\(s (50)\)$s(50)$ — 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 \(s(50) \le 7.57142857142858\)$s(50)\le 7.57142857142858$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound \(8\)$8$.

External intake, 2026-09-28. wand125’s point-and-mixed source reports \(s (50) \ge 37 / 5 = 7.4\)$s(50)\ge 37/5=7.4$, from a D4-expanded density of 553 rectangle orbits with total mass \(4999999/100000 < 50\)$4999999/100000<50$, core side \(9977/10000\)$9977/10000$ and a 201-node angle net, accepted there by a research copy of Tokoharu’s interval checker that accepts coverage at least 1 instead of \(10001/10000\)$10001/10000$ and checks each net node’s unit-square centre domain. It exceeds Green’s reported bound by more than \(0.0825\)$0.0825$, and the review of 28 September found both changes to Tokoharu’s checker legitimate and no blocking finding. This repository’s exact audit checks the premises, and the complete coverage replay here on 29 September returned the shipped 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.

Open. The best known packing gives \(s(50) \le 7.57142858\)$s(50)\le 7.57142858$, and the verified lower bound is \(s(50) \ge 37/5 = 7.4\)$s(50)\ge 37/5=7.4$, from wand125’s mixed rectangle-density certificate, leaving a gap of \(0.1714\)$0.1714$.

The verified upper bound is a ceiling

verified_upper_bound for this case is \(7.57142857142858\)$7.57142857142858$, 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 \(7.57142857142857\)$7.57142857142857$ two fields above it, by \(1 \times 10^{-14}\)$1\times {10}^{-14}$.

It is not the value of \(s(50)\)$s(50)$ and not a different packing: it is the certificate’s own side, \(7.5714285714285714287\ldots\)$7.5714285714285714287\dots$, rounded up at the fourteen decimals the catalogue prints. The catalogue gives the packing’s side exactly, as \(7 + (4/7)\)$7+(4/7)$, and the certificate’s side lies \(7.6 \times 10^{-20}\)$7.6\times {10}^{-20}$ 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 rational, so an exact certificate of the packing at that side would reach it, a rational one where the packing’s exact point is rational. 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 50 squares, each a rational centre and a rational \(t = \tan(\theta/2)\)$t=\tan (\theta /2)$, in a square of side \(7.5714285714285714287\ldots\)$7.5714285714285714287\dots$, \(1.4 \times 10^{-15}\)$1.4\times {10}^{-15}$ above the side the Kingbird catalogue prints, \(7.57142857142857\)$7.57142857142857$. Square for square, its pose lies within \(2.3 \times 10^{-4}\)$2.3\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 6 squares move by more than \(1 \times 10^{-8}\)$1\times {10}^{-8}$, all of them squares 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 \(s(50) \le 7.57142857142858\)$s(50)\le 7.57142857142858$, 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 Thomas Schadt in 2025, via simulated annealing.

The lower bound

The certificate is a D4-expanded density of 553 rectangle orbits with no point mass, of total mass \(4999999/100000 < 50\)$4999999/100000<50$, at core side \(9977/10000\)$9977/10000$ on 201 net half-angles of step \(83/40000\)$83/40000$, such that the shrunken core of every rotated unit square at each net direction captures mass at least \(1\)$1$. The source’s research copy of Tokoharu’s interval checker, which accepts coverage at least \(1\)$1$ and checks each net node’s unit-square centre domain, decides the oblique directions, and exact integer tables the axis direction.

The verified field rests on a complete replay here (E-n050-wand125-mixed-740-source-replay, V3/C3): the source’s driver and checker, unchanged, on the pinned bundle, all 201 directions on 29 September, each equal to the shipped record. It runs the source’s own algorithm, so it confirms the source’s run rather than deciding coverage a second way. The checker refuses two mutated copies of the source’s \(n = 37\)$n=37$ certificate of the same kind, and the review of 28 September found both changes to Tokoharu’s checker legitimate. The certificate’s mass is below 51 too, so it gives \(s(51) \ge 37/5\)$s(51)\ge 37/5$ as well.

Earlier lower bounds

Until 2 October 2026 the verified lower bound was Nagamochi’s general closed form, now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this \(n\)$n$, 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.

Before the 2026-09-28 intake the reported lower bound was the expression \(3 \sqrt{14} / 25 + 2 \sqrt{2} + 101 / 25\)$3\sqrt{14}/25+2\sqrt{2}+101/25$ (approximately \(7.317426011159\)$7.317426011159$). Friedman’s DS7 survey, Theorem 9, k=7, reports this bound at n=50; reference [8] is Green’s private communication (2000). The source proof has not been recovered, and the report stays as evidence. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at \(k = 7\)$k=7$ that point pattern leaves a unit square empty (review). The exact specialization and comparison are retained by audit_ds7_lower_bounds.

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.