T-026: $s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$

V3 C3 lower bound confirmed superseded by T-060

2026-09-09 established · Levy · $n = 11$

$s (11) \geq 955000$ *sqrt(518400042893309449)/179696714646249 = 3.826447410572939, proved by an exact dilation-limit corollary of T-025's threshold certificate re-certified on a finer direction net.

The 584 point atoms and 320 threshold atoms of the retained 191/50 threshold certificate, on the same D4-symmetric sites and with every weight multiplied by 500000000/498684619, form a threshold certificate of the retained form at side 191/50 on the 1440-step net with shrunken side 249507/250000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell charge exactly 1. The rescaled total budget is 5483661432/498684619 = 10.996251384, still strictly below eleven.

For every positive rational q with q^2 below 32400002680831840562500000000/32290909254655439869209770001, common scaling of the point atoms, the threshold atoms' points, the container side and the shrunken side preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 191/50. Rational density and upward embedding exclude every real side below the supremum, which proves the displayed lower bound. The direct certificate family covers strict rational sides below the supremum; the density-and-embedding step supplies the exact >= conclusion at the supremum.

Significance, composition and next rung

Significance: The rubric's S5 anchor is movement on a central open case, and this raises the proved lower bound for the smallest open case by about 0.0064474 beyond T-025's 191/50 and by about 0.0375930 beyond Stromquist's 2 + 4/sqrt(5), leaving the interval [3.826447410572939, 3.877084] and a bound gap of about 0.0506362.

The movement is the shrink a finer net admits under Condition 4, measured on the frozen T-025 atoms; no atom was re-optimised and no new theorem was needed, since Conditions 1, 1', 2' and 3 are untouched by common scaling and Condition 5' is preserved by inverse dilation of placements exactly as Condition 5 is.

For this fixed certificate, the rescaled budget is 5483661432/498684619 = 10.996251384 against the eleven a certificate may not reach, leaving about 0.0037 of budget. The measured 720- and 1440-step nets share the same crossing shrink and least unscaled charge. These finite measurements leave the effect of another net, a different core side, re-optimized weights, changed sites, and richer charge atoms unresolved.
Composition: Derived from a full replay of the 1440-step threshold source's Conditions 1, 1', 2', 3, 4 and 5' by the threshold theorem's own exact sweep, scaling equivariance for Conditions 1 and 1', invariance of Conditions 2' and 3, inverse-dilation preservation of Condition 5', and the sharpened containment lemma T-022 proved by an exact rational-square comparison; rational density and upward embedding supply the limit step.

The confirmation rests on the two machine decisions of the source certificate, which fail differently: the exact event-cell sweep of sqpack.fractional.threshold, whose dense-grid and slab evaluations must agree at every direction, and the interval branch and bound of sqpack.fractional.threshold_interval over the doubled net, which never invokes the D4 argument.

One adversarial AI review is retained and mapped, of the self-contained claim, by a separate agent from the implementation authors; it covers the threshold count, exact sweep, dilation, and endpoint inference. It is not a human oversight record, so the result is C3. T-025's earlier review covers the threshold theorem and T-022's covers the dilation argument; the new review checks their composition with the T-026 values and embedded data.

The 720-step certificate retained beside it crosses at the same shrink and reaches 955000*sqrt(129600042893309449)/89874194646249 = 3.825347845913112, which isolates the remaining 0.0010996 as the effect of D halving alone.
Next rung: The rescaled 1440-step budget is 5483661432/498684619, leaving 1869377/498684619 below eleven. The measured 720- and 1440-step certificates have the same crossing shrink and least unscaled charge. These finite measurements do not decide another net or prove that frozen-atom refinement cannot improve the limit. Re-certification on another net, re-optimization on a finer net, changed site generation, and richer charge atoms remain distinct hypotheses. The retained plateau duals with depth above one are useful separation inputs but not globally feasible obstruction witnesses.

The self-contained T-026 claim embeds the standard-library threshold verifier, the 1440-step certificate, and the exact dilation record; its mapped source-distinct review is the one adversarial AI review retained. V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; rung 5 would require a proof-assistant formalization reviewed by human experts.
Novelty: apparently-novel Not found in the recorded search, subject to its stated gaps

The case

Case record

3.877

345

3.3174.317

\sqrt{n}

\sqrt{n} + 1

Proven

$s (11) =$ $3.877084$

optimal
exact
rigid

Citation record n-011

lowerAhmed after Levy, Kleddamag 2026, GitHub (confirmed T-060)

upperTrump 1979, Squares in Squares (confirmed T-011)

The case record

LowerUpper

Verified3.87708359…3.87708359…

Reported3.87708359…3.87708359…

Gap0 solved: the verified bounds meet

Results on the case

23 results in the register on $n = 11$ , oldest first, each with what it established and how it stands now.

1979 published T-011
Trump's 1979 packing is exactly valid, so $s (11) \leq 3.877083590022814 \dots$
V3 C3 upper bound confirmed
Trump · Trump 2023 · register
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-060
Nagamochi · Nagamochi 2005 · source · register
2026-08-24 established T-010
$s (11) \geq 2 + 4 / \sqrt{5}$ , by a repair of Stromquist 2003's Figure 14 point set
V3 C3 lower bound confirmed superseded by T-060
Levy after Stromquist · register
2026-09-04 established T-018
$s (11) \geq 381 / 100 = 3.81$
V3 C3 lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · 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; chelokot · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-06 established T-022
$s (11) \geq 38100 \sqrt{8100042893309449} / 899996306539 = 3.8100257 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · source · register
2026-09-08 established T-023
Conditional exclusion: no eleven-square packing in the four-owner branch at $q = 96 / 25$
V3 C3 case exclusion confirmed superseded in part by T-060
Levy · source · register
2026-09-09 established T-024
$s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · source · register
2026-09-09 established T-025
$s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate
V3 C3 lower bound confirmed superseded by T-060
Levy · source · register
2026-09-09 established T-026 this result
$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy · source · register
2026-09-20 established T-031
The octagon corner class (threshold $1 / 2$ ) holds no eleven-square packing at side $96 / 25$
V3 C3 case exclusion confirmed superseded by T-060
Levy · register
2026-09-22 established T-033
$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$
V3 C3 lower bound confirmed superseded by T-060
Levy · source · register
2026-09-22 published T-037
$s (11) > 31 / 8 = 3.875$
V3 C3 lower bound confirmed superseded by T-060
Kleddamag after Levy, Guzhou0806, Mira · Kleddamag n11 2026 · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-22 published T-047
$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$
V3 C3 lower bound confirmed superseded by T-060
Tokoharu after Levy, wand125, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 2026 · packet · source · review · register
2026-09-24 established T-035
Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within rho of his pose
V3 C3 case exclusion confirmed
Levy · register
2026-09-24 established T-036
Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry
V3 C2 restricted optimality confirmed superseded in part by T-060 and T-112
Levy · 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
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-059
Reported equality of 12028 n11 row minima reproduced by a complete bound replay
V3 C3 audit confirmed
wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-060
Trump's eleven-square packing is globally optimal
V3 C3 optimality confirmed
Ahmed after Levy, Kleddamag · Ahmed n11 optimality 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-29 published T-061
$s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$
V3 C3 lower bound confirmed superseded by T-060
Wang, Li after Kleddamag, Levy · Wang Li n11 2026 · packet · source 1 · source 2 · review · 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-060
Karakuş · Karakuş 2026 · source · register
2026-10-06 established T-112
Trump's packing is the only optimal packing of eleven squares, up to symmetry
V3 C2 uniqueness confirmed
Levy after Ahmed · packet · register
2026-10-07 published T-124
Reported non-strict local minima for 178 source configurations
V0 C0 restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

Links

On this site: Case record, $n = 11$ · Frontier row, $n = 11$ · T-026 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-026 in results.yaml, line 2018
Evidence: E-n011-threshold-net1440-certificate · E-n011-threshold-net1440-interval-decision · E-n011-threshold-net1440-dilation-limit · E-n011-threshold-net720-certificate · E-n011-threshold-net720-dilation-limit
Proofs and certificates: certificate certificate-191-50-net1440.json · certificate t-026-dilation-limit-corollary.json · proof t-026-dilation-limit-proof.md · certificate certificate-191-50-net720.json · certificate t-026-net720-dilation-limit-corollary.json
Sources: Burns–Massaccesi n17
Artifacts: 15 artifacts and controls
cases/n11_threshold_certificate/certificate-191-50-net1440.json cases/n11_threshold_certificate/t-026-dilation-limit-corollary.json cases/n11_threshold_certificate/verify_claim.py cases/n11_threshold_certificate/t-026-verifiable-claim-dilation-limit.md cases/n11_threshold_certificate/certificate-191-50-net720.json cases/n11_threshold_certificate/t-026-net720-dilation-limit-corollary.json cases/n11_threshold_certificate/t-026-dilation-limit-proof.md devtools/measure_threshold_net_refinement.py devtools/dilation_corollary.py campaign/series/series-000-smoke-and-calibration/results/agenda-034/lane-a2-threshold-certificate-on-finer-nets.md tests/test_dilation_corollary.py tests/test_measure_threshold_net_refinement.py tests/test_decide_threshold_certificate.py tests/test_verify_threshold_claim.py tests/test_verify_threshold_claim_oracle.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-026: s(11)≥955000518400042893309449​/179696714646249=3.8264474…s(11)≥955000518400042893309449/179696714646249=3.8264474…

The case

Results on the case

Links

T-026: $s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$