$n=28$$n=28$

$s(28)$$s(28)$ — 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}$$\mathbb{Q}$ by two exact checkers: it proves $s(28)\le 5.824444616674059297016873205016$$s(28)\le 5.824444616674059297016873205016$, 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 $5.82444461667406$$5.82444461667406$. Until then the verified upper bound was exactly $5.82444461667406$$5.82444461667406$, 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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(28)\le 5.82444461667406$$s(28)\le 5.82444461667406$, the verified upper bound from 6 to 12 October 2026 (T-101). Until then the verified upper bound was the trivial grid bound $6$$6$.

External intake, 2026-10-06. Hosono’s check2 source reports $s(28)\ge 1147/200=5.735$$s(28)\ge 1147/200=5.735$ (T-107), from a density of 531 rectangles of total mass $2799999/100000<28$$2799999/100000<28$, on a net the certificate declares: core side $999/1000$$999/1000$ and 416 half-angle tangents of step $1/1001$$1/1001$. Since $999/1000\cdot (1+1/1001)<1$$999/1000·(1+1/1001)<1$, every unit square contains a core at a net angle strictly in its interior. The source ships no record of its C++ checker for it: it accepts it at every direction with its copy of this repository’s sqverify_fast, changed to read a declared net. It is above the reported $2289/400$$2289/400$ (T-068) by $0.0125$$0.0125$. This repository’s clean-room verifier sqverify-fast decided it here on 6 October 2026 at all 416 directions of its net, and refused two mutants scaled below coverage one: confirmed, re-implemented sharing the producer’s components (the source’s check is a copy of the same crate), 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, 2026-10-01. Hosono’s rectangle-density source reports $s(28)\ge 2289/400=5.7225$$s(28)\ge 2289/400=5.7225$, since superseded (T-107), with total mass $2799/100=27.99<28$$2799/100=27.99<28$, 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 was also verified until 2026-10-06, when the certificate above superseded it in both lanes. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. Hosono’s rectangle-density source reports a direct $143/25=5.72$$143/25=5.72$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $2799/100=27.99<28$$2799/100=27.99<28$, accepted by Tokoharu’s unchanged interval checker. The replayed n27 certificate carried $28/5=5.6$$28/5=5.6$ here by monotonicity, the verified lower bound until the 2026-10-01 intake above. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. Hosono’s rectangle-density source reports a direct $1139/200=5.695$$1139/200=5.695$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass $2799/100=27.99<28$$2799/100=27.99<28$, accepted by Tokoharu’s unchanged interval checker. The replayed n27 certificate carried $28/5=5.6$$28/5=5.6$ here by monotonicity, the verified lower bound until the 2026-10-01 intake above.

The complete replay of Tokoharu’s n26 rectangle-density certificate previously verified $s(28)\ge 1377/250=5.508$$s(28)\ge 1377/250=5.508$ by monotonicity, until the replayed n27 certificate superseded it. Tokoharu’s README says parts of the work were produced with AI assistance under human direction. Friedman DS7 reports Green’s $5.511708697746$$5.511708697746$ lower bound, now weaker than both lanes; its cited private-communication proof has not been recovered.

Open. The best known packing gives $s(28)\le 5.82444462$$s(28)\le 5.82444462$; the verified lower bound is $1147/200=5.735$$1147/200=5.735$, from Hosono’s check2 certificate on a declared net (T-107, 2026-10-06, V3/C3), which superseded its rectangle-density certificate of 1 October at $2289/400=5.7225$$2289/400=5.7225$ (T-074), leaving a gap of about $0.0894$$0.0894$ against that reported upper construction.

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 28 squares in a square of side $5.824444616674059297016873205016$$5.824444616674059297016873205016$, 7.0×10−16$7.0\times {10}^{-16}$ 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; none moves by more than 1×10−3$1\times {10}^{-3}$, and none turns by more than 1×10−6$1\times {10}^{-6}$ as a sine. It draws the same picture as the witness, so the atlas drawing does not change.

Each pair and each wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(28)\le 5.824444616674059297016873205016$$s(28)\le 5.824444616674059297016873205016$. 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 $5.82444461667406$$5.82444461667406$, 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 28 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $5.8244446166740592971\dots$$5.8244446166740592971\dots$, 9.3×10−15$9.3\times {10}^{-15}$ above the side the Kingbird catalogue prints, $5.82444461667405$$5.82444461667405$. 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(28)\le 5.82444461667406$$s(28)\le 5.82444461667406$, 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

Found by Ellsworth in 2025, via simulated annealing. The catalogue flags it rigid: no continuous deformation is possible, so the contact conditions pin the side length exactly as an algebraic number. Its side length is algebraic of degree 6 over $\mathbb{Q}$$\mathbb{Q}$. Its Galois group over $\mathbb{Q}$$\mathbb{Q}$ is S6, computed here from the recorded minimal polynomial with sympy.galois_group, and S6 is not solvable. The side length is therefore not expressible in radicals, so the empty exact_form is a permanent property of this case rather than a transcription gap.

The lower bound

The earlier external report, [Friedman DS7], gives the lower-bound expression 65​/5+22​$6\sqrt{5}/5+2\sqrt{2}$ for $s(28)$$s(28)$ (approximately $5.511708697746$$5.511708697746$). Friedman’s DS7 survey, Theorem 10, k=5, reports this bound at n=28; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified lower bound is $1147/200=5.735$$1147/200=5.735$, from Hosono’s check2 certificate of 6 October on a declared net (T-107), decided here by sqverify-fast at every direction of its net; its rectangle-density certificate of 1 October, $2289/400=5.7225$$2289/400=5.7225$ (T-074), held the field from 2026-10-02 until then. Before that the verified lower bound was Hosono’s $28/5$$28/5$, inherited from the fully replayed n27 rectangle-density certificate, whose mass $26999/1000$$26999/1000$ is below 28, by monotonicity, about $0.0883$$0.0883$ above Green’s literal report, while its direct n28 certificate at $143/25$$143/25$, which replaced the $1139/200$$1139/200$ certificate on 2026-09-28, was the reported lower bound. A method-distinct rectangle coverage implementation would add confidence and improve generalization, but it is not a missing premise of the verified fixed bound.

Corrected 2 October 2026: Nagamochi’s Lemma 1 is false (Karakuş 2026), so his closed form below is now a reported bound whose published proof is incomplete (review). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s earlier general closed form remains part of the evidence history and applies to every $N\ge 4$$N\ge 4$:

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.