$n=128$$n=128$

$s(128)$$s(128)$ — 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}$$\mathbb{Q}$ by two exact checkers: it proves $s(128)\le 11.825091968213687029149282424753443193$$s(128)\le 11.825091968213687029149282424753443193$, the verified upper bound (T-155) rounded up at 36 places, which the record still prints as $11.82509196821369$$11.82509196821369$. Until then the verified upper bound was exactly $11.82509196821369$$11.82509196821369$, the side of Daniel’s exact certificate of this packing rounded up at the printed precision (T-101).

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=128$$n=128$, $k=11$$k=11$ and interval 10 give $a(128)=11.410198096219834384485926334692\dots$$a(128)=11.410198096219834384485926334692\dots$, so the reported lower bound here is now $s(128)\ge 11.410198096219$$s(128)\ge 11.410198096219$, rounded down, above Nagamochi’s closed form, $11.34408043278$$11.34408043278$. 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.

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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(128)\le 11.82509196821369$$s(128)\le 11.82509196821369$, the verified upper bound from 6 to 12 October 2026 (T-101). Until then the verified upper bound was the trivial grid bound $12$$12$.

Open. The best known packing gives $s(128)\le 11.82509197$$s(128)\le 11.82509197$, and the strongest lower bound independently verified here is $11.328203$$11.328203$ from Karakuş’s general theorem, leaving a gap of $0.4969$$0.4969$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

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, and this project’s same-side re-solve then fixed the result’s tight contacts over $\mathbb{Q}$$\mathbb{Q}$ at the witness’s own side. The result is an exact rational packing of the same 128 squares in a square of side $11.825091968213687029149282424753443193$$11.825091968213687029149282424753443193$ (rounded up at 36 places), 3.0×10−15$3.0\times {10}^{-15}$ below the verified upper bound the record held before, and 1.2×10−19$1.2\times {10}^{-19}$ below the side of Daniel’s certificate of this packing (T-101): it is the witness’s own side. 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(128)\le 11.825091968213687029149282424753443193$$s(128)\le 11.825091968213687029149282424753443193$, the verified upper bound (T-155) rounded up at 36 places. The exact side, a rational, is the exact form of the case’s verified_upper_bound, which prints it as $11.82509196821369$$11.82509196821369$, 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 128 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $11.8250919682136870293\dots$$11.8250919682136870293\dots$, 7.0×10−15$7.0\times {10}^{-15}$ above the side the Kingbird catalogue prints, $11.82509196821368$$11.82509196821368$. 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. 1 square moves by more than 1×10−8$1\times {10}^{-8}$, one the source lists as carrying no force; every other square’s centre rounds to the witness’s. 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(128)\le 11.82509196821369$$s(128)\le 11.82509196821369$, 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 2024, via an unrecorded method. Improved by Ellsworth in December 2024. Improved by Cantrell and Ellsworth in January 2025. Its side length is algebraic of degree 40 over $\mathbb{Q}$$\mathbb{Q}$.

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.