$n=40$$n=40$

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-013 S3 V3 C3 Levy · 2026-08-31 · n = 40

  Goebel's $n=40$$n=40$ packing: seven verified first-order flexes, each refused at second order

  Claim and records

  Claim

  Goebel's $n=40$$n=40$ packing is infinitesimally flexible -- seven verified independent first-order flexes turn the sixteen-square tilted block -- and every retained flex is refused at second order by a verified non-negative self-stress, exactly over Q(sqrt 2), so no first-order argument can establish rigidity here.

  Significance

  The first separation of infinitesimal from finite rigidity in this literature's cases: the sources assert rigidity without the distinction, and this refines rather than contradicts them.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 40registerevidence 1

  n = 40registerevidence 1

• T-044 S3 V3 C3 Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 17 cases

  Weighted point lower bounds for ten counts in $n=26\dots 72$$n=26\dots 72$, plus seven from the same files

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

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

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

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

• T-058 S2 V3 C3 Hosono 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 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases

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

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

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

• 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-133 S3 V3 C3 Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · 2026-10-10 · n = 40

  $s(40)>335427/50000=6.70854$$s(40)>335427/50000=6.70854$, by a clipped-corner transfer on a 401-direction net

  Claim and records

  Claim

  $s(40)>335427/50000$$s(40)>335427/50000$ = 6.70854, by Guzhou0806, published on 10 October 2026 as release n40-670854-20261010 of Guzhou0806/n40-square-packing and reported on jlevy/squares#485. It raises the reported and verified lower bound at $n=40$$n=40$ by 427/50000 = 0.00854 over T-068's 67/10, from the same certificate. The release also gives the weaker $s(40)>67000$$s(40)>67000$√(6400006889)/798988091 = 6.7084890815…, which needs no density peak.
  
  The certificate is Hosono's rect_n40_L67 rectangle density of T-068, unchanged, of mass 3999/100 at core side 9977/10000. Every closed square of that side inside [0, 67/10]^2, at each of 401 half-angle tangents of step 83/80000, twice as fine as T-068's net of 201, captures mass at least 10001/10000.
  
  A continuous-angle transfer carries this to every unit square scaled into that container: the net direction above a square's angle gives a reference square inside it when the angle is close enough, and otherwise the one below it does, less the density's essential supremum, at most 2818711359413/10^9, times a bound on the area of the reference's four clipped corners. Each square then captures at least 99979/100000, and forty of them exceed the mass by 1/625, so no packing fits at side 335427/50000; compactness makes the bound strict.
  
  The source decided the nodal statement with this repository's sqverify-fast, vendored unchanged, at all 401 directions, and the finite steps with its own exact Python, in a local run and a GitHub Actions run.
  
  Both parts were decided here on 10 October 2026. sqverify-fast, built from reviewed source, verified all 401 directions on an input regenerated from the retained density, every row equal to both release receipts apart from timing, and refused two mass mutants: confirmed, reproduced with the producer's code, since the release ran the same crate.
  
  This repository's exact audit of the transfer, which shares no code with the release's finite checks, recomputed the density's mass, symmetry and essential supremum and every step of the chain, and refused four altered parameters: the finite steps were independently re-implemented. The 10 October review re-derived the transfer, the clipped-corner bound and the strict endpoint and found no blocking defect. No second method decided the nodal statement.
  
  Guzhou0806 after Hosono, Tokoharu and Levy, n40-square-packing. The source credits the density and the original verification method to Hosono, Tokoharu and their cited predecessors, and says Guzhou0806's contribution, the finer-net verification, the strict core transfer and the clipped-corner composition, came from AI-assisted research; the issue says AI assisted the exploration, proof drafting, implementation, testing, packaging and submission under the author's direction.

  Significance

  The strongest lower bound on record at $n=40$$n=40$, 0.00854 above T-068 on T-068's own certificate. The transfer, a continuous-angle argument from a finer net with a clipped-corner loss bound and the density's peak, is new to this record and applies with a finer net to any format T certificate it holds, but it is one count and not yet a bound family. S3 by the precedent of T-099, a finer net on a certificate kind the record holds, with the technique noted, as the 10 October review suggested.

  Novelty

  previously-published Present in an identified source

  Records

  n = 40registerevidence 1evidence 2evidence 3source 1source 2review

  n = 40registerevidence 1evidence 2evidence 3source 1source 2review

• T-145 S3 V3 C3 Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · 2026-10-10 · n = 40

  $s(40)>1340000/199529=6.7158157\dots$$s(40)>1340000/199529=6.7158157\dots$, by a continuous-pose interval kernel

  Claim and records

  Claim

  $s(40)>1340000/199529$$s(40)>1340000/199529$ = 6.715815746082023…, by Guzhou0806, published on 10 October 2026 as release n40-continuous-1340000-199529-20261010 of Guzhou0806/n40-square-packing and reported on jlevy/squares#485. It raises T-133's 335427/50000 by 72586117/9976450000 = 0.0072757…, from the same certificate.
  
  The certificate is Hosono's rect_n40_L67 rectangle density of T-068, unchanged, of mass 3999/100. Every closed square of side 199529/200000 inside [0, 67/10]^2, at every angle and every legal centre, captures mass at least 4999/5000, and 40 × 4999/5000 − 3999/100 = 1/500, so no packing fits at side 1340000/199529; compactness makes the bound strict.
  
  A new interval kernel covers the continuous pose domain, centre and half-angle tangent together, in 43 closed slabs of t = tan(θ/2) over [10^-6, 83/200], by outward-rounded binary64 branch and bound with a concave-section integral bound and translation-derivative bounds; an exact integer certificate for horizontal squares of side 997643/1000000, which lie inside every side-B square at t ≤ 10^-6, covers the rest. No net, no transfer and no density peak are needed.
  
  The source decided both parts with its two C++ kernels, in a local run and a GitHub Actions run.
  
  Both were decided here on 10 October 2026: the two kernels, built from the retained source into the CI run's binaries, ran the axis certificate and all 43 slabs, every output equal to both release receipts apart from timing, and refused every synthetic density whose least capture is below the threshold: confirmed, reproduced with the producer's code.
  
  This repository's exact audit, which shares no code with the release's, regenerated both kernel inputs and decided every finite step, and sqverify-fast decided the near-axis statement a second way on the original density. The 10 October review re-derived the kernel's lemmas and found no blocking defect. No second method decided the continuous cover.
  
  Guzhou0806 after Hosono, Tokoharu and Levy, n40-square-packing. The source credits the density to Hosono (as wand125), the rectangle-density method and the translation-derivative structure to Tokoharu, says the concave-section bound is informed by this repository's sqverify_fast, and gives Guzhou0806's contribution, the continuous-pose extension, the smaller-square coverage, the near-axis patch and the composition, as AI-assisted research; the comment says AI assisted the exploration, proof drafting, implementation, tests, packaging and the update under the author's direction.

  Significance

  The strongest lower bound on record at $n=40$$n=40$, 0.00728 above T-133 on the same certificate. The technique, an interval cover of the continuous pose domain with the angle as an interval dimension, is new to this record, dispenses with the net, the transfer and the density peak, and applies as it stands to every format T certificate the record holds, with the side and threshold as its parameters; it is one count and not yet a bound family. S3 by the precedent of T-133, with the technique noted, as the 10 October review suggests.

  Novelty

  previously-published Present in an identified source

  Records

  n = 40registerevidence 1evidence 2evidence 3evidence 4source 1source 2review

  n = 40registerevidence 1evidence 2evidence 3evidence 4source 1source 2review

• T-157 S3 V3 C3 Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · 2026-10-12 · n = 40

  $s(40)>3359/500=6.718$$s(40)>3359/500=6.718$, by T-145's continuous-pose kernel on a density with 163 new supports

  Claim and records

  Claim

  $s(40)>3359/500$$s(40)>3359/500$ = 6.718, by Guzhou0806, published on 11 October 2026 as release n40-new-supports-6718-20261011 of Guzhou0806/n40-square-packing and reported on jlevy/squares#485. It raises T-145's 1340000/199529 by 217911/99764500 = 0.00218425..., from a new certificate.
  
  The certificate is a new rational density of 685 positive D4 rectangle orbits, 522 of them rows of Hosono's rect_n40_L67 pool and 163 generated outside it, every weight new, of mass 3999/100. Every closed square of side 3350/3359 inside [0, 67/10]^2, at every angle and every legal centre, captures mass at least 4999/5000, and 40 x 4999/5000 - 3999/100 = 1/500, so no packing fits at side 3359/500; compactness makes the bound strict.
  
  The two kernels are T-145's, byte for byte: the continuous-pose interval kernel over the same 43 closed slabs of t = tan(theta/2) in [10^-6, 83/200], and the exact integer axis certificate, here at side 249329659/250000000, which lies inside every side-B square at t <= 10^-6. The source decided both parts in a local run and a GitHub Actions run.
  
  Both were decided here on 12 October 2026: the two kernels, built from the retained source into the CI run's binaries, ran the axis certificate and all 43 slabs, every output equal to both release receipts apart from timing, and refused every synthetic density whose least capture is below the threshold: confirmed, reproduced with the producer's code.
  
  This repository's exact audit, which shares no code with the release's, decided the density's orbit facts and every finite step and regenerated both kernel inputs to the digests both receipts name; sqverify-fast decided the near-axis statement a second way on the density itself and verified the 401-direction nodal statement at the new side. The 12 October review found no blocking defect. No second method decided the continuous cover.
  
  Guzhou0806 after Hosono, Tokoharu and Levy, n40-square-packing. The source credits the original density and pool to Hosono (as wand125), the translation-derivative structure to Tokoharu, says the concave-section calculations draw on this repository's sqverify_fast, and gives Guzhou0806's AI-assisted contribution as the new supports, the weight optimization, the continuous-cover certificate and the support-family obstruction, with AI assistance in exploration, code, tests, proof writing and publication.

  Significance

  The strongest lower bound on record at $n=40$$n=40$, 0.00218 above T-145, by T-145's technique on a density with 163 new supports and new weights; a single-count raise on an existing technique, the kernel's reuse at the same count and not a bound family. S3 by the precedent of T-099, T-100 and T-145, as the 12 October review keeps it.

  Novelty

  previously-published Present in an identified source

  Records

  n = 40registerevidence 1evidence 2evidence 3evidence 4source 1source 2review

  n = 40registerevidence 1evidence 2evidence 3evidence 4source 1source 2review

• T-201 S3 V0 C0 Hosono after Tokoharu and Levy · 2026-10-17 · n = 40

  Mixed rectangle-measure lower bound $s(40)\ge 1347/200$$s(40)\ge 1347/200$ reported, on a 2,073-direction declared net

  Claim and records

  Claim

  A rectangle measure of wand125/square-packing, published on 16 October 2026 and reported in jlevy/squares#446, is reported to prove $s(40)\ge 1347/200$$s(40)\ge 1347/200$ = 6.735.
  
  The certificate, mixed_n40_L6735, is a density of 707 uniform rectangles in D4 orbits, and no point mass, of total mass 3999999/100000, on a net its candidate declares: core side 4999/5000 and 2,073 half-angle tangents of step 1/5002, the net of T-132, T-142 and T-143. Since 4999/5000 (1 + 1/5002) = 25009997/25010000 < 1, and the last tangent 1036/2501 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 2,073 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, in two complete runs, and a control with every mass scaled by 197/200 refused at 32 sampled directions.
  
  The value is above T-157's 3359/500 = 6.718, the verified and reported lower bound at $n=40$$n=40$, by 17/1000.
  
  Here the maintained readers accepted its exact mass, declared-net and check2 premises from the retained release asset. No complete replay has run here, so the value stands as reported.
  
  wand125 after Tokoharu and Levy, square-packing. The source says parts of the work were produced with AI assistance under human direction.

  Significance

  Scored as a claim: if a complete replay passes, it is the strongest lower bound on record at $n=40$$n=40$, 17/1000 above the reported lane, by the certificate kind and declared net of T-132, T-142 and T-143, with no new technique, at S3 as T-176 to T-178 are.

  Novelty

  previously-published Present in an identified source

  Records

  n = 40registerevidence 1

  n = 40registerevidence 1

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