$n=266$$n=266$

$s(266)$$s(266)$ — 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=266$$n=266$, $k=16$$k=16$ and interval 7 give $a(266)=16.382131212207268779454260030055\dots$$a(266)=16.382131212207268779454260030055\dots$, so the reported lower bound here is now $s(266)\ge 16.382131212207$$s(266)\ge 16.382131212207$, rounded down, above Nagamochi’s closed form, $16.32970971675$$16.32970971675$. 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.

Held for the owner, 15 October 2026; lifted 16 October 2026. T-174 at $16.8230287507564760$$16.8230287507564760$, 1.68×10−19$1.68\times {10}^{-19}$ below this case’s verified ceiling, 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. The case keeps its house until T-193’s adoption, staged under think-ot5c, moves it to T-193’s side, below T-174’s, once the writer takes an algebraic side. T-193’s exact algebraic side at this count, a root of a degree-32 integer polynomial 1.68×10−19$1.68\times {10}^{-19}$ below this case’s verified ceiling, is confirmed here and accepted by the review of 16 October 2026.

New exact construction, 7 October 2026. The complete #399 certificate proves $s(266)\le 16.8230287507564760$$s(266)\le 16.8230287507564760$ (T-119). The display is the least upward sixteen-place ceiling of the full rational side. It replaces the atlas pose; earlier source reports and certifications below describe their original, different geometries.

Previously, 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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(266)\le 16.82306208283781$$s(266)\le 16.82306208283781$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound $17$$17$.

Open. Previously, the best known packing gives $s(266)\le 16.82306209$$s(266)\le 16.82306209$, and the strongest lower bound independently verified here is $16.319292$$16.319292$ from Karakuş’s general theorem, leaving a gap of $0.5038$$0.5038$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

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 266 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $16.8230620828378046464\dots$$16.8230620828378046464\dots$, 4.6×10−15$4.6\times {10}^{-15}$ above the side the Kingbird catalogue prints, $16.82306208283780$$16.82306208283780$. 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). Previously, that proves $s(266)\le 16.82306208283781$$s(266)\le 16.82306208283781$, 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.”

For the earlier packing, its verified upper bound and 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 earlier packing

Found by Ellsworth in 2024, via an unrecorded method. Its side length is algebraic of degree 32 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.

The three new arrangements

Daniel supplies this complete rational centre/half-angle packing at exact side $8411514375378237985354391855627/500000000000000000000000000000$$8411514375378237985354391855627/500000000000000000000000000000$. Both independent project implementations accept every square, wall and pair; both reject complete-roster duplicate-square and outside-container controls. The full inputs and native results are retained in the separate packet. A separately prompted mathematical review independently reran all nine jobs. This establishes feasible upper bounds at V3/C3. Arrangement novelty, KKT, local minimality, rigidity, global optimality and human oversight are not established. The earlier #375 source namespace and T-098/T-101 certificates remain unchanged.