$n=257$$n=257$

$s(257)$$s(257)$ — 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}$$\mathbb{Q}$ by two exact checkers: it proves $s(257)\le 16.535533905932737622169577149584$$s(257)\le 16.535533905932737622169577149584$, 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 $16.53553390593274$$16.53553390593274$. Until then the verified upper bound was exactly $16.53553390593274$$16.53553390593274$, 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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(257)\le 16.53553390593274$$s(257)\le 16.53553390593274$, the verified upper bound from 6 to 12 October 2026 (T-101). Until then the verified upper bound was the trivial grid bound $17$$17$.

Open. The best known packing gives $s(257)\le 16.53553391$$s(257)\le 16.53553391$, and the strongest lower bound independently verified here is $16.032224$$16.032224$ from Karakuş’s general theorem, leaving a gap of $0.5033$$0.5033$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The verified upper bound is a ceiling

verified_upper_bound for this case is $16.535533905932737622169577149584$$16.535533905932737622169577149584$, exactly, proved by Daniel’s exact certificate of this packing (E-evand-exact-ceilings-2026-10-05-exact-replay, T-101) and by the exact regularized certificate of it (E-regularized-bounds-2026-10-12-exact-decision, T-155), and the record prints it as $16.53553390593274$$16.53553390593274$. As printed it is larger than the best known $16.53553390593273$$16.53553390593273$ two fields above it, by 1×10−14$1\times {10}^{-14}$.

It is not the value of $s(257)$$s(257)$ and not a different packing: it is the certificate’s own exact side. The catalogue gives the packing’s side exactly, as 13+(5/2)2​$13+(5/2)\sqrt{2}$, and the certificate’s side lies 1.7×10−19$1.7\times {10}^{-19}$ above that, 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 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 257 squares in a square of side $16.535533905932737622169577149584$$16.535533905932737622169577149584$, 2.4×10−15$2.4\times {10}^{-15}$ 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; 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(257)\le 16.535533905932737622169577149584$$s(257)\le 16.535533905932737622169577149584$. 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 $16.53553390593274$$16.53553390593274$, 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 257 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $16.5355339059327376222\dots$$16.5355339059327376222\dots$, 7.6×10−15$7.6\times {10}^{-15}$ above the side the Kingbird catalogue prints, $16.53553390593273$$16.53553390593273$. Square for square, its pose lies within 1.5×10−4$1.5\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 21 squares move by more than 1×10−8$1\times {10}^{-8}$, all of them squares the source lists as carrying no force; every other square moves by at most 8.9×10−16$8.9\times {10}^{-16}$. 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(257)\le 16.53553390593274$$s(257)\le 16.53553390593274$, 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.”

The packing

Found by Ellsworth in 2025. The recorded construction method is an extension of a smaller record.

The lower bound

The strongest lower bound independently verified in this record is 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.

Source-reported bounds are recorded separately from this independently verified theorem.

The reported lower-bound expression for $s(257)$$s(257)$ is 5742​/257+3343/257$574\sqrt{2}/257+3343/257$ (approximately $16.16637581635$$16.16637581635$). Friedman’s DS7 survey, Theorem 9, k=16, reports this bound at n=257; 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=16 that point pattern leaves a unit square empty (devtools.check_green_ds7). This is a source-reported lower bound; it does not replace the verified bound above. 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.