-evand-exact-ceilings-2026-10-05-exact-replay, T-101). It is larger than the best known \(7.84666719284348\)$7.84666719284348$ two fields above it, by \(1 \times 10^{-14}\)$1\times {10}^{-14}$.

It is not the value of \(s(54)\)$s(54)$ and not a different packing: it is the certificate’s own side, \(7.8466671928434897831\ldots\)$7.8466671928434897831\dots$, rounded up at the fourteen decimals the catalogue prints. The catalogue gives the packing’s side exactly, as \(7 - (1 / 2) \sqrt{2} + \sqrt{1 + \sqrt{2}}\)$7-(1/2)\sqrt{2}+\sqrt{1+\sqrt{2}}$, and the certificate’s side lies \(7.8 \times 10^{-20}\)$7.8\times {10}^{-20}$ above that: its squares are rational and each is kept clear of its neighbours and the walls, so it bounds the exact side from above and does not reach it. That side is irrational, so no rational certificate reaches it. The mathematics blocker in the frontmatter records the difference. Read reported_upper_bound for the best known side length.

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 54 squares, each a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$, in a square of side \(7.8466671928434897831\ldots\)$7.8466671928434897831\dots$, \(9.8 \times 10^{-15}\)$9.8\times {10}^{-15}$ above the side the Kingbird catalogue prints, \(7.84666719284348\)$7.84666719284348$. 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 \(s(54) \le 7.84666719284349\)$s(54)\le 7.84666719284349$, 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 packing

Found by David W. Cantrell in 2005, via a hand construction. Later improved by Joe DeVincentis.

The lower bound

The verified field rests on a complete decision here (E-n054-wand125-mixed-7685-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 5 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 \(3069/400\)$3069/400$ (E-wand125-rectangle-2026-10-01-source-replay).

From 2 October until that decision the verified lower bound was wand125’s rectangle-density certificate of 1 October above (T-074). It superseded wand125’s exact weighted point certificate for \(n = 53\)$n=53$, replayed here in full (T-044) and carried to \(n = 54\)$n=54$ by monotonicity: deleting one square from a packing of 54 leaves a packing of 53, so \(s(54) \ge s(53) \ge 369/50 = 7.38\)$s(54)\ge s(53)\ge 369/50=7.38$, which was the strongest registered bound once Nagamochi’s formula left the verified lane. Karakuş’s general bound gives only \(7.373863\)$7.373863$ here.

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 (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s value had masked the registered bound above. The review’s text would put Karakuş’s weaker bound here; the record keeps the strongest registered verified bound instead.

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.