$n=264$$n=264$

$s(264)$$s(264)$ — open

Hosono’s closed form, 14 October 2026. Hosono’s closed form (wand125, 13 October 2026, T-180) states $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, from a measure of line segments on 34 intervals of the fractional part. At $n=264$$n=264$, $k=16$$k=16$ and interval 6 give $a(264)=16.308403094438469074118669059935\dots$$a(264)=16.308403094438469074118669059935\dots$, so the reported lower bound here is now $s(264)\ge 16.308403094438$$s(264)\ge 16.308403094438$, rounded down, above Nagamochi’s closed form, $16.26433752247$$16.26433752247$. The source replayed its certificate with its own checker; no record of it has been replayed here, so the verified lower bound is unchanged. The source states that its search, certificates, checkers and text were produced with AI assistance (Claude and Codex), under the direction of and reviewed by Hosono.

House adoption, 16 October 2026. Polkovskyi’s tighter certificate, after PingYou Ltd’s, of Ostashev’s composition of his 38-square rectangle with a 93-square piece of Daniel’s exact optimum of de Winter’s n = 126 packing (T-186) is now this case’s house: its exact certificate proves $s(264)\le 16.77473513238783285$$s(264)\le 16.77473513238783285$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.68×10−19$1.68\times {10}^{-19}$. The earlier ceiling, displayed outward as $16.7747351323878329$$16.7747351323878329$ from Ostashev’s packing (T-158), applied until this adoption and stays retained as history (the certificate).

House adoption, 12 October 2026. Ostashev’s composition of his 38-square rectangle with a 93-square piece of Daniel’s exact optimum of de Winter’s n = 126 packing (T-158) was this case’s house until 16 October 2026: its exact certificate proves $s(264)\le 16.7747351323878329$$s(264)\le 16.7747351323878329$, the reported and the verified upper bound, 3.44×10−3$3.44\times {10}^{-3}$ below the ceiling the case held until then, $16.77817459305202$$16.77817459305202$ from the catalogue’s packing (T-101, T-155), which stays retained as history (the certificate).

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}$$\mathbb{Q}$ by two exact checkers: it proves $s(264)\le 16.778174593052022768577069729084$$s(264)\le 16.778174593052022768577069729084$, a bound Daniel’s exact certificate of this packing already proves at the same side (T-101). The verified upper bound was that side exactly until 12 October 2026, resting on both certificates (T-101, T-155), and the record printed it as $16.77817459305203$$16.77817459305203$. Until then the verified upper bound was exactly $16.77817459305203$$16.77817459305203$, 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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(264)\le 16.77817459305203$$s(264)\le 16.77817459305203$, 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. Previously, the best known packing gave $s(264)\le 16.7781746$$s(264)\le 16.7781746$, and the strongest lower bound independently verified here is $16.255951$$16.255951$ from Karakuş’s general theorem, leaving a gap of $0.5222$$0.5222$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The earlier verified ceiling

Until 12 October 2026 the verified upper bound here was $16.778174593052022768577069729084$$16.778174593052022768577069729084$, exactly, proved by Daniel’s exact certificate of this packing (E-evand-exact-ceilings-2026-10-05-exact-replay, T-101) and by the exact regularized certificate of it (E-regularized-bounds-2026-10-12-exact-decision, T-155), and the record prints it as $16.77817459305203$$16.77817459305203$. As printed it was larger than the catalogue’s printed side $16.77817459305202$$16.77817459305202$, then the best known, by 1×10−14$1\times {10}^{-14}$.

It was not the value of $s(264)$$s(264)$ and not a different packing: it is the certificate’s own exact side. The catalogue gives the packing’s side exactly, as 9+(11/2)2​$9+(11/2)\sqrt{2}$, 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. A mathematics blocker in the frontmatter recorded the difference until Ostashev’s certificate (T-158) replaced both lanes.

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 264 squares in a square of side $16.778174593052022768577069729084$$16.778174593052022768577069729084$, 7.2×10−15$7.2\times {10}^{-15}$ 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(264)\le 16.778174593052022768577069729084$$s(264)\le 16.778174593052022768577069729084$, the verified upper bound until 12 October 2026. 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 was the exact form of the case’s verified_upper_bound until 12 October 2026, which cited both certificates (T-101, T-155) and printed it as $16.77817459305203$$16.77817459305203$, 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 264 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $16.7781745930520227686\dots$$16.7781745930520227686\dots$, 2.8×10−15$2.8\times {10}^{-15}$ above the side the Kingbird catalogue prints, $16.77817459305202$$16.77817459305202$. 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(264)\le 16.77817459305203$$s(264)\le 16.77817459305203$, 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 an unrecorded author, via a hand construction.

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.

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.

Daniil Ostashev’s certificate (T-158)

Ostashev’s composition of his 38-square rectangle with a 93-square piece of Daniel’s exact optimum of de Winter’s n = 126 packing, reported on jlevy/squares#528, at an exact rational side, displayed upward as $16.7747351323878329$$16.7747351323878329$ (T-158). The certificate gives a rational side and, for each of the 264 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. No route here is the source’s code, so the replay is independently re-implemented. The review of 12 October 2026 accepted the replay for finite feasibility. That proves $s(264)\le 16.7747351323878329$$s(264)\le 16.7747351323878329$, the verified upper bound until 16 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The issue says the author set up an AI-assisted research loop and drove it with his own ideas, the rectangle library and the composition being his, and that AI agents wrote and ran the search, composition and verification code.

The house it replaces, $16.77817459305202$$16.77817459305202$ from the catalogue’s packing (T-101, T-155), stays retained with its own replay as history. T-174 at $16.7747351323878329$$16.7747351323878329$, 1.68×10−19$1.68\times {10}^{-19}$ below this house, is confirmed here and accepted by the review of 14 October 2026. From 15 October 2026 until the owner’s house rule of 16 October 2026 lifted the hold, PR #574’s houses were held for the owner, since their witnesses would add about 18.5 MB to a checkout and their custody inputs about 5.8 MB to every worker’s snapshot.

Mykhailo Polkovskyi’s certificate (T-186)

Polkovskyi’s tighter certificate, after PingYou Ltd’s, of Ostashev’s composition of his 38-square rectangle with a 93-square piece of Daniel’s exact optimum of de Winter’s n = 126 packing, reported in his repository on 14 October 2026, with no issue, at an exact rational side, displayed upward as $16.77473513238783285$$16.77473513238783285$ (T-186). The certificate gives a rational side and, for each of the 264 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls, and a third exact route, half-extent containment and exact intersection area, decided it again. No route here is the source’s code, so the replay is independently re-implemented. The review of 16 October 2026 accepted the replay for finite feasibility. That proves $s(264)\le 16.77473513238783285$$s(264)\le 16.77473513238783285$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is Polkovskyi’s tightening of the packing T-174 records, measured here pose by pose as that packing (T-186’s regularizations), so the packing keeps its finder’s credit and Polkovskyi is credited with the certificate.

The commit makes no statement about AI assistance; the author’s issue 593 discloses Gemini assistance with solver scripting, precision checks and formatting.

The house it replaces, $16.7747351323878329$$16.7747351323878329$ from Ostashev’s packing (T-158), stays retained with its own replay as history. The other certificates confirmed here at this count, T-174 at $16.7747351323878329$$16.7747351323878329$, stay verified and are not selected.