$n=292$$n=292$

$s(292)$$s(292)$ — open

House adoption, 11 October 2026. Daniel’s display regularization of Chaoweeraprasit’s SQUISH packing (T-146) is now this case’s house: its exact certificate proves $s(292)\le 17.5913781454976982$$s(292)\le 17.5913781454976982$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 5.87×10−3$5.87\times {10}^{-3}$. The earlier ceiling, displayed outward as $17.597249391156465040471442005510391858592626557$$17.597249391156465040471442005510391858592626557$ from Couzo’s packing, improved by Rehwaldt (T-117), applied until this adoption and stays retained as history (the certificate).

Before the 7 October refinement, the best known published packing, Couzo’s (T-056), had the displayed ceiling $17.597249391199519$$17.597249391199519$, and the strongest lower bound independently verified here is $17.090660$$17.090660$ from Karakuş’s general theorem, leaving a gap of $0.5066$$0.5066$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The packing

Couzo’s square-packing reports a packing of side $17.597249391199519$$17.597249391199519$ for this count, dated 27 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side $17.601260370442159$$17.601260370442159$, is dated 26 September 2026. The repository names no method and no tolerance, and itself states no AI assistance; its author said on issue #227 that he found the 102 and 103 packings “with the help of Claude”.

This repository certifies it exactly. The retained decimal pose rounds to an exact rational packing at centre dilation 1, of side $17.5972493911995189091\dots$$17.5972493911995189091\dots$, no larger than the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That replay established the earlier verified ceiling $17.597249391199519$$17.597249391199519$; it says nothing about optimality. Interval arithmetic on the printed pose itself, with each angle’s true cosine and sine and no rational rounding, decides every pair and wall again and gives the same verified upper bound (interval route).

Casson’s square-packing, dated 23 September 2026 and made with the help of Claude as its README says, reports a packing of side $17.60256849240573729$$17.60256849240573729$ for this count, larger than Couzo’s by $0.00531910120621829$$0.00531910120621829$. Casson’s is the earlier of the two by the timestamps this record’s priority notes keep: Couzo’s first packing for this count, of side $17.601260370442159$$17.601260370442159$, is dated 26 September 2026. The record states both dates and infers nothing about whether either packing derives from the other.

The previous best known packing

Before this intake the best known packing was the Kingbird catalogue’s, of side $17.60257141234448$$17.60257141234448$.

Found by an unrecorded author, via extension of a smaller record. Its side length is algebraic of degree 8 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.

The reported lower-bound expression for $s(292)$$s(292)$ is 834​/145+22​+2031/145$8\sqrt{34}/145+2\sqrt{2}+2031/145$ (approximately $17.15703136722$$17.15703136722$). Friedman’s DS7 survey, Theorem 9, k=17, reports this bound at n=290; 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=17 that point pattern leaves a unit square empty (devtools.check_green_ds7). Inherited at n=292 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.

The 7 October rational refinement

Rehwaldt reports the finite rational witness (T-117) at exact side $17597249391156465040471442005510391858592626557/1000000000000000000000000000000000000000000000$$17597249391156465040471442005510391858592626557/1000000000000000000000000000000000000000000000$, whose complete terminating decimal is $17.597249391156465040471442005510391858592626557$$17.597249391156465040471442005510391858592626557$. The pinned source refines Couzo’s existing construction with disclosed OpenAI Codex assistance. The complete finite witness is now exact-replayed here at V3/C3 after scoped review and public recovery of all original scientific evidence. Earlier ceilings and verification remain historical evidence for their own geometries. No unrestricted lower bound, optimality or rigidity claim is added. The earlier rigidity screen belongs to the earlier geometry and is preserved in the packet’s prior-state record; the current numerical translation screen is separate from the exact feasibility evidence.

Evan Daniel’s certificate (T-146)

Daniel’s display regularization of Chaoweeraprasit’s SQUISH packing, reported in his regularized record lists, on no issue, at exact side $4397844536374424539902715795867/250000000000000000000000000000$$4397844536374424539902715795867/250000000000000000000000000000$, displayed upward as $17.5913781454976982$$17.5913781454976982$ (T-146). The certificate gives a rational side and, for each of the 292 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(292)\le 17.5913781454976982$$s(292)\le 17.5913781454976982$, 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-136 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, $17.597249391156465040471442005510391858592626557$$17.597249391156465040471442005510391858592626557$ from Couzo’s packing, improved by Rehwaldt (T-117), stays retained with its own replay as history. The other certificates confirmed here at this count, T-136 at $17.5913781454979008$$17.5913781454979008$, stay verified and are not selected. T-174 at $17.5913781454976982$$17.5913781454976982$, 1.76×10−19$1.76\times {10}^{-19}$ 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-6 integer polynomial 1.76×10−19$1.76\times {10}^{-19}$ below this house, is confirmed here and accepted by the review of 16 October 2026.