T-073: Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$

V3 C3 lower bound confirmed

2026-10-02 published · wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 6 cases, $n = 83$ to 105

Two measures of points, segments and rectangles that wand125/square-packing-bounds published on 2 October 2026 prove $s (101) \geq 257 / 25$ = 10.28 and $s (83) \geq 187 / 20$ = 9.35. A packing of 102 to 105 squares contains one of 101, so the first also gives $s (102), s (103), s (104), s (105) \geq 257 / 25$ .

Each certificate is a measure of a kind this record has not registered before: point masses, segments of uniform density and rectangles of uniform density, given in D4 orbits with exact rational geometry, of total mass n - 1/100000, at core side 9977/10000 on 201 net half-angles of step 83/40000. It has 333 point, 897 segment and 4 rectangle orbits at $n = 101$ , and 86, 222 and 774 at $n = 83$ .

The source reports each accepted at all 201 net angles, the axis included, by code/unified_linear_verify.cpp, the verifier of its superseded $s (50) \geq 147 / 20$ certificate: an outward-rounded interval branch and bound over centre boxes in one quadrant of the centre domain, which the measure's quarter-turn symmetry makes stand for the whole, counting points in the closed core, segments by the length inside and rectangles by the area inside. It says each full replay was run again from the published tarball, by the same implementation and not an independent one.

At $n = 101$ the value exceeds Green's reported 2 sqrt(2) - 1 + (810 + 18 sqrt(5))/101 = 10.2467362..., by more than 0.0332, and so it does at $n = 102$ to 104, where this record held that value by monotonicity; at $n = 105$ it exceeds Nagamochi's 10.2736.... At $n = 83$ it exceeds Green's reported 9.2667335..., held here by monotonicity from $n = 82$ , by 0.083266...; the source's "more than 0.0833" is measured from 9.2667, below Green's value, and is false, though the bound is unaffected.

Here the source was retained, its exact premises were recomputed from the retained bytes, and the replay's checks before its first angle were run on both pinned tarballs, and a negative control at $n = 101$ refused two mutated measures. Each certificate passed a complete replay here of the source's unchanged checker and replay functions on its pinned tarball, $n = 101$ 's on 2 October 2026 and $n = 83$ 's on 3 October: all 201 directions, each returning the certificate's own record. The replays run the source's own algorithm and are not an independent decision of coverage. $n = 101$ 's five counts are also registered as replayed in their own entry (T-080).

wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in jlevy/squares#294 and its comment of 2 October 2026. The source says parts of the work were produced with AI assistance under human direction.

Significance, composition and next rung

Significance: Raised the verified lower bound at five counts, $n = 101$ to 105 (T-080), and confirmed $n = 83$ 's value below the replayed 937/100 there; when registered, the strongest lower bounds on record at six counts, past Green's reported values at $n = 83$ and $n = 101$ and past Nagamochi's at $n = 105$ , by a certificate kind new to this record that adds points and segments to the rectangle densities. Further sizes from one generator at S3. The 2 October review of the certificates confirmed S3: the kind is new here, but the technique is the measure argument with Tokoharu's net and checker, extended by two capture bounds.
Composition: One claim per certificate, each on its own reported entry; $n = 102$ to 105 take the $n = 101$ certificate directly, its mass being below them, so no step is added. Each replay runs the source's C++ checker unchanged: one interval-certified method, replayed here, C3. Coverage is decided by that checker alone, which the 2 October review read function by function. $n = 83$ 's replay entry is cited here; $n = 101$ 's is entered once, in T-080, which carries $n = 101$ to 105 in the verified lane, so a case's citation names one result.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record; one adversarial review is retained, read before the replays. A second machine method would be a method-distinct decision of rotated coverage for a linear measure; the replays here run the source's own checker. linear-control n83 would put a negative control on that certificate itself; the checker's controls are $n = 101$ 's.
Novelty: previously-published Present in an identified source

The cases

This result concerns 6 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
83	9.370000	9.634758	0.26475764…	open=	frontier n-083.md
101	10.280000	10.535534	0.25553390…		frontier n-101.md
102		10.605829	0.32582869…		frontier n-102.md
103		10.679233	0.39923204…		frontier n-103.md
104		10.707107	0.42710678…		frontier n-104.md
105		10.790677	0.51067657…		frontier n-105.md

Results on these cases

19 results in the register on these cases, oldest first

2005 published T-007 · 6 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 on these cases, superseded by T-073, T-075 and T-080
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 6 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
2026-09-24 published T-089 · case 83
$s (83) \leq 9.63475764863195$ and $s (87) \leq 9.83881526994915$ , Chang's packings, certified exactly
upper bound confirmed superseded by T-101
Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis · Chang n83 2026-09-24 · Chang n87 2026-09-24 · packet · packet · register
2026-09-27 published T-056 · cases 102, 103, 105
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-058 · case 83
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
method limit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083 · 6 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 on these cases, superseded by T-073, T-075 and T-080
Karakuş · Karakuş 2026 · source · register
2026-10-02 published T-073 this result · 6 of these cases
Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-02 published T-075 · case 83
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register
2026-10-02 published T-080 · 5 of these cases
Linear-measure lower bound replayed at $n = 101 \dots 105$
lower bound confirmed
wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-05 published T-098 · cases 102, 103
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 83, 101, 104
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-117 · case 105
Exact rational ceiling refinements at $n = 105$ , 292
upper bound confirmed on these cases, superseded by T-125
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-120 · case 102
Reported s(102) ≤ alpha_102 with a degree-8 exact side form
upper bound recorded superseded by T-125
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-129 · case 105
Three dated Daniel certificate reports with historical source custody
upper bound recorded superseded by T-125
Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register
2026-10-08 published T-125 · cases 102, 103, 105
Complete rational construction reports at 25 counts
upper bound confirmed
ry-xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-128 · case 105
Eight complete rational refinements from Francisco Couzo
upper bound confirmed
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · case 105
Five follow-up rational refinements from Francisco Couzo
upper bound confirmed
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-138 · cases 103, 105
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
upper bound confirmed
Mishapolk after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-146 · cases 102, 103
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-073 in the results table

On GitHub, at main

Register: T-073 in results.yaml, line 6761
Evidence: E-n083-wand125-linear-935-report · E-n083-wand125-linear-935-source-replay · E-n101-wand125-linear-1028-report
Proofs and certificates: certificate candidate.json.gz · proof continuous-density-certificate.ja.md · audit review-2026-10-02-wand125-linear-certificates-and-n76.md · certificate candidate.json.gz
Sources: wand125 linear certificates 2026-10-02 (its own site, retained copy)
Source packet: resources/web/wand125-linear-certificates-2026-10-02/README.md
Artifacts: 10 artifacts and controls
resources/web/wand125-linear-certificates-2026-10-02/README.md resources/web/wand125-linear-certificates-2026-10-02/square-packing-bounds/certificates/mixed_n101_L1028/README.md resources/web/wand125-linear-certificates-2026-10-02/square-packing-bounds/certificates/mixed_n83_L935/README.md resources/web/wand125-linear-certificates-2026-10-02/square-packing-bounds/certificates/mixed_n101_L1028/code/unified_linear_verify.cpp resources/web/wand125-linear-certificates-2026-10-02/receipts/linear-audit.json resources/web/wand125-linear-certificates-2026-10-02/receipts/n83/merged.json resources/web/wand125-linear-certificates-2026-10-02/receipts/n101/merged.json devtools/audit_wand125_linear.py resources/web/wand125-linear-certificates-2026-10-02/receipts/n101/control.json tests/test_wand125_linear_certificates.py
Case file: each case’s file is linked from its row above

T-073: Linear-measure lower bounds replayed at n=83n=83 and n=101…105n=101…105

The cases

Results on these cases

Links

T-073: Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$