T-157: $s (40) > 3359 / 500 = 6.718$ , by T-145's continuous-pose kernel on a density with 163 new supports

V3 C3 Lower bound confirmed

2026-10-11 published · Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · $n = 40$

$s (40) > 3359 / 500$ = 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, composition and next rung

Significance: The strongest lower bound on record at $n = 40$ , 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.
Composition: One certificate, Guzhou0806's 685-orbit density, and three load-bearing parts. The continuous cover of the 43 slabs is decided by the release's continuous_pose.cpp, interval-certified, the producer's code, rebuilt here from reviewed source into the CI run's binary and replayed in full. The near-axis statement at side 249329659/250000000 is decided by the release's axis_integer_grid.cpp, exact, rebuilt into the CI run's binary and replayed here, and again by sqverify-fast's exact axis-vertex sweep on the density itself, undominated, which shares no code with it.

The finite steps, the density's orbit facts, both inputs' identity, the near-axis containment, the slab partition, the threshold and the counting margin, are decided by this repository's exact audit, independently re-implemented with respect to exact.py, and are recorded as structure.

The result takes the part closest to the producer's code: confirmed, reproduced with the producer's code. C3 rests on the complete replay of both kernels and its controls, and the kernels' lemmas on the 10 October review of T-145, which read the same bytes. No second method decides the continuous cover; the 401-direction run of sqverify-fast at the new side is a necessary condition it passed. Hosono's published sqverify_continuous (think-xzob), once reviewed and replayed here, would make the cover independently re-implemented and move the mark, not the rung.
Next rung: V4 and C4 need a second retained adversarial AI review by a distinct reviewer (the 11 October final review covers T-145's kernel and density, not this density) and a human oversight record. Beside the rung: a decision of the continuous cover by code other than the producer's, which Hosono's published sqverify_continuous (think-xzob: retention, a soundness review, about 2.3 CPU-hours here) or a first-party interval kernel with the angle as an interval dimension (GC-4 of the 10 October review) would supply; either would make the result independently re-implemented and leave the rung where it is.
Novelty: previously-published Present in an identified source

The case

Case record

6.716.828

678

6.3257.325

\sqrt{n}

\sqrt{n} + 1

Proven

$6.71800$ $\leq s (40) \leq$ $6.828428$

exact
rigid

Citation record n-040

lowerGuzhou0806 after Hosono et al. 2026, GitHub (confirmed T-157)

upperGöbel 1979, Squares in Squares

Open

optimality

The case record

LowerUpper

Verified

\frac{3359}{500}

4 + 2 \sqrt{2}

Reported

\frac{3359}{500}

4 + 2 \sqrt{2}

Gap

2 \sqrt{2} - \frac{1359}{500}

≈ 0.11042712…

Results on the case

14 results in the register on $n = 40$ , oldest first, each with what it established and how it stands now.

2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 Lower bound incomplete on this case, superseded by T-157
Nagamochi · Nagamochi 2005 · source · register
2026-08-30 established T-013
Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order
V3 C3 Rigidity confirmed
Levy · register
2026-09-04 published T-085
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-22 published T-044
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
V3 C3 Lower bound confirmed superseded by T-157
Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-27 published T-045
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
V3 C3 Lower bound confirmed superseded by T-157
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
V0 C0 Lower bound recorded superseded by T-157
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-29 published T-058
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
V3 C3 Method limit confirmed
Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed on this case, superseded by T-157
Karakuş · Karakuş 2026 · source · register
2026-10-01 published T-068
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
V3 C3 Lower bound confirmed on this case, superseded by T-157
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
V3 C3 Lower bound confirmed on this case, superseded by T-157
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-10 published T-133
$s (40) > 335427 / 50000 = 6.70854$ , by a clipped-corner transfer on a 401-direction net
V3 C3 Lower bound confirmed superseded by T-157
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 clipped corner 2026-10-10 · packet · packet · source 1 · source 2 · review · register
2026-10-10 published T-145
$s (40) > 1340000 / 199529 = 6.7158157 \dots$ , by a continuous-pose interval kernel
V3 C3 Lower bound confirmed superseded by T-157
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 continuous pose 2026-10-10 · packet · packet · source 1 · source 2 · review · register
2026-10-11 published T-157 this result
$s (40) > 3359 / 500 = 6.718$ , by T-145's continuous-pose kernel on a density with 163 new supports
V3 C3 Lower bound confirmed
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 new supports 2026-10-11 · packet · source 1 · source 2 · review · register
2026-10-16 published T-201
Mixed rectangle-measure lower bound $s (40) \geq 1347 / 200$ reported, on a 2,073-direction declared net
V0 C0 Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

Links

On this site: Case record, $n = 40$ · Frontier row, $n = 40$ · T-157 in the results table

On GitHub, at main

Register: T-157 in results.yaml, line 19488
Evidence: E-n040-guzhou-new-supports-report · E-n040-guzhou-new-supports-kernel-replay · E-n040-guzhou-new-supports-finite-audit · E-n040-guzhou-new-supports-crate-axis
Proofs and certificates: certificate certified_candidate.json.gz · proof PROOF.md · audit review-2026-10-12-n40-new-supports-6718.md · proof SOUNDNESS.md
Sources: Guzhou0806 n40 new supports 2026-10-11 (its own site, retained copy)
Source packet: resources/web/guzhou-n40-new-supports-2026-10-11/README.md
Artifacts: 23 artifacts and controls in 10 directories. Complete artifact list in the result entry. docs/project/reviews packing/devtools packing/resources/web/guzhou-n40-new-supports-2026-10-11 packing/resources/web/guzhou-n40-new-supports-2026-10-11/receipts packing/resources/web/guzhou-n40-new-supports-2026-10-11/release-ci/results packing/resources/web/guzhou-n40-new-supports-2026-10-11/source packing/resources/web/guzhou-n40-new-supports-2026-10-11/source/certificate packing/resources/web/guzhou-n40-new-supports-2026-10-11/source/results packing/resources/web/guzhou-n40-new-supports-2026-10-11/source/verifier packing/tests
Case file: frontier/n-040.md (verified lower, verified upper, reported lower, reported upper)

T-157: s(40)>3359/500=6.718s(40)>3359/500=6.718, by T-145's continuous-pose kernel on a density with 163 new supports

The case

Results on the case

Links

T-157: $s (40) > 3359 / 500 = 6.718$ , by T-145's continuous-pose kernel on a density with 163 new supports