$n=209$$n=209$

s(209) — Square Packing Case

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=209$$n=209$, $k=14$$k=14$ and interval 27 give $a(209)=14.534080438075709123990508478096\dots$$a(209)=14.534080438075709123990508478096\dots$, so the reported lower bound here is now $s(209)\ge 14.534080438075$$s(209)\ge 14.534080438075$, rounded down, above Nagamochi’s closed form, $14.49073756323$$14.49073756323$. 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 refinement of Couzo’s refinement of Chaoweeraprasit’s SQUISH packing (T-174) is now this case’s house: its exact certificate proves $s(209)\le 14.94622322644645106$$s(209)\le 14.94622322644645106$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.49×10−19$1.49\times {10}^{-19}$. The earlier ceiling, displayed outward as $14.9462232264464511$$14.9462232264464511$ from Chaoweeraprasit’s packing, improved by Couzo (T-140), applied until this adoption and stays retained as history (the certificate).

House adoption, 11 October 2026. Couzo’s refinement of Chaoweeraprasit’s SQUISH packing (T-140) was this case’s house until 17 October 2026: its exact certificate proves $s(209)\le 14.9462232264464511$$s(209)\le 14.9462232264464511$, the reported and the verified upper bound, 3.39×10−3$3.39\times {10}^{-3}$ below the ceiling the case held until then, $14.9496179522003981$$14.9496179522003981$ from Chaoweeraprasit’s packing, improved by Gupta (T-127), which stays retained as history (the certificate).

The Upper Bound

Previously, Chaoweeraprasit (itsnaka), using SQUISH, reports s(209)≤Sn​$s(209)\le S_n$, with Sn​=3201017081708390588522178805764785398243001234645794991233471​$S_n=\frac{4785398243001234645794991233471}{320101708170839058852217880576}$ in SQUISH’s ten-packing release. The certificate gives rational centres and half-angle parameters for 209 unit squares. The source’s original finite decimal display is $14.9496179522017929$$14.9496179522017929$; the rational fraction states the claimed bound exactly.

The new packing improves the earlier reported side $14.953939011857122362887224108935$$14.953939011857122362887224108935$ from [evand exact optima 2026-10-05]. This repository confirmed the upper bound by independently re-implemented exact-rational checks: native exact_verify and devtools.check_rational_witness_independent accept every pair and wall of the new packing and reject both geometric controls. The verified display is $14.9496179522017921$$14.9496179522017921$, the exact source side rounded upward at sixteen decimals. Both accepted reviews cover the shared half-angle conversion and the complete roster. The submitted certificates establish feasible upper bounds and make no optimality or rigidity claim. Separately, this repository’s numerical translation screen finds a nontrivial feasible motion in the retained configuration. That numerical assessment concerns this configuration alone; it does not settle the optimum or all packings at this side.

The author credits Ellsworth’s records and tools and states that SQUISH’s solver, verification and issue were written with Claude Opus 5.5 under his direction, management, review and steering. The source packet retains derived packing facts and source metadata.

Earlier Packing

Couzo found the earlier packing; Daniel improved it. Its reported side was $14.953939011857122362887224108935$$14.953939011857122362887224108935$. The earlier certified ceiling $14.953939011857122362887224108935$$14.953939011857122362887224108935$ was the verified upper bound before the SQUISH certificate (T-098), supported by Daniel’s certificate packet. This repository replayed Daniel’s two source checkers and independently decided the rational certificate with its own exact-rational routes; the packet records the shared premises and receipts. Any optimality or rigidity observations about that packing concern its own pose.

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 $37374044880500995118216936421/2500000000000000000000000000$$37374044880500995118216936421/2500000000000000000000000000$, displayed upward as $14.9496179522003981$$14.9496179522003981$ (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.

Francisco Couzo’s certificate (T-140)

Couzo’s refinement of Chaoweeraprasit’s SQUISH packing, reported on jlevy/squares#488, at exact side $14946223226446451055256712327571/1000000000000000000000000000000$$14946223226446451055256712327571/1000000000000000000000000000000$, displayed upward as $14.9462232264464511$$14.9462232264464511$ (T-140). The certificate gives a rational side and, for each of the 209 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. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The review of 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(209)\le 14.9462232264464511$$s(209)\le 14.9462232264464511$, the verified upper bound until 17 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The author says the search code and the request were written with Claude under his direction.

The house it replaces, $14.9496179522003981$$14.9496179522003981$ from Chaoweeraprasit’s packing, improved by Gupta (T-127), stays retained with its own replay as history. The other certificates confirmed here at this count, T-136 at $14.9462236544879195$$14.9462236544879195$ and T-113 at $14.9496179522017921$$14.9496179522017921$, stay verified and are not selected. T-174 at $14.9462232264464511$$14.9462232264464511$, 1.49×10−19$1.49\times {10}^{-19}$ 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.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s refinement of Couzo’s refinement of Chaoweeraprasit’s SQUISH packing, reported in pull request #574, at an exact rational side, displayed upward as $14.94622322644645106$$14.94622322644645106$ (T-174). The certificate gives a rational side and, for each of the 209 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(209)\le 14.94622322644645106$$s(209)\le 14.94622322644645106$, 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-140 records, that packing by the source’s statement, not decided here (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, $14.9462232264464511$$14.9462232264464511$ from Chaoweeraprasit’s packing, improved by Couzo (T-140), stays retained with its own replay as history. The other certificates confirmed here at this count, T-136 at $14.9462236544879195$$14.9462236544879195$, T-127 at $14.9496179522003981$$14.9496179522003981$ and T-113 at $14.9496179522017921$$14.9496179522017921$, stay verified and are not selected.