$n=258$$n=258$

$s(258)$$s(258)$ — open

House adoption, 11 October 2026. Daniel’s display regularization of Chaoweeraprasit’s SQUISH packing as Mishapolk refined it (T-146) is now this case’s house: its exact certificate proves $s(258)\le 16.5634480021337894$$s(258)\le 16.5634480021337894$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 5.36×10−12$5.36\times {10}^{-12}$. The earlier ceiling, displayed outward as $16.5634480021391540$$16.5634480021391540$ from Chaoweeraprasit’s packing (T-115), applied until this adoption and stays retained as history (the certificate).

The SQUISH Update

Chaoweeraprasit, using SQUISH, reported on 7 October 2026 a bound tighter than the case then held $s(258)\le 16.5634480021391540$$s(258)\le 16.5634480021391540$, with exact side $582774517581285/35184372088832$$582774517581285/35184372088832$. This update is confirmed at V3/C3 by the complete retained dual exact replay and separately prompted scoped AI reviews. It establishes a feasible upper bound; optimality and human oversight have not been established. The author reports the seed as Couzo’s s(259), squares removed. The revision-specific packet retains the source print $16.5634480021391539$$16.5634480021391539$ separately from the safe decimal ceiling. The atlas drawing below uses the update’s exact rational geometry, checked for the drawing independently of the frontier assurance. Previously, Earlier certified geometries, evidence and reviews remain retained as historical provenance, separate from this update’s confirmation.

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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(258)\le 16.57106781186548$$s(258)\le 16.57106781186548$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound $17$$17$.

Open. Previously, the earlier verified packing gives $s(258)\le 16.57106782$$s(258)\le 16.57106782$, and the strongest lower bound independently verified here is $16.064382$$16.064382$ from Karakuş’s general theorem, leaving a gap of $0.5067$$0.5067$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

Earlier catalogue ceiling

The earlier certified ceiling was $16.57106781186548$$16.57106781186548$, proved by Daniel’s exact rational certificate of this packing (E-evand-exact-ceilings-2026-10-05-exact-replay, T-101). It is larger than the earlier catalogue print $16.57106781186547$$16.57106781186547$, by 1×10−14$1\times {10}^{-14}$.

It is not the value of $s(258)$$s(258)$ and not a different packing: it is the certificate’s own side, $16.5710678118654752442\dots$$16.5710678118654752442\dots$, rounded up at the fourteen decimals the catalogue prints. The catalogue gives the packing’s side exactly, as (19/2)+52​$(19/2)+5\sqrt{2}$, and the certificate’s side lies 1.7×10−19$1.7\times {10}^{-19}$ 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 irrational, so no rational certificate reaches it. The earlier catalogue-geometry blocker in the frontmatter records that difference. The current reported_upper_bound instead names the tighter, confirmed SQUISH update.

The earlier catalogue side $16.57106781186547$$16.57106781186547$ had also been conjectured optimal in this record. The smaller update geometry rules out that conjecture; it is retained here as history, and no current optimum is conjectured.

The exact certificate

Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings the register then listed 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 258 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $16.5710678118654752442\dots$$16.5710678118654752442\dots$, 5.2×10−15$5.2\times {10}^{-15}$ above the side the Kingbird catalogue prints, $16.57106781186547$$16.57106781186547$. Rounded to binary64, square for square, its pose is the one the atlas previously pictured 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}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). Previously, that proves $s(258)\le 16.57106781186548$$s(258)\le 16.57106781186548$, 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 an unrecorded author, via 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(258)$$s(258)$ is 5742​/257+3343/257$574\sqrt{2}/257+3343/257$ (approximately $16.16637581635$$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). Inherited at n=258 by monotonicity. 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.

Evan Daniel’s certificate (T-146)

Daniel’s display regularization of Chaoweeraprasit’s SQUISH packing as Mishapolk refined it, reported in his regularized record lists, on no issue, at exact side $16563448002133789379765816191273/1000000000000000000000000000000$$16563448002133789379765816191273/1000000000000000000000000000000$, displayed upward as $16.5634480021337894$$16.5634480021337894$ (T-146). The certificate gives a rational side and, for each of the 258 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 11 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(258)\le 16.5634480021337894$$s(258)\le 16.5634480021337894$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate regularizes for display the packing T-138 records, at the side the source reports as that packing’s exact optimum, so the packing stays with its finder and Daniel is credited with the regularization and the certificate. That the two are one packing is the source’s claim, measured here and not decided (T-146’s regularizations).

The source’s credits say the repository’s work was produced by Claude (Anthropic) under human direction.

The house it replaces, $16.5634480021391540$$16.5634480021391540$ from Chaoweeraprasit’s packing (T-115), stays retained with its own replay as history. The other certificates confirmed here at this count, T-138 at $16.563448002138$$16.563448002138$ and T-125 at $16.5639734753165640$$16.5639734753165640$, stay verified and are not selected. T-174 at $16.5634480021337894$$16.5634480021337894$, 1.66×10−19$1.66\times {10}^{-19}$ 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. The case keeps this 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-8 integer polynomial 1.66×10−19$1.66\times {10}^{-19}$ below this house, is confirmed here and accepted by the review of 16 October 2026.