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-045 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 15 cases

  Rectangle-density lower bounds replayed at 15 counts in \(n = 18\ldots78\)$n=18\dots 78$

• T-046 S3 V0 C0 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 48 cases

  Rectangle-density lower bounds reported for 48 counts in \(n = 18\ldots95\)$n=18\dots 95$

• T-047 S4 V3 C3 Tokoharu after Levy, wand125, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 7 cases

  \(s(11) \ge 381/100\)$s(11)\ge 381/100$; \(s(n) \ge 1377/250\)$s(n)\ge 1377/250$ for \(n = 26\ldots28\)$n=26\dots 28$; \(s(n) \ge 571/100\)$s(n)\ge 571/100$ for \(n = 29\ldots31\)$n=29\dots 31$

• 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-068 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases

  Rectangle-density lower bounds verified at 34 counts in \(n = 19\ldots95\)$n=19\dots 95$

• T-074 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 31 cases

  Rectangle-density lower bounds replayed at 31 counts in \(n = 19\ldots95\)$n=19\dots 95$

• 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 \(b > 2\)$b>2$

• T-109 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-06 · n = 30

  Mixed rectangle-measure lower bound verified at \(n = 30\)$n=30$, on a declared net

  Claim and records

  Claim

  A rectangle density of wand125/square-packing-bounds, checked at coverage one on a net it declares and published on 6 October 2026, proves \(s(30) \ge 11767/2000\)$s(30)\ge 11767/2000$ = 5.8835.
  
  The certificate is a density of 402 uniform rectangles in D4 orbits, and no point mass, of total mass 2999999/100000, on the net T-099's certificate declares: core side 1999/2000 and 832 half-angle tangents of step 1/2006. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 is past tan(pi/8), every unit square contains a concentric core at a net angle strictly in its interior.
  
  The source reports every such core capturing mass at least one at all 832 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.
  
  The value is above 47/8 = 5.875, the source's rectangle certificate rect_n30_L5875 and the reported (T-068) and verified (T-074) lower bound at \(n = 30\)$n=30$ before it, by 0.0085.
  
  The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 832 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 347 and 831, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.
  
  wand125 after Tokoharu and Levy, square-packing-bounds. No registration was requested: an intake pass of the source found it. The source says parts of the work were produced with AI assistance under human direction.

  Composition

  One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 832 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.

  Next rung

  V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 832 directions (devtools.audit_wand125_declared_net cpp-sample, about 59 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.

  Significance

  The strongest verified lower bound on record at \(n = 30\)$n=30$, by 0.0085 above T-074, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.

  Novelty

  previously-published Present in an identified source

  Records

  n = 30registerevidence 1evidence 2sourcereview

  n = 30registerevidence 1evidence 2sourcereview

In the frontier survey On GitHub