$n=131$$n=131$

$s(131)$$s(131)$ — 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=131$$n=131$, $k=11$$k=11$ and interval 27 give $a(131)=11.545544735188575285759371528017\dots$$a(131)=11.545544735188575285759371528017\dots$, so the reported lower bound here is now $s(131)\ge 11.545544735188$$s(131)\ge 11.545544735188$, rounded down, above Nagamochi’s closed form, $11.4880884817$$11.4880884817$. 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 tighter certificate of Chaoweeraprasit’s SQUISH packing (T-174) is now this case’s house: its exact certificate proves $s(131)\le 11.949659588035860397$$s(131)\le 11.949659588035860397$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.09×10−31$1.09\times {10}^{-31}$. The earlier ceiling, displayed outward as $11.9496595880358604$$11.9496595880358604$ from Chaoweeraprasit’s packing (T-136), applied until this adoption and stays retained as history (the certificate).

House adoption, 11 October 2026. Chaoweeraprasit’s SQUISH packing (T-136) was this case’s house until 17 October 2026: its exact certificate proves $s(131)\le 11.9496595880358604$$s(131)\le 11.9496595880358604$, the reported and the verified upper bound, 1.49×10−3$1.49\times {10}^{-3}$ below the ceiling the case held until then, $11.9511500449119512$$11.9511500449119512$ from Xu’s packing (T-125), which stays retained as history (the certificate).

Previously, the best known published packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(131)\le 11.9549168302161244102\dots$$s(131)\le 11.9549168302161244102\dots$, and the strongest lower bound independently verified here is $11.465856$$11.465856$ from Karakuş’s general theorem, leaving a gap of $0.4891$$0.4891$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The earlier exact packing

Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing at its exact optimum (T-098): the same 131 squares in a square of side $11.9549168302161244102\dots$$11.9549168302161244102\dots$, 2.9×10−12$2.9\times {10}^{-12}$ below the side Couzo prints. Square for square, its pose lies within 4.2×10−12$4.2\times {10}^{-12}$ of the binary64 pose the atlas then pictured for this count. The known-best witness this record listed then was that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That previously proved $s(131)\le 11.9549168302161244102\dots$$s(131)\le 11.9549168302161244102\dots$, the verified upper bound at that intake; 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(131)$$s(131)$. 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 $11.954916830219048$$11.954916830219048$ for this count, dated 27 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side $11.956521791347278$$11.956521791347278$, 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 $11.9549168302190485663\dots$$11.9549168302190485663\dots$, 5.663×10−16$5.663\times {10}^{-16}$ above the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\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(131)\le 11.954916830219049$$s(131)\le 11.954916830219049$, 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 $11.95652190193193043$$11.95652190193193043$ for this count, larger than Couzo’s by $0.00160507171288243$$0.00160507171288243$. 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 $11.956521791347278$$11.956521791347278$, 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 UnitSquare Project’s, of side $11.956543108124773261501115489978287643498579738$$11.956543108124773261501115489978287643498579738$, below the Kingbird catalogue’s $11.95654869347733$$11.95654869347733$.

The UnitSquare Project’s 29 July 2026 release improves the public Ellsworth parent by $0.0000055853525567384986$$0.0000055853525567384986$. The release classifies the result as a construction-only upper bound and says it used outward-rounded interval arithmetic, a 300-digit zero-tolerance recomputation, and an independent published checker. The public release does not include the interval boxes, governed receipt, or replayable checker needed to inspect the formal claim. This repository recorded the value as reported. The formal lane held the exact $12\times 12$$12\times 12$ grid construction until this intake.

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 $119511500449119511500449/10000000000000000000000$$119511500449119511500449/10000000000000000000000$, displayed upward as $11.9511500449119512$$11.9511500449119512$. 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.

Nate Chaoweeraprasit’s certificate (T-136)

Chaoweeraprasit’s SQUISH packing, reported on jlevy/squares#481, at an exact rational side, displayed upward as $11.9496595880358604$$11.9496595880358604$ (T-136). The certificate gives a rational side and, for each of the 131 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(131)\le 11.9496595880358604$$s(131)\le 11.9496595880358604$, 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 under the author’s direction.

The house it replaces, $11.9511500449119512$$11.9511500449119512$ from Xu’s packing (T-125), stays retained with its own replay as history. The other certificates confirmed here at this count, T-128 at $11.9511053894146775$$11.9511053894146775$ and T-138 at $11.951105389418$$11.951105389418$, stay verified and are not selected. T-174 at $11.9496595880358604$$11.9496595880358604$, 1.09×10−31$1.09\times {10}^{-31}$ 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 tighter certificate of Chaoweeraprasit’s SQUISH packing, reported in pull request #574, at an exact rational side, displayed upward as $11.949659588035860397$$11.949659588035860397$ (T-174). The certificate gives a rational side and, for each of the 131 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(131)\le 11.949659588035860397$$s(131)\le 11.949659588035860397$, 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-136 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, $11.9496595880358604$$11.9496595880358604$ from Chaoweeraprasit’s packing (T-136), stays retained with its own replay as history. The other certificates confirmed here at this count, T-128 at $11.9511053894146775$$11.9511053894146775$, T-138 at $11.951105389418$$11.951105389418$ and T-125 at $11.9511500449119512$$11.9511500449119512$, stay verified and are not selected.