$n=126$$n=126$

s(126) — 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=126$$n=126$, $k=11$$k=11$ and interval 6 give $a(126)=11.313323850997974097849202861841\dots$$a(126)=11.313323850997974097849202861841\dots$, so the reported lower bound here is now $s(126)\ge 11.313323850997$$s(126)\ge 11.313323850997$, rounded down, above Nagamochi’s closed form, $11.24695076595$$11.24695076595$. 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, 11 October 2026. Daniel’s reconstruction and exact certificate of Xu’s arrangement (T-139) is now this case’s house: its exact certificate proves $s(126)\le 11.7426406871192852$$s(126)\le 11.7426406871192852$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 9.25×10−11$9.25\times {10}^{-11}$. The earlier ceiling, displayed outward as $11.7426406872117427$$11.7426406872117427$ from Xu’s packing (T-125), applied until this adoption and stays retained as history (the certificate).

The SQUISH Update

Previously, nate Chaoweeraprasit, using SQUISH, reports a tighter bound $s(126)\le 11.7733036066072403$$s(126)\le 11.7733036066072403$, with exact side $3196078786847066046655612475207/271468306062659398105946914816$$3196078786847066046655612475207/271468306062659398105946914816$. 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(105), grafted into n’s Kingbird record. The revision-specific packet retains the source print $11.7733036066072394$$11.7733036066072394$ 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.

Separately, this repository’s numerical translation screen finds a nontrivial feasible motion in the current update geometry drawn by the atlas. Previously, that numerical assessment concerns this update pose alone; it does not settle the optimum or describe the earlier certified pose.

The Earlier Verified Upper Bound

Previously, nate Chaoweeraprasit (itsnaka), using SQUISH, reports s(126)≤Sn​$s(126)\le S_n$, with Sn​=4101530661069023219504623124484828972106460286237253237344483​$S_n=\frac{4828972106460286237253237344483}{410153066106902321950462312448}$ in SQUISH’s ten-packing release. The certificate gives rational centres and half-angle parameters for 126 unit squares. The source’s original finite decimal display is $11.7735852916961079$$11.7735852916961079$; the rational fraction states the claimed bound exactly.

The earlier SQUISH packing improves the earlier reported side $11.774735132387832842560264587322$$11.774735132387832842560264587322$ from [evand exact optima 2026-10-05]. This repository confirmed the upper bound by independently re-implemented exact-rational checks: native exact_verify and devtools.check_rational_witness_independent accept every pair and wall of the earlier packing and reject both geometric controls. The verified display is $11.7735852916961071$$11.7735852916961071$, 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.

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

de Winter found the earlier packing; Daniel improved it. Its reported side was $11.774735132387832842560264587322$$11.774735132387832842560264587322$. The earlier certified ceiling $11.774735132387832842560264587322$$11.774735132387832842560264587322$ 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 Xu construction

Xu reports (T-125) a construction at exact side $14678300859014678300859/1250000000000000000000$$14678300859014678300859/1250000000000000000000$, displayed upward as $11.7426406872117427$$11.7426406872117427$. The pinned factual packet retains the entire source input and complete actual deciding outcomes.

The complete rational source certificate already includes the source’s stated dilation. No additional dilation or coordinate rounding is applied here.

Complete native replay, required full-roster controls and scoped independent review confirm finite feasibility at V3/C3. The source discloses LLM assistance. No global optimum, local-minimum theorem, rigidity or novelty is established. The source’s 191-touching-pair assertion for n51 is unconfirmed and is not admitted.

Evan Daniel’s certificate (T-139)

Daniel’s reconstruction and exact certificate of Xu’s arrangement, reported in his repository’s search/trio126 directory, on no issue, at exact side $11742640687119285146522492579501/1000000000000000000000000000000$$11742640687119285146522492579501/1000000000000000000000000000000$, displayed upward as $11.7426406871192852$$11.7426406871192852$ (T-139). The certificate gives a rational side and, for each of the 126 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 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(126)\le 11.7426406871192852$$s(126)\le 11.7426406871192852$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source’s commit names Claude Opus 5.5 as co-author.

The house it replaces, $11.7426406872117427$$11.7426406872117427$ from Xu’s packing (T-125), stays retained with its own replay as history. The other certificates confirmed here at this count, T-115 at $11.7733036066072403$$11.7733036066072403$ and T-113 at $11.7735852916961071$$11.7735852916961071$, stay verified and are not selected. T-174 at $11.7426406871192852$$11.7426406871192852$, 1.17×10−19$1.17\times {10}^{-19}$ 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-2 integer polynomial 1.17×10−19$1.17\times {10}^{-19}$ below this house, is confirmed here and accepted by the review of 16 October 2026.