$n=153$$n=153$

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-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-101 S2 V3 C3 Daniel after Couzo, de Winter, Ellsworth, Levy · 2026-10-06 · 77 cases

  Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

• T-114 S3 V3 C3 Chaoweeraprasit after Ellsworth · 2026-10-07 · n = 153

  $s(153)\le 7250614903299225/562949953421312$, from a SQUISH rational packing

  Claim and records

  Claim

  Chaoweeraprasit (itsnaka), using SQUISH, establishes $s(153)$$s(153)$ <=
  
  7250614903299225/562949953421312, about 12.8796793733329640, in the supplement to
  
  jlevy/squares#401 on 7 October 2026, about 0.0019874 below the earlier reported upper bound
  
  here. Each certificate supplies rational centres and rational half-angle parameters for
  
  closed unit squares in the rational container.
  
  The claim establishes only upper bounds, with no optimality or rigidity conclusion. The
  
  author credits Ellsworth’s records and tools and states human-directed Claude Opus 5.5
  
  assistance.
  
  This repository confirmed the upper bounds by independently re-implemented exact-rational
  
  checks: sqpack.witness.exact_verify and devtools.check_rational_witness_independent accept
  
  every pair and container wall of each certificate, with both geometric controls rejected.

  Significance

  New packings reduce the recorded upper bounds at $n=153$$n=153$ by about 0.00199, substantive case-bound improvements. They settle no exact optimum and introduce no proved general packing theorem.

  Novelty

  previously-published Present in an identified source

  Records

  n = 153registerevidence 1evidence 2

  n = 153registerevidence 1evidence 2

• T-125 S3 V3 C3 Xu · 2026-10-08 · 25 cases

  Complete rational construction reports at 25 counts

• T-127 S3 V3 C3 Gupta after Chaoweeraprasit, Daniel · 2026-10-08 · 17 cases

  Fourteen rational refinements, seventeen complete source cases

• T-136 S3 V3 C3 Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · 2026-10-10 · 17 cases

  Seventeen exact rational packings from SQUISH's third request

• T-146 S2 V3 C3 Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · 2026-10-10 · 15 cases

  Exact rational certificates at fifteen counts from Daniel's regularized record lists

• T-148 S3 V3 C3 PingYou Ltd after Xu, Daniel, Levy · 2026-10-11 · n = 153

  A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle

  Claim and records

  Claim

  One complete rational source certificate by PingYou Ltd, reported on jlevy/squares#504, proves a finite upper bound at the exact side it states: $s(153)\le 6435006925203522001385040447/500000000000000000000000000$ = 12.870013850407044002770080894.
  
  The certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in SQUISH's JSON rows; its side and centres already carry the guard dilation 5000000000001/5000000000000 it states, and every square keeps side 1.
  
  The author reports that three exact Fraction checkers of its own, by vertex projections, by centre and support-radius separating axes and by rational polygon clipping, accept all 11,628 pairs and every wall; that its reconstruction script rebuilds the certificate exactly from an explicit rational construction table and recomputes the guard; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept its corners Witness/v2.
  
  It reports that a SymPy script checks the symbolic identity behind the chosen angle and isolates the root of u^7-9u^6+3u^5-u^4-u^3+u^2-3u+1, and that the construction table comes from an exploratory selection and a floating-point solve that the release does not repeat. Those statements are the author's; no source program ran here.
  
  All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 11 October 2026 by both maintained exact routes, from an exact fact derived from the pinned certificate, and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided it with six controls and found the fact to be the packing the pinned certificate and the pinned witness state. The source ran the two maintained routes on its own witness, but neither is its generator's code, and it did not run the third route, so the replay is independently re-implemented. The 11 October review found no blocking defect.
  
  The side was 9.67e-03 below the ceiling the case held (T-127), and is below Daniel's T-146 by exactly 0.002015998674135970992348937968, the figure the issue gives, and below SQUISH's T-136 by 2.02e-03. On 11 October 2026 the case record adopted this entry as the house at 153, its reported and verified upper bounds moving together, T-127's certificate kept as history and T-146 and T-136 verified and not selected; private-worker custody (RV-6) was closed in the same change (packing/devtools/house-custody-2026-10-11.json).
  
  Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) for the n153 construction, as the source asks: by its account, Xu's n70 construction (issue 432) and its reflection, composed by the boundary-patch approach of the source's n207 (issue 503, T-147) with a different choice of surviving squares and contacts and reclosed at a new angle, giving 121 upright squares and tilted populations of 17 and 15.
  
  The n70 geometry it started from is Daniel's exact n70 certificate (https://github.com/evand/square-packing, the regularized n-070 certificate T-146 retains), credited to him for its exact retention. The source credits the previous n153 packing to Chaoweeraprasit (SQUISH, issue 481) and its refined exact certificate to Daniel (T-146), as comparisons, and claims neither the n70 packing nor patch composition in general.
  
  That lineage is the source's account of its construction and was not measured by pose: no placement of the n70 lands more than 31 of its squares within 1e-9 of a square of this packing, nor more than one exactly, which fits the source's account of contacts reclosed at a new angle.
  
  The source and the issue make no statement about AI assistance; the issue omits the template's section on it. The same author published it on 11 October 2026, minutes after the n207 release of issue 503 (T-147), under the same workflow. That release as first published, and its issue as first posted, said that the verification, the comparison review and the package preparation were assisted by OpenAI Codex; minutes later a revision titled "Correct attribution in submission materials" removed that sentence, and the issue was edited to match, with no reason given. Asked on issue 503 which account stands for both releases, the author answered there on 12 October 2026: "Both solutions were AI assisted by 'PY', an unreleased agentic AI offering by PingYou Ltd."

  Significance

  A smaller finite construction side at $n=153$$n=153$, 9.67e-03 below the case ceiling and 2.02e-03 below the smallest pending side, a new construction that reclosed the contacts of boundary patches of an existing packing at a new angle; no lower bound or optimum.

  Novelty

  previously-published Present in an identified source

  Records

  n = 153registerevidence 1evidence 2

  n = 153registerevidence 1evidence 2

• T-166 S1 V3 C3 PingYou Ltd after Xu, Daniel, Levy · 2026-10-14 · n = 153

  A tighter rational realization of PingYou Ltd's packing of 153 squares

  Claim and records

  Claim

  One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(153)\le 12.870013850407034002770080892$$s(153)\le 12.870013850407034002770080892$.
  
  All three retained jobs, the positive with its 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 certificate with six controls each and held the derived fact to the pinned file it is derived from, fetched at the pin. No route here is the source's code; the source's runs of this repository's two maintained routes on its corners witnesses are its own and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.
  
  The author reports that three exact checkers, its own rational polygon-projection checker and Daniel's unchanged verify_cert.py and verify_cert2.py, accept all 153 squares, 612 wall conditions and 11,628 pairs; that its package replays the recorded rational construction and checks the lossless conversion to its corners Witness/v2; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept that witness. Those statements are the author's; no source program ran here.
  
  Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with the n153 construction and this refinement of its certificate, as the source asks; by the source's account the construction starts from Xu's n70 construction (ry-xu, issue 432), taken from Daniel's exact n70 certificate, its contacts reclosed at a new angle. The source names SQUISH's n153 (issue 481) only as its earlier comparison. The pinned source makes no statement about AI assistance; the author's statement on issue 503 of 12 October, 'Both solutions were AI assisted by "PY", an unreleased agentic AI offering by PingYou Ltd.', covers the n153 construction.
  
  Measured pose by pose by that review, it is PingYou Ltd's T-148 packing with two squares moved, by at most 0.090. At 153 Polkovskyi's issue 573 (T-173) is smaller and has held the case since 15 October 2026.

  Significance

  The same packing at less padding, 1.00e-14 below the case ceiling, as T-155; no lower bound or optimum.

  Novelty

  previously-published Present in an identified source

  Records

  n = 153registerevidence 1evidence 2

  n = 153registerevidence 1evidence 2

• T-173 S2 V3 C3 Polkovskyi after Xu, Chaoweeraprasit, Levy · 2026-10-14 · n = 153, 207

  Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there

  Claim and records

  Claim

  Two complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(153)\le 12.87001385040448$$s(153)\le 12.87001385040448$ and $s(207)\le 14.880960371558$$s(207)\le 14.880960371558$.
  
  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 both certificates this entry cites with six controls each and held each derived fact to the pinned file it is derived from, fetched at the pin. No route here is the source's code; the author's runs of this repository's sqpack verifier, from copies outside the pinned tree, 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 this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, accept all six certificates the issue requests in exact rational arithmetic. Those statements are the author's; no source program ran here.
  
  Credit Polkovskyi (Mishapolk), as he asks on issue 593, with the certificates. The issue calls 153 and 207 new arrangements from basin hopping with tilted boundary corridors. Measured pose by pose by the 14 October review against PingYou Ltd's packings: at 207 every square pairs with one of T-147's within 2.8e-04, three turned by at most 0.012 degrees, a re-solve of PingYou's packing and not a new arrangement by pose; at 153 it is T-148's design with 100 squares moved beyond 1e-03 and six beyond a tenth, another arrangement of PingYou's design. Whether to credit them as new arrangements is the owner's decision 1 on pull request 602. It discloses that Gemini was used for the search implementation, verification orchestration and issue preparation, directed by the author.
  
  On 15 October 2026 the case records adopted this entry as the house at 153 and 207, each case's reported and verified upper bounds moving together and PingYou Ltd's certificates (T-148, T-147) kept as history, with T-166 and T-167 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). PR 574's algebraic certificates (T-174) are smaller at both counts, by 1.6e-14 and 2.4e-12, and rest on one route; the rule of 15 October left them out for the owner's decision (RP-3 of the review of PR 574), and the house rule of 16 October 2026 adopts them instead, staged under think-ot5c until the writer takes an algebraic side (devtools.select_houses.PENDING_ADOPTION).

  Significance

  Another arrangement of PingYou Ltd's design at 153 and a re-solve of its packing at 207, 2.56e-12 and 1.49e-09 below the case ceilings, as T-159 and T-150 are; no lower bound or optimum.

  Novelty

  previously-published Present in an identified source

  Records

  n = 153n = 207registerevidence 1evidence 2

  n = 153n = 207registerevidence 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-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

In the frontier survey On GitHub