$n=260$$n=260$

$s(260)$$s(260)$ — open

House adoption, 22 October 2026. Daniel’s display regularization of Couzo’s packing from a search seeded with Daniel’s exact certificate of de Winter’s n = 260 packing (T-225) is now this case’s house: its exact certificate proves $s(260)\le 16.6457532505198122$$s(260)\le 16.6457532505198122$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 2.51×10−11$2.51\times {10}^{-11}$. The earlier ceiling, displayed outward as $16.6457532505449560$$16.6457532505449560$ from Couzo’s packing (T-223), applied until this adoption and stays retained as history (the certificate).

House adoption, 21 October 2026. Couzo’s packing from a search seeded with Daniel’s exact certificate of de Winter’s n = 260 packing (T-223) was this case’s house until 22 October 2026: its exact certificate proves $s(260)\le 16.6457532505449560$$s(260)\le 16.6457532505449560$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 2.56×10−5$2.56\times {10}^{-5}$. The earlier ceiling, displayed outward as $16.6457788041813798$$16.6457788041813798$ from de Winter’s packing (T-196), applied until this adoption and stays retained as history (the certificate).

Reported certificate, 2026-10-18. Couzo reported a 260-square certificate at container side $16.645753250545$$16.645753250545$, rounded upward from the retained exact certificate, in issue #678. It reached this repository after de Winter’s certificate, which the source credits for the construction, and its side is 2.56×10−5$2.56\times {10}^{-5}$ below his. Before its review, the exact rational side was registered at V0/C0 in the result register. Until the adoption of 21 October 2026, both case upper lanes, the house best packing (de Winter’s T-196) and its drawing stayed as they were, and the rigidity and arrangement notes below described that packing.

House adoption, 20 October 2026. de Winter’s packing (T-196) was this case’s house until 21 October 2026: its exact certificate proves $s(260)\le 16.6457788041813798$$s(260)\le 16.6457788041813798$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.11×10−2$1.11\times {10}^{-2}$. The earlier exact ceiling, $416421356237309504880168872421/25000000000000000000000000000$$416421356237309504880168872421/25000000000000000000000000000$ from Ellsworth’s packing (T-155), applied until this adoption and stays retained as history; the reported upper bound it replaces, $16.65685424949238$$16.65685424949238$ from [Kingbird], lies 1.11×10−2$1.11\times {10}^{-2}$ above the new side (the certificate).

Reported certificate, 2026-10-17. de Winter reported a 260-square certificate at container side $16.645778804182$$16.645778804182$, rounded upward from the retained exact certificate. The exact rational side is registered in the result register. Its V0/C0 registration left both case upper lanes, the house witness and its drawing unchanged until the house adoption of 20 October 2026, and until then the rigidity and arrangement notes below described that retained house. On 20 October 2026 both of this project’s maintained exact routes decided the certificate and refused its two controls, which raised T-196 to V3/C3, independently re-implemented: no route is the source’s code. Couzo’s later #678 reported a smaller side here.

Exact regularized certificate, 2026-10-12. This project’s regularization pass, Daniel’s exact solver and regularizer run here on this register’s witness, produced an exact rational packing of this arrangement, decided here over $\mathbb{Q}$$\mathbb{Q}$ by two exact checkers: it proves $s(260)\le 16.65685424949238019520675489684$$s(260)\le 16.65685424949238019520675489684$, the verified upper bound until 20 October 2026 (T-155), which the record printed as $16.65685424949239$$16.65685424949239$. Until then the verified upper bound was exactly $16.65685424949239$$16.65685424949239$, the side of Daniel’s exact certificate of this packing rounded up at the printed precision (T-101).

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}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(260)\le 16.65685424949239$$s(260)\le 16.65685424949239$, the verified upper bound from 6 to 12 October 2026 (T-101). Until then the verified upper bound was the trivial grid bound $17$$17$.

Open. Previously, the best known packing gave $s(260)\le 16.65685425$$s(260)\le 16.65685425$, and the strongest lower bound independently verified here is $16.128499$$16.128499$ from Karakuş’s general theorem, leaving a gap of $0.5284$$0.5284$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The earlier verified ceiling

Until 20 October 2026 the verified upper bound here was $16.65685424949238019520675489684$$16.65685424949238019520675489684$, exactly, proved by the exact regularized certificate of this packing (E-regularized-bounds-2026-10-12-exact-decision, T-155), and the record prints it as $16.65685424949239$$16.65685424949239$. As printed it was larger than the catalogue’s printed side $16.65685424949238$$16.65685424949238$, then the best known, by 1×10−14$1\times {10}^{-14}$.

It was not the value of $s(260)$$s(260)$ and not a different packing: it is the certificate’s own exact side. The catalogue gives the packing’s side exactly, as 11+42​$11+4\sqrt{2}$, and the certificate’s side lies 1.2×10−30$1.2\times {10}^{-30}$ above that, so it bounds the exact side from above and does not reach it. That side is irrational, so no rational certificate reaches it. A mathematics blocker in the frontmatter recorded the difference until de Winter’s certificate (T-196) replaced both lanes.

The regularized exact certificate

This project’s regularization pass of 11 and 12 October 2026 ran Daniel’s exact solver and regularizer (evand/square-packing, MIT, vendored here) on this register’s witness for this count, with the conservative angle policy and coincide placement, and this project’s same-side re-solve then fixed the result’s tight contacts over $\mathbb{Q}$$\mathbb{Q}$ at the witness’s own side. The result is an exact rational packing of the same 260 squares in a square of side $16.65685424949238019520675489684$$16.65685424949238019520675489684$, 9.8×10−15$9.8\times {10}^{-15}$ below the verified upper bound the record held before, and 1.7×10−19$1.7\times {10}^{-19}$ below the side of Daniel’s certificate of this packing (T-101): it is the witness’s own side. Measured against the witness under the box’s eight symmetries, its squares pair one to one with the witness’s; none moves by more than 1×10−3$1\times {10}^{-3}$, and none turns by more than 1×10−6$1\times {10}^{-6}$ as a sine. It draws the same picture as the witness, so the atlas drawing does not change.

Each pair and each wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(260)\le 16.65685424949238019520675489684$$s(260)\le 16.65685424949238019520675489684$, the verified upper bound until 20 October 2026 (T-155), then the exact form of the case’s verified_upper_bound, which printed it as $16.65685424949239$$16.65685424949239$, the decimal it printed before and an upper display of it. It says nothing about optimality, and the packing stays its finders’.

The earlier exact certificate

Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register lists 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 260 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $16.6568542494923801954\dots$$16.6568542494923801954\dots$, 2.0×10−16$2.0\times {10}^{-16}$ above the side the Kingbird catalogue prints, $16.65685424949238$$16.65685424949238$. Square for square, its pose lies within 1.5×10−4$1.5\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 46 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 2.6×10−14$2.6\times {10}^{-14}$. 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. At this count it started from the witness after the source’s own local squeeze, which it has not published; that bears on reproducing the solve and not on checking the certificate.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(260)\le 16.65685424949239$$s(260)\le 16.65685424949239$, the verified upper bound until 12 October 2026: 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 Ellsworth in 2024. The recorded construction method is an 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.

The reported lower-bound expression for $s(260)$$s(260)$ is 5742​/257+3343/257$574\sqrt{2}/257+3343/257$ (approximately $16.16637581635$$16.16637581635$). Friedman’s DS7 survey, Theorem 9, k=16, reports this bound at n=257; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at k=16 that point pattern leaves a unit square empty (devtools.check_green_ds7). Inherited at n=260 by monotonicity. This is a source-reported lower bound; it does not replace the verified bound above. The exact specialization and comparison are retained by audit_ds7_lower_bounds.

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.

Joost de Winter’s certificate (T-196)

de Winter’s packing, reported on jlevy/squares#668, at exact side $16645778804181379786535601/1000000000000000000000000$$16645778804181379786535601/1000000000000000000000000$, displayed upward as $16.6457788041813798$$16.6457788041813798$ (T-196). The certificate gives a rational side and, for each of the 260 unit squares, a rational centre and the cosine and sine of its rotation; 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 20 October 2026 accepted the replay for finite feasibility. That proves $s(260)\le 16.6457788041813798$$s(260)\le 16.6457788041813798$, the verified upper bound until 21 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The issue says the search, numerical optimization, rational certificates, checkers and figures were made with OpenAI Codex agents under de Winter’s direction, and that Claude (Anthropic) replayed the checkers and prepared the repository for publication.

The issue says the search started from this repository’s catalogue, whose credits for the former packings apply, and built asymmetric Young-corner fills around slipped tilted ribbons, then restored the supports and angles; the three counts are one construction family, and the 229-square packing descends from the 260-square one. None of this lineage is measured here (T-196).

The house it replaces, $416421356237309504880168872421/25000000000000000000000000000$$416421356237309504880168872421/25000000000000000000000000000$ from Ellsworth’s packing (T-155), stays retained with its own replay as history.

Francisco Couzo’s certificate (T-223)

Couzo’s packing from a search seeded with Daniel’s exact certificate of de Winter’s n = 260 packing, reported on jlevy/squares#678, at exact side $146417594016514621500994169/8796093022208000000000000$$146417594016514621500994169/8796093022208000000000000$, displayed upward as $16.6457532505449560$$16.6457532505449560$ (T-223). The certificate gives a rational side and, for each of the 260 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, sqpack’s predicates, is the corner verifier the source reports running, and the packet’s filtered replay reads the certificate once for both routes, through one parser, half-angle map and pair filter; the source’s own checker, run unchanged from its retained copy, accepted it too. The review of 20 October 2026 accepted the replay for finite feasibility. That proves $s(260)\le 16.6457532505449560$$s(260)\le 16.6457532505449560$, the verified upper bound until 22 October 2026: confirmed, re-implemented, sharing the producer’s components. It says nothing about optimality, rigidity or a local minimum.

The issue states nothing about AI assistance; the header of the source’s checker names Codex, and the author’s issues of 13 and 14 October 2026 disclose OpenAI Codex and Claude Code assistance.

The source’s manifest credits de Winter’s #668 packing as the seed at 260, through Daniel’s exact-form certificate of it in evand/square-packing, which the issue’s comment of 19 October names and the intake retained. Measured here under the box’s symmetries, 162 of the 260 squares lie within 10⁻⁴ of a square of that seed and 55 moved by more than 10⁻²: not a refinement of the seed in place (T-223).

The house it replaces, $16.6457788041813798$$16.6457788041813798$ from de Winter’s packing (T-196), stays retained with its own replay as history.

Evan Daniel’s certificate (T-225)

Daniel’s display regularization of Couzo’s packing from a search seeded with Daniel’s exact certificate of de Winter’s n = 260 packing, reported in his regularized record lists, on no issue, at exact side $4161438312629953048686574163329/250000000000000000000000000000$$4161438312629953048686574163329/250000000000000000000000000000$, displayed upward as $16.6457532505198122$$16.6457532505198122$ (T-225). The certificate gives a rational side and, for each of the 260 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 the 21 October review’s clean-room checker decided it again on two routes of its own, in integers alone and in rationals with exact intersection area on the pairs whose boxes overlap; its sharp controls at 10−30${10}^{-30}$ found both maintained routes exact at the far wall, the near wall and the least-gap pair. No route here is the source’s code, so the replay is independently re-implemented. The review of 21 October 2026 accepted the replay for finite feasibility. That proves $s(260)\le 16.6457532505198122$$s(260)\le 16.6457532505198122$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate regularizes for display the packing T-223 records, at the side the source reports as that packing’s exact optimum, so the packing stays with its finder and Daniel is credited with the regularization and the certificate. That the two are one packing is the source’s claim, measured here and not decided (T-225’s regularizations).

The source’s credits say the repository’s work was produced by Claude (Anthropic) under human direction, and the commit that publishes the certificate names Claude Opus 5.5 as co-author and says Daniel approved it.

The source names Couzo’s #678 packing (T-223) as the one it regularizes and calls the certificate not a new bound: Daniel solved that packing to its exact side, 2.51×10−11$2.51\times {10}^{-11}$ below #678’s certificate, and regularized it for display at that side. Measured by the import and again by the 21 October review, 207 of the 260 squares lie within 10−9${10}^{-9}$ of T-223’s and none turns; under no symmetry of the box do nearest centres pair the two one to one, and an optimal assignment slides 41 squares beyond 10−3${10}^{-3}$ and 20 beyond a tenth, at most 0.643 (T-225).

The house it replaces, $16.6457532505449560$$16.6457532505449560$ from Couzo’s packing (T-223), stays retained with its own replay as history. The other certificates confirmed here at this count, T-196 at $16.6457788041813798$$16.6457788041813798$, stay verified and are not selected.