$n=239$$n=239$

$s(239)$$s(239)$ — 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=239$$n=239$, $k=15$$k=15$ and interval 27 give $a(239)=15.531440832262968642134276202010\dots$$a(239)=15.531440832262968642134276202010\dots$, so the reported lower bound here is now $s(239)\ge 15.531440832262$$s(239)\ge 15.531440832262$, rounded down, above Nagamochi’s closed form, $15.49137674618$$15.49137674618$. 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, 17 October 2026. PingYou Ltd’s exact re-solve of Chaoweeraprasit’s SQUISH packing composed from SQUISH’s n = 209 packing as Gupta refined it and its own n = 240 packing (T-174) is now this case’s house: its exact certificate proves $s(239)\le 15.94844332730049991$$s(239)\le 15.94844332730049991$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.49×10−31$1.49\times {10}^{-31}$. The earlier ceiling, displayed outward as $15.9484433273005000$$15.9484433273005000$ from Chaoweeraprasit’s packing (T-151), applied until this adoption and stays retained as history (the certificate).

House adoption, 12 October 2026. Chaoweeraprasit’s SQUISH packing, composed from SQUISH’s n = 209 packing as Gupta refined it and its own n = 240 packing (T-151) was this case’s house until 17 October 2026: its exact certificate proves $s(239)\le 15.9484433273005000$$s(239)\le 15.9484433273005000$, the reported and the verified upper bound, 8.70×10−4$8.70\times {10}^{-4}$ below the ceiling the case held until then, $15.9493131697276962$$15.9493131697276962$ from Chaoweeraprasit’s packing, improved by Gupta (T-127), which stays retained as history (the certificate).

The SQUISH Update

Previously, Chaoweeraprasit, using SQUISH, reports a tighter bound $s(239)\le 15.9493131697291908$$s(239)\le 15.9493131697291908$, with exact side $4489332553000483/281474976710656$$4489332553000483/281474976710656$. 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(210), grafted into n’s Kingbird record. The revision-specific packet retains the source print $15.9493131697291908$$15.9493131697291908$ 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. Earlier certified geometries, evidence and reviews remain retained as historical provenance, separate from this update’s confirmation.

Open. Previously, the earlier verified packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(239)\le 15.9538193334807799155\dots$$s(239)\le 15.9538193334807799155\dots$, and the strongest lower bound independently verified here is $15.474979$$15.474979$ from Karakuş’s general theorem, leaving a gap of $0.4788$$0.4788$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The exact optimum

Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing at its exact optimum (T-098): the same 239 squares in a square of side $15.9538193334807799155\dots$$15.9538193334807799155\dots$, 3.9×10−12$3.9\times {10}^{-12}$ below the side Couzo prints. Square for square, its pose lies within 1.5×10−4$1.5\times {10}^{-4}$ of the binary64 pose the atlas previously pictured for this count. 3 squares move by more than 1×10−8$1\times {10}^{-8}$, all of them squares the source lists as carrying no force; every other square moves by at most 5.4×10−12$5.4\times {10}^{-12}$. The previous atlas witness was that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That previously proved $s(239)\le 15.9538193334807799155\dots$$s(239)\le 15.9538193334807799155\dots$, the verified upper bound; it says nothing about optimality. The source reports the exact point as a KKT local minimum: multipliers that keep it in equilibrium, a reduced Hessian positive definite once its exact flat motions are set aside (its per-count report), and no first-order descent across corner-to-corner contacts, each computed numerically at that point; none of that is verified here, and it bears on this packing alone, not on $s(239)$$s(239)$. 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

Couzo’s square-packing reports a packing of side $15.953819333484685$$15.953819333484685$ for this count, dated 26 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side $15.954681985629890$$15.954681985629890$, is dated 24 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 $15.9538193334846845335\dots$$15.9538193334846845335\dots$, no larger than the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That proves $s(239)\le 15.953819333484685$$s(239)\le 15.953819333484685$, which was the verified upper bound until Daniel’s exact optimum above replaced it; 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 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 $15.95623903092502971$$15.95623903092502971$ for this count, larger than Couzo’s by $0.00241969744034471$$0.00241969744034471$. 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 $15.954681985629890$$15.954681985629890$, is dated 24 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 $15.95643304058435$$15.95643304058435$.

Found by Ellsworth in 2026, via simulated annealing.

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 Gupta precision refinement

Gupta refines the earlier SQUISH construction at exact rational side $1993664146215962019789087062079/125000000000000000000000000000$$1993664146215962019789087062079/125000000000000000000000000000$, displayed upward as $15.9493131697276962$$15.9493131697276962$ (T-127). The complete factual packet retains every original certificate, complete comparator and actual deciding outcome for all seventeen source cases, including three withdrawals.

Complete dual-route native replay, full-roster controls and separately prompted source and production review confirm finite feasibility at V3/C3 using independently re-implemented deciding code. The repository routes share parsing, half-angle conversion, Fraction arithmetic and SAT methodology; the recorded code relation is to the source producer’s verification, not to each other. The full source poses are converted without dilation or rounding. The source credits Chaoweeraprasit for SQUISH and Daniel for the optimizer. No author program is executed here. Feasibility establishes no optimizer, local-minimum, rigidity, novelty, priority, global-optimality, formal proof or human-oversight assurance.

Nate Chaoweeraprasit’s certificate (T-151)

Chaoweeraprasit’s SQUISH packing, composed from SQUISH’s n = 209 packing as Gupta refined it and its own n = 240 packing, reported on jlevy/squares#500, at an exact rational side, displayed upward as $15.9484433273005000$$15.9484433273005000$ (T-151). The certificate gives a rational side and, for each of the 239 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 accepted the replay for finite feasibility. That proves $s(239)\le 15.9484433273005000$$s(239)\le 15.9484433273005000$, the verified upper bound until 17 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source states that SQUISH’s solver, its verification and the issue were written with Claude Opus 5.5, and that the Mondrian composition method was formulated by the author and implemented with Claude Opus 5.5, under his direction.

The house it replaces, $15.9493131697276962$$15.9493131697276962$ from Chaoweeraprasit’s packing, improved by Gupta (T-127), stays retained with its own replay as history. The other certificates confirmed here at this count, T-115 at $15.9493131697291908$$15.9493131697291908$, stay verified and are not selected. T-174 at $15.9484433273005000$$15.9484433273005000$, 1.49×10−31$1.49\times {10}^{-31}$ 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.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s exact re-solve of Chaoweeraprasit’s SQUISH packing composed from SQUISH’s n = 209 packing as Gupta refined it and its own n = 240 packing, reported in pull request #574, at an exact rational side, displayed upward as $15.94844332730049991$$15.94844332730049991$ (T-174). The certificate gives a rational side and, for each of the 239 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. The source ran both maintained routes itself on its corners witnesses; those runs are its own, and the replay here is independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(239)\le 15.94844332730049991$$s(239)\le 15.94844332730049991$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is PingYou Ltd’s tightening of the packing T-151 records, measured here pose by pose as that packing (T-174’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The source makes no statement about AI assistance in this pull request; for its #503 and #504 certificates the author said on 12 October 2026 that they were AI assisted by ‘PY’, an unreleased agentic AI offering by PingYou Ltd.

The house it replaces, $15.9484433273005000$$15.9484433273005000$ from Chaoweeraprasit’s packing (T-151), stays retained with its own replay as history. The other certificates confirmed here at this count, T-127 at $15.9493131697276962$$15.9493131697276962$ and T-115 at $15.9493131697291908$$15.9493131697291908$, stay verified and are not selected.