$n=261$$n=261$

$s(261)$$s(261)$ — 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=261$$n=261$, $k=16$$k=16$ and interval 5 give $a(261)=16.192480533769752717717472451742\dots$$a(261)=16.192480533769752717717472451742\dots$, so the reported lower bound here is now $s(261)\ge 16.192480533769$$s(261)\ge 16.192480533769$, rounded down, above a smaller count’s bound by monotonicity, $16.16637581635$$16.16637581635$. 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 Barnett’s refinement of Xu’s packing (T-174) is now this case’s house: its exact certificate proves $s(261)\le 16.67877987534290982$$s(261)\le 16.67877987534290982$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.99×10−21$1.99\times {10}^{-21}$. The earlier ceiling, displayed outward as $16.6787798753429099$$16.6787798753429099$ from Xu’s packing, improved by Barnett and PingYou Ltd (T-169), applied until this adoption and stays retained as history (the certificate).

House adoption, 15 October 2026. PingYou Ltd’s exact re-solve of Barnett’s refinement of Xu’s packing (T-169) was this case’s house until 17 October 2026: its exact certificate proves $s(261)\le 16.6787798753429099$$s(261)\le 16.6787798753429099$, the reported and the verified upper bound, 2.98×10−13$2.98\times {10}^{-13}$ below the ceiling the case held until then, $16.6787798753432078$$16.6787798753432078$ from Xu’s packing, improved by Barnett (T-150), which stays retained as history (the certificate).

House adoption, 11 October 2026. Barnett’s refinement of Xu’s packing (T-150) was this case’s house until 15 October 2026: its exact certificate proves $s(261)\le 16.6787798753432078$$s(261)\le 16.6787798753432078$, the reported and the verified upper bound, 7.35×10−11$7.35\times {10}^{-11}$ below the ceiling the case held until then, $16.6787798754166788$$16.6787798754166788$ from Xu’s packing (T-125), which stays retained as history (the certificate).

Exact certificate, 2026-10-06. Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing, decided here over $\mathbb{Q}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it previously proved $s(261)\le 16.68292682926830$$s(261)\le 16.68292682926830$, the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound $17$$17$.

Previously, the packing gave $s(261)\le 16.68292683$$s(261)\le 16.68292683$, and the strongest lower bound independently verified here is $16.160459$$16.160459$ from Karakuş’s general theorem, leaving a gap of $0.5225$$0.5225$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The earlier verified ceiling

The verified ceiling at that intake was $16.68292682926830$$16.68292682926830$, proved by Daniel’s exact rational certificate of this packing (E-evand-exact-ceilings-2026-10-05-exact-replay, T-101). It was larger than the then-reported $16.68292682926829$$16.68292682926829$, by 1×10−14$1\times {10}^{-14}$.

It is not the value of $s(261)$$s(261)$ and not a different packing: it is the certificate’s own side, $16.6829268292682926831\dots$$16.6829268292682926831\dots$, rounded up at the fourteen decimals the catalogue prints. The catalogue gives the packing’s side exactly, as $16+(28/41)$$16+(28/41)$, and the certificate’s side lies 1.7×10−19$1.7\times {10}^{-19}$ above that: its squares are rational and each is kept clear of its neighbours and the walls, so it bounds the exact side from above and does not reach it. That side is rational, so an exact certificate of the packing at that side would reach it, a rational one where the packing’s exact point is rational. That certificate gap was the upper-bound blocker at that intake. Read reported_upper_bound for the best known side length.

The exact certificate

Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register then listed as best known, solved from its own witnesses, and offered those that do not lower a printed side as a replay of the existing bounds, asking for nothing to be registered from them. The certificate for this count holds the same 261 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $16.6829268292682926831\dots$$16.6829268292682926831\dots$, 2.7×10−15$2.7\times {10}^{-15}$ above the side the Kingbird catalogue prints, $16.68292682926829$$16.68292682926829$. Square for square, its pose lies within 2.9×10−4$2.9\times {10}^{-4}$ of the binary64 pose the atlas then pictured for this count. 6 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’s centre rounds to the witness’s. 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.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That previously proved $s(261)\le 16.68292682926830$$s(261)\le 16.68292682926830$, the verified upper bound at that intake: the certificate’s side rounded up at the fourteen decimals the catalogue prints, as the record writes any certificate of a printed side. It says nothing about optimality. 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

Found by an unrecorded author, via extension of a smaller record.

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 $83393899377083393899377/5000000000000000000000$$83393899377083393899377/5000000000000000000000$, displayed upward as $16.6787798754166788$$16.6787798754166788$. 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.

Robert Barnett’s certificate (T-150)

Barnett’s refinement of Xu’s packing, reported on jlevy/squares#505, at an exact rational side, displayed upward as $16.6787798753432078$$16.6787798753432078$ (T-150). The certificate gives a rational side and, for each of the 261 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(261)\le 16.6787798753432078$$s(261)\le 16.6787798753432078$, the verified upper bound until 15 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The author discloses OpenAI Codex assistance with the solver, the certificate and checker code, source review and preparation.

The house it replaces, $16.6787798754166788$$16.6787798754166788$ from Xu’s packing (T-125), stays retained with its own replay as history.

PingYou Ltd’s certificate (T-169)

PingYou Ltd’s exact re-solve of Barnett’s refinement of Xu’s packing, reported on jlevy/squares#567, at an exact rational side, displayed upward as $16.6787798753429099$$16.6787798753429099$ (T-169). The certificate gives a rational side and, for each of the 261 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(261)\le 16.6787798753429099$$s(261)\le 16.6787798753429099$, the verified upper bound until 17 October 2026: 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-150 records, measured here pose by pose as that packing (T-169’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The pinned source makes no statement about AI assistance.

The house it replaces, $16.6787798753432078$$16.6787798753432078$ from Xu’s packing, improved by Barnett (T-150), stays retained with its own replay as history. The other certificates confirmed here at this count, T-125 at $16.6787798754166788$$16.6787798754166788$, stay verified and are not selected. T-174 at $16.6787798753429099$$16.6787798753429099$, 1.99×10−21$1.99\times {10}^{-21}$ 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 Barnett’s refinement of Xu’s packing, reported in pull request #574, at an exact rational side, displayed upward as $16.67877987534290982$$16.67877987534290982$ (T-174). The certificate gives a rational side and, for each of the 261 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(261)\le 16.67877987534290982$$s(261)\le 16.67877987534290982$, 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-169 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, $16.6787798753429099$$16.6787798753429099$ from Xu’s packing, improved by Barnett and PingYou Ltd (T-169), stays retained with its own replay as history. The other certificates confirmed here at this count, T-150 at $16.6787798753432078$$16.6787798753432078$ and T-125 at $16.6787798754166788$$16.6787798754166788$, stay verified and are not selected.