Bounds

Results in the register

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

  \(s(n) \ge \min(\lceil\sqrt{n}\rceil, \sqrt{n - 2\lfloor\sqrt{n}\rfloor + 1} + 1)\)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for \(4 \le n \le 324\)$4\le n\le 324$

• T-058 S2 V3 C3 wand125 after Tokoharu, Daniel · 2026-09-29 · 100 cases

  Rectangle-certificate ceiling α·UB(n) proved for \(n = 1\)$n=1$..100; B·UB(n) on 64 grid rows

• T-076 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-03 · n = 82

  Linear-measure lower bound replayed at \(n = 82\)$n=82$

  Claim and records

  Claim

  A measure of points, segments and rectangles that wand125/square-packing-bounds published on 2 October 2026 proves \(s(82) \ge 233/25\)$s(82)\ge 233/25$ = 9.32. The measure is of the kind and checker of T-073: point masses, segments of uniform density and rectangles of uniform density, in D4 orbits with exact rational geometry, of total mass 82 - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000, with 86 point, 222 segment and 774 rectangle orbits. They are those of T-073's \(n = 83\)$n=83$ certificate, every coordinate scaled by exactly 932/935, with the masses solved again.
  
  The source reports it accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp, byte for byte the verifier of T-073, in 153,579,479 nodes, and says the full replay was run again from the published tarball by the same implementation. The value exceeds Green's reported 2 sqrt(2) + (247 + 12 sqrt(2))/41 = 9.2667335..., the value this record reported at \(n = 82\)$n=82$, by 0.0532664..., and does not rest on Green's argument, which jlevy/squares#308 reports the published material does not establish for k >= 4. Its mass is below 83, where T-075's 937/100 is higher.
  
  It passed a complete replay here on 3 October 2026 of the source's unchanged checker and replay functions on its pinned tarball: all 201 directions, each returning the certificate's own record. The replay runs the source's own algorithm and is not an independent decision of coverage.
  
  wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 2 October 2026 on jlevy/squares#294. The source says parts of the work were produced with AI assistance under human direction.

  Composition

  One claim from one certificate, on its reported entry and its replay entry. The replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone, which the 2 October review of the linear certificates read function by function.

  Next rung

  V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; the review of these certificates was the retaining lane's own, and the checker's separately prompted review is T-073's. A second machine method would be a method-distinct decision of rotated coverage for a linear measure; the replay here runs the source's own checker. linear-control n82 would put a negative control on this certificate itself; the checker's controls are \(n = 101\)$n=101$'s.

  Significance

  Raised the verified lower bound at \(n = 82\)$n=82$ from Nagamochi's closed form past Green's reported value, the one count from 82 to 85 where the record still reported Green's value, by a certificate of T-073's kind that does not rest on Green's argument. A further size from one generator and checker, at S3 as T-073 is; the 2 October review of the afternoon certificates proposed S3, which the replay leaves standing. Nagamochi's closed form has been a reported bound since Karakuş's finding (T-085, merged 2026-10-03); on the merged record the verified floor this replay raised at \(n = 82\)$n=82$ was Karakuş's general bound (T-083), lower still.

  Novelty

  previously-published Present in an identified source

  Records

  n = 82registerevidence 1evidence 2sourcereview

  n = 82registerevidence 1evidence 2sourcereview

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

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

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

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

In the frontier survey On GitHub