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

2026-10-16 published · Hosono after Tokoharu and Levy · $n = 40$

A rectangle measure of wand125/square-packing, published on 16 October 2026 and reported in jlevy/squares#446, is reported to prove $s (40) \geq 1347 / 200$ = 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 $n = 40$ , 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, composition and next rung

Significance: Scored as a claim: if a complete replay passes, it is the strongest lower bound on record at $n = 40$ , 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.
Composition: One reported certificate on its reported entry. The source's check is a copy of this repository's sqverify_fast crate with one change, so a sqverify-fast replay here would be re-implemented sharing the producer's components, as T-104's is, and no second implementation stands beside it.
Next rung: C3 needs a complete sqverify-fast census at all 2,073 directions of the retained candidate, with its whole-net control receipt (the 99/100 mutant at every direction, FC-1) and a review that reads this certificate, by the checklist of the 6 October declared-net soundness review. The census runs in think-f40c, priced at about 11.37 to 13.06 CPU-hours from the source's receipt.
Unfinished confirmations: C3 (think-f40c): a complete sqverify-fast census at all 2,073 directions of the retained candidate, with its whole-net control receipt (the 99/100 mutant at every direction, FC-1) and a review that reads this certificate, by the checklist of the 6 October declared-net soundness review. About 11.37 CPU-hours, up to about 13.06 (priced 2026-10-17).
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
$s (40) > 3359 / 500 = 6.718$ , 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 this result
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-201 in the results table

On GitHub, at main

Register: T-201 in results.yaml, line 28073
Evidence: E-wand125-fine-net-n40-6735-report
Proofs and certificates: certificate candidate.json.gz
Sources: wand125 fine-net lower bounds 2026-10-16
Source packet: resources/web/wand125-fine-net-lower-bounds-2026-10-16/README.md
Artifacts: 7 artifacts and controls
resources/web/wand125-fine-net-lower-bounds-2026-10-16/README.md resources/web/wand125-fine-net-lower-bounds-2026-10-16/acquisition/sources.json resources/web/wand125-fine-net-lower-bounds-2026-10-16/acquisition/release-assets.json resources/web/wand125-fine-net-lower-bounds-2026-10-16/release/certificates/mixed_n40_L6735/candidate.json.gz resources/web/wand125-fine-net-lower-bounds-2026-10-16/release/certificates/mixed_n40_L6735/check2/receipt.json resources/web/wand125-fine-net-lower-bounds-2026-10-16/receipts/premises-n40.json devtools/fine_net_followup.py
Case file: frontier/n-040.md (verified lower, verified upper, reported lower, reported upper)

T-201: Mixed rectangle-measure lower bound s(40)≥1347/200s(40)≥1347/200 reported, on a 2,073-direction declared net

The case

Results on the case

Links

T-201: Mixed rectangle-measure lower bound $s (40) \geq 1347 / 200$ reported, on a 2,073-direction declared net