T-144: Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported

V0 C0 lower bound recorded

2026-10-10 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 5 cases, $n = 122$ to 126

A measure of points, segments and rectangles that wand125/square-packing published on 10 October 2026, reported on no issue, is reported to prove $s (122) \geq 563 / 50$ = 11.26. Its mass is below 122, and a packing of 123 to 126 squares contains one of 122, so it also gives $s (123), s (124), s (125), s (126) \geq 563 / 50$ : five counts in all. At $n = 127$ and beyond the record already holds more.

The measure, mixed_n122_L1126, is 502 point, 1,268 segment and 3 rectangle orbits under D4, with exact rational geometry, of total mass 12199999/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000: the measure kind, net and checker of T-080. The source reports it accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp, byte for byte T-080's checker, and a complete replay of its tarball on a fresh machine.

The value exceeds Green's reported 11.2292808755... at $n = 122$ to 125, by more than 0.0307, and Nagamochi's 11.2469507659... at $n = 126$ , and the verified Karakus strip bounds at all five.

Here the T-080 route's readers accepted its exact premises: the measure, its D4 invariance, digest, net, centre domains, checker and code identity, the 201 records, the source's audit, the proof bundle's 810 files and every angle's input. Two angles replayed here by the source's replay function and unchanged checker, from this repository's retained copies, returned the certificate's own records: a diagnostic, not a replay. 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 raises the reported and verified lower bounds at five counts, $n = 122$ to 126, past Green's reported values and Nagamochi's closed form, by T-080's certificate kind, net and checker at a new size, with no new technique, at S3 as T-080 is. V0/C0 says how far it stands, and the score gates nothing.
Composition: One reported certificate on its reported entry. $n = 123$ to 126 take it directly, its mass being below them, so no step is added. A replay here would run the source's checker unchanged, as T-080's did: one interval-certified method, and no second implementation stands beside it.
Next rung: C3 needs a complete replay of all 201 angles by the T-080 route, with the certificate registered in devtools.audit_wand125_linear so that linear-replay writes resumable receipts, the route's linear-control receipt and a review that reads this certificate, as the 2 October review read T-080's. The two-angle sample prices the replay and control at about 11.2 CPU-hours (the author's own replay suggests up to about 23). The owner set its budget on 11 October 2026, and the replay is queued in think-nd7g after the regularization work (jlevy/squares#497).
Unfinished confirmations: C3 (think-nd7g): a complete replay of all 201 angles by the T-080 route, with the certificate registered in devtools.audit_wand125_linear so that linear-replay writes resumable receipts, the route's linear-control receipt and a review that reads this certificate, as the 2 October review read T-080's. About 11.24 CPU-hours, up to about 23 (priced 2026-10-10).
Novelty: previously-published Present in an identified source

The cases

This result concerns 5 cases, too many to draw one by one. Each is listed with the film’s bounds, the proved lower bound and the best known side, and links to its case record, where its packing and number line are drawn.

n	Proved lower	Best known	Gap	Status	Records
122	11.047511	11.535534	0.48802235…	open=	frontier n-122.md
123	11.094810	11.591379	0.49656809…		frontier n-123.md
124	11.141898	11.656855	0.51495592…		frontier n-124.md
125	11.188779	11.707107	0.51832761…		frontier n-125.md
126	11.235455	11.742641	0.50718541…		frontier n-126.md

Results on these cases

13 results in the register on these cases, oldest first

2005 published T-007 · 5 of these cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
lower bound incomplete
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 5 of these cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
correction confirmed
Karakuş; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-056 · case 123
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
upper bound confirmed superseded by T-125
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 5 of these cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098 · cases 123, 126
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
upper bound confirmed on these cases, superseded by T-125
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · cases 122, 124, 125
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
upper bound confirmed
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · case 126
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
upper bound confirmed on these cases, superseded by T-125
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · cases 123, 126
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
upper bound confirmed on these cases, superseded by T-125
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-08 published T-125 · cases 123, 126
Complete rational construction reports at 25 counts
upper bound confirmed
ry-xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · case 123
Fourteen rational refinements, seventeen complete source cases
upper bound confirmed on these cases, superseded by T-125
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-09 published T-139 · case 126
A more precise certificate of Ryan Xu's 126-square arrangement, from Evan Daniel
upper bound confirmed
Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register
2026-10-10 published T-144 this result · 5 of these cases
Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported
lower bound recorded
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register
2026-10-10 published T-146 · case 123
Exact rational certificates at fifteen counts from Evan Daniel's regularized record lists
upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Mishapolk, Levy · Daniel regularized lists 2026-10-10 · packet · register

Links

On this site: The frontier survey · T-144 in the results table

On GitHub, at main

Register: T-144 in results.yaml, line 15355
Evidence: E-wand125-linear-n122-1126-report
Proofs and certificates: certificate reported-n122.json
Sources: wand125 linear n122 2026-10-10 (its own site, retained copy)
Source packet: resources/web/wand125-linear-n122-2026-10-10/README.md
Artifacts: 7 artifacts and controls
resources/web/wand125-linear-n122-2026-10-10/README.md resources/web/wand125-linear-n122-2026-10-10/reported-n122.json resources/web/wand125-linear-n122-2026-10-10/source-manifest.json resources/web/wand125-linear-n122-2026-10-10/receipts/linear-premises.json resources/web/wand125-linear-n122-2026-10-10/receipts/linear-sample-diagnostic.json devtools/fine_net_followup.py devtools/audit_wand125_linear.py
Case file: each case’s file is linked from its row above

T-144: Linear-measure lower bound s(n)≥563/50=11.26s(n)≥563/50=11.26 for n=122…126n=122…126, reported

The cases

Results on these cases

Links

T-144: Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported