T-200: Mixed rectangle-measure lower bound $s (38) \geq 2631 / 400$ reported, on a 2,073-direction declared net

V0 C0 Lower bound recorded

2026-10-16 published · Hosono after Tokoharu and Levy · $n = 38$

A rectangle measure of wand125/square-packing, published on 16 October 2026 and reported in jlevy/squares#446, is reported to prove $s (38) \geq 2631 / 400$ = 6.5775.

The certificate, mixed_n38_L65775, is a density of 654 uniform rectangles in D4 orbits, and no point mass, of total mass 3799999/100000, 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 1309/200 = 6.545, the reported (T-068) and verified (T-074) lower bound at $n = 38$ , by 13/400.

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 $n = 38$ , 13/400 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 7.46 to 8.57 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 7.46 CPU-hours, up to about 8.57 (priced 2026-10-17).
Novelty: previously-published Present in an identified source

The case

Case record

6.546.707

678

6.1647.164

\sqrt{n}

\sqrt{n} + 1

Proven

$6.54500$ $\leq s (38) \leq$ $6.707107$

exact

Citation record n-038

lowerHosono after Tokoharu, Levy et al. 2026, GitHub (confirmed T-074)

upperGöbel 1979, Squares in Squares

Open

optimality

The case record

LowerUpper

Verified

\frac{1309}{200}

6 + \frac{1}{2} \sqrt{2}

Reported

\frac{1309}{200}

6 + \frac{1}{2} \sqrt{2}

Gap

\frac{\sqrt{2}}{2} - \frac{109}{200}

≈ 0.16210678…

Results on the case

9 results in the register on $n = 38$ , 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-068 and T-074
Nagamochi · Nagamochi 2005 · source · 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
2026-09-27 published T-046
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
V0 C0 Lower bound recorded superseded by T-068 and T-074
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-28 published T-070
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
V3 C3 Lower bound confirmed superseded by T-068 and T-074
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · 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-068 and T-074
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
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
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-16 published T-200 this result
Mixed rectangle-measure lower bound $s (38) \geq 2631 / 400$ 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, $n = 38$ · Frontier row, $n = 38$ · T-200 in the results table

On GitHub, at main

Register: T-200 in results.yaml, line 27970
Evidence: E-wand125-fine-net-n38-65775-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_n38_L65775/candidate.json.gz resources/web/wand125-fine-net-lower-bounds-2026-10-16/release/certificates/mixed_n38_L65775/check2/receipt.json resources/web/wand125-fine-net-lower-bounds-2026-10-16/receipts/premises-n38.json devtools/fine_net_followup.py
Case file: frontier/n-038.md (verified lower, verified upper, reported lower, reported upper)

T-200: Mixed rectangle-measure lower bound s(38)≥2631/400s(38)≥2631/400 reported, on a 2,073-direction declared net

The case

Results on the case

Links

T-200: Mixed rectangle-measure lower bound $s (38) \geq 2631 / 400$ reported, on a 2,073-direction declared net