$n=241$$n=241$

Bounds

Results in the register

• T-007 S3 V0 C1 Nagamochi · 2026-08-31 · 321 cases

  s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for $4\le n\le 324$$4\le n\le 324$

• T-056 S3 V3 C3 Couzo · 2026-09-29 · 49 cases

  Smaller packings for 49 counts from $n=68$$n=68$ to $307$$307$, each certified two independent ways

• T-083 S3 V3 C3 Karakuş · 2026-10-02 · 301 cases

  s(n)≥1/2+n−⌊n​⌋+1/4​$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare $8\le n\le 324$$8\le n\le 324$

• T-085 S3 V3 C3 Karakuş; Vlasenko · 2026-10-02 · 315 cases

  Nagamochi 2005, Lemma 1 is false for every container with $a>3$$a>3$ and $b>2$$b>2$

• T-098 S2 V3 C3 Daniel after Couzo, de Winter, Ellsworth, Levy · 2026-10-05 · 48 cases

  Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

• T-163 S4 V2 C1 Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · 2026-10-14 · n = 211, 241, 273, 307

  s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang

  Claim and records

  Claim

  Fang states two theorems of a note, draft v0.6 of 13 October 2026, which is not in his repository: Theorem 1, s(n^2 - n + 1) < n for every integer n >= 15; and Theorem 2, s(n^2 - n + F(n)) < n with F(n) = 1 + max{p + 2q : 16p + 21q <= n - 15}. The proof glues a squeezable 17 x 16 rectangle of 257 squares, or a 22 x 21 rectangle of 443 that adds two, onto a square packing (Lemma 3, tools/glue.py) and inducts on n from finitely many certificates; the construction gives side n - 1/20000 at every n >= 15.
  
  Inside the case corpus the theorem's inventory certifies $s(211)\le 14.99995$$s(211)\le 14.99995$, $s(241)\le 15.99995$$s(241)\le 15.99995$, $s(273)\le 16.99995$$s(273)\le 16.99995$ and $s(307)\le 17.99995$$s(307)\le 17.99995$, the counts n^2 - n + F(n) for $n=15$$n=15$ to 18. Each is above its case's bound (T-149, T-098, T-098, T-136), so these four finite bounds are superseded; nothing supersedes the theorem itself, which no other result implies at any n.
  
  The 14 October review re-derived both theorems from the retained material and decided every finite piece the chains use, and rebuilt all 76 inventory certificates byte for byte; both maintained exact routes decided the six retained ones, at 211, 241, 273, 307, 308 and 932, with their controls. The note itself has not been read.
  
  Credit the theorem and its certificates to Fang (https://github.com/TheSnakeFang/squarepack-certs). The gluing lemma is Arslanov, Mustafin and Shangitbayev's decomposition (Electronic Journal of Combinatorics 28(4), P4.22, 2021); the base packings are de Winter's 211, Ellsworth's and Stead's 241 and 273, the AMS21 and Ellsworth packings of m^2 - m squares, and the author's own; the rectangles grow from Ellsworth's, Stead's and Couzo's packings; the certificate format and the pieces' certificates are Daniel's. The search, the verifier and the bundle were written and run with Claude (Anthropic), directed and reviewed by the author.

  Significance

  A bound family for every n >= 15 with an explicit construction: the deficiency bound c*(k) <= k - 1 - F(k) improves Arslanov, Mustafin and Shangitbayev's k - 1 at every k >= 15 and Ryu's at 15 <= k <= 78 and at k = 86; it settles no case n <= 324, and its four in-corpus finite bounds trail the cases.

  Novelty

  previously-published Present in an identified source

  Records

  n = 211n = 241n = 273n = 307registerevidence 1evidence 2

  n = 211n = 241n = 273n = 307registerevidence 1evidence 2

• T-174 S1 V3 C3 PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · 2026-10-14 · 68 cases

  Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

• T-175 S3 V3 C3 Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · 2026-10-14 · n = 211, 241, 304

  New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly

  Claim and records

  Claim

  Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(211)\le 149936941865832465706359181817612514235389167555807511399813149936941865832465706359181817612514235389167555807511399813/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$, $s(241)\le 9991958905656775259135410873409821230409663792812755550161009991958905656775259135410873409821230409663792812755550161/625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ and $s(304)\le 358588555161503499313326522779782297248782400899192783432303585885551615034993133265227797822972487824008991927834323/20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$.
  
  All 12 retained jobs, four certificates with their duplicate-square and outside-container controls, were decided again here on 14 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the three certificates this entry cites with six controls each and held each retained file to its upstream bytes, fetched at the pin. No route here is the source's code, and the source ran none of them, so the replay is independently re-implemented. The 14 October review found no blocking defect.
  
  The author reports that each packing passes its own C++ checker (sqverify --precise --certify, MPFR then GMP rationals) at delta = 1e-60, that sqverify --check-cert certifies the shipped certificates in GMP rationals with no dilation and no tolerance, and that a separately written 80-digit mpmath cross-check finds least separations of 5e-21 at 211 and 241 and 4.995e-21 at 176 and 304; the issue states that all three checkers are the author's, written with AI assistance, the first two sharing the C++ parser and geometry.
  
  It also states that each side is the exact first-order optimum of its contact topology, with nonnegative KKT multipliers at a 60-digit solution reached by two routes that agree, and the load-bearing squares scaled by 1 + 1e-20. Those statements are the author's; no source program ran here.
  
  Credit Ostashev (https://github.com/dostashev/square-packing) with the packings and their certificates, which the issue says are new arrangements.
  
  At 176, 211 and 241 a packing whose tilted squares form two bands was re-hardened on a GPU with every square soft and the box held (soft-to-rigid, after Nakajima's idea), and the two bands merged into one V: at 176 seeded from the author's earlier composition of issue 528's 38-square piece and issue 562's 44-square piece, at 211 from Daniel's exact optimum of de Winter's packing, and at 241 from Daniel's exact optimum of Couzo's packing, the case's house. At 304 the same re-hardening relaxed the author's composition of a 55-square (h + 8) x h piece and a 68-square (h + 7) x h piece without forming a V. Each was solved to the exact optimum of its contact topology.
  
  The 211 is a different packing from the author's T-149, which it replaces in his repository. The author asks that the seeds' authors (Couzo, Daniel, de Winter) and the composition pieces' sources be acknowledged.
  
  The issue states AI assistance: an AI-assisted research loop the author drives with his own ideas, merging the bands at the junction being his; AI agents wrote and ran the search and verification code, and an agent extended the move from his compositions to published two-band packings.
  
  Measured pose by pose by that review, the 211 and 304 are new arrangements and the 241 rebuilds the seed's design, every square paired and 15 moved beyond a tenth. On 15 October 2026 the case record at 304 adopted this entry as its house, its reported and verified upper bounds moving together and Couzo's certificate (T-152) kept as history; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). At 211 and 241 Couzo's issue 586 (T-182) is smaller and holds the cases.

  Significance

  Three new arrangements, 1.38e-04 to 2.64e-03 below the case ceilings, as T-158 is; no lower bound or optimum.

  Novelty

  previously-published Present in an identified source

  Records

  n = 211n = 241n = 304registerevidence 1evidence 2

  n = 211n = 241n = 304registerevidence 1evidence 2

• T-180 S4 V0 C1 Hosono · 2026-10-14 · 264 cases

  Closed-form lower bound $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, reported

• T-181 S3 V3 C3 Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · 2026-10-14 · n = 211, 241, 302

  Exact rational certificates at 211, 241 and 302 from Couzo

  Claim and records

  Claim

  Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(211)\le 5735936523425659529936799651/382560041700500000000000000$, $s(241)\le 56654293892568867634658195801/3543746727775500000000000000$ and $s(302)\le 17.870013850404464344638853539285$$s(302)\le 17.870013850404464344638853539285$.
  
  The same release states $s(339)\le 4247372425158980137376328899/224403568199500000000000000$, $s(379)\le 19.953531531602552149382778262085$$s(379)\le 19.953531531602552149382778262085$, $s(380)\le 71405320581108789876116351057/3574993313028500000000000000$, $s(381)\le 19.974977397656713743580918961689$$s(381)\le 19.974977397656713743580918961689$ and $s(382)\le 94198950843216391147752843823/4712633609411500000000000000$, beyond the case corpus; the source register tracks those as beyond-horizon rows.
  
  All 21 retained jobs of the issue-583 packet and all 42 of the issue-586 packet, each certificate with its duplicate-square and outside-container controls, were decided again here on 14 and 15 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the three certificates this entry cites with six controls each and held each retained file to its upstream bytes, fetched at the pin. No route here is the source's code; the author's runs of this repository's sqpack verifier are his and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.
  
  The certificates at 211, 241, 339, 379, 380 and 382 are those at the revision of franciscouzo/square-packing the issue pins, which issue 586's revision replaced; the couzo-issue583-2026-10-13 packet retains them. The 302 and 381 are unchanged at issue 586's revision, where the couzo-certificates-2026-10-13 packet retains them.
  
  The author reports that every certificate of issue 583 passes this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, in exact rational arithmetic; that 154, 211, 241, 302 and 381 are exact optima from Daniel's exact contact solver; and that 339, 379, 380 and 382 are rational witnesses of their decimal packings. Issue 570's 154 is this revision's n154.cert, byte for byte. Those statements are the author's; no source program ran here.
  
  Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Daniel's 154 arrangement of issue 552, continued through the author's browser refinement, native basin hopping and exact finishing; Ostashev's 211 and 241 arrangements of issue 575, continued with native search and refinement.
  
  Beyond the case corpus, 339 and 379 continue the author's L-border constructions of issue 506, and 380 and 382 continue Fang's published arrangements (TheSnakeFang/squarepack-certs). Daniel's optimizer, exact contact solver and verifier, Ellsworth's search and refinement ideas, and this project's sqpack verifier were used.
  
  The author states that OpenAI Codex assisted the search implementation, the verification orchestration and the requests, that earlier search work used Claude Code, and that he directed the work.
  
  Measured pose by pose by that review, the 211 and 241 continue Ostashev's packings of issue 575 (T-175) and the 302 continues PingYou Ltd's arrangement (T-164), each paired square for square. On 15 October 2026 the case record at 302 adopted the 302, as the issue-586 packet retains it, as its house, its reported and verified upper bounds moving together and Daniel's regularization of Chaoweeraprasit's packing (T-146) kept as history, with T-164 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). At 211 and 241 issue 586's certificates (T-182) are smaller and hold the cases.

  Significance

  Continuations of earlier packings at $n=211$$n=211$, 241 and 302, 1.02e-03 to 2.78e-03 below the case ceilings, as T-158 is; no lower bound or optimum.

  Novelty

  previously-published Present in an identified source

  Records

  n = 211n = 241n = 302registerevidence 1evidence 2evidence 3evidence 4

  n = 211n = 241n = 302registerevidence 1evidence 2evidence 3evidence 4

• T-182 S3 V3 C3 Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · 2026-10-14 · n = 154, 211, 241

  Exact rational certificates at 154, 211 and 241 from Couzo

  Claim and records

  Claim

  Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(154)\le 12.921206577132631050071241337979$$s(154)\le 12.921206577132631050071241337979$, $s(211)\le 14.993558913974006171333205657646$$s(211)\le 14.993558913974006171333205657646$ and $s(241)\le 15.987117095150238758321758129189$$s(241)\le 15.987117095150238758321758129189$.
  
  The same release states $s(339)\le 7843211074020413644369172117/414429522933500000000000000$, $s(379)\le 3465647601101642309045474807/173685927478500000000000000$, $s(380)\le 10836602188256857315809318661/542872123174250000000000000$ and $s(382)\le 16910059023533815728370360359/846022698554500000000000000$, beyond the case corpus; the source register tracks those as beyond-horizon rows.
  
  All 42 retained jobs of the packet, its fourteen certificates with their duplicate-square and outside-container controls, were decided again here on 14 and 15 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the three certificates this entry cites with six controls each and held each retained file to its upstream bytes, fetched at the pin. No route here is the source's code; the author's runs of this repository's sqpack verifier are his and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.
  
  The author reports that every certificate of issues 583 and 586 passes this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, in exact rational arithmetic; that 154, 211, 241, 302 and 381 are exact optima from Daniel's exact contact solver; and that 339, 379, 380 and 382 are rational witnesses of their decimal packings with centres dilated by 1 + 1e-11 at 339 and 1 + 1e-12 at the others. Issue 586 updates issue 583 at 154, 339, 379, 380 and 382 and tightens the certificates of the 211 and 241 arrangements, which the record lists as regularizations of issue 583's (T-181). Those statements are the author's; no source program ran here.
  
  Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Daniel's 154 arrangement of issue 552, itself descended from the author's earlier 155 with one square removed, continued through the author's browser refinement, native basin hopping and exact finishing; Ostashev's 211 and 241 arrangements of issue 575, continued with native search and refinement.
  
  Beyond the case corpus, 339 and 379 continue the author's L-border constructions of issue 506 (339 last through the browser's fq search, 379 through native SLP refinement), and 380 and 382 continue Fang's published arrangements (TheSnakeFang/squarepack-certs) through browser and native searches. Daniel's optimizer, exact contact solver and verifier, Ellsworth's search and refinement ideas, and this project's sqpack verifier were used.
  
  The author states that OpenAI Codex assisted the search implementation, the verification orchestration and the requests, that earlier search work used Claude Code, and that he directed the work.
  
  Measured pose by pose by that review, the 211 and 241 are issue 583's packings (T-181), no square moved beyond 1e-05, and the 154 moves issue 583's design well beyond a refinement, 19 squares beyond a tenth. On 15 October 2026 the case records adopted this entry as the house at 154, 211 and 241, each case's reported and verified upper bounds moving together and the earlier houses kept as history, Daniel's at 154 (T-160), Ostashev's at 211 (T-149) and Daniel's exact optimum of Couzo's packing at 241 (T-098), with T-172, T-175 and T-181 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).
  
  Polkovskyi's issue 593 states the same poses at 211 and 241 and asks to be recorded jointly; the owner decides that credit (RQ-1).

  Significance

  Continuations at $n=211$$n=211$ and 241 and a further arrangement at 154, 1.02e-03 to 2.78e-03 below the case ceilings, as T-158 is; no lower bound or optimum.

  Novelty

  previously-published Present in an identified source

  Records

  n = 154n = 211n = 241registerevidence 1evidence 2

  n = 154n = 211n = 241registerevidence 1evidence 2

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