T-024: $s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$

V3 C3 lower bound confirmed superseded by T-060

2026-09-09 established · Levy after Burns, Massaccesi · $n = 11$

$s (11) \geq 3175000$ *sqrt(518400042893309449)/598960960743657 = 3.816609502788862, proved by an exact dilation-limit corollary of T-018's retained atoms re-certified on a finer direction net.

The 1121 atoms of the retained 381/100 certificate, on the same D4-symmetric sites and with every weight multiplied by 200000/198931, form a certificate of the retained form at side 381/100 on the 1440-step net with shrunken side 2494953/2500000, decided from its frozen bytes by the exact event-cell sweep and by the interval branch and bound, which agree at least cell mass exactly 1.

For every positive rational q with q^2 below 3240000268083184056250000000000/3228788092454681596330191602841, common scaling preserves the source hypotheses other than containment, and the exact sharpened containment lemma rules out a packing at side q times 381/100. 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 0.0065838 beyond T-022 and by about 0.0066095 beyond the T-018 certificate side, for about 0.0277551 beyond Stromquist in total. The movement comes from the shrink that a finer net admits under Condition 4, measured on the frozen T-018 atoms; it is not a new certificate architecture, no atom was re-optimised, and the gap to Trump's packing is still about 0.0604741. The same measurement shows where this route ends: the exact depth-one ceiling family at 191/50 caps every one-body point certificate at unit side 3.8288, and these certificates sit about 0.011 below it.
Composition: Derived from a full replay of the 1440-step source certificate's five conditions, scaling equivariance for Condition 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 source certificate is decided by two distinct machine methods, the exact event-cell sweep and the interval branch and bound on its frozen bytes. The load-bearing dilation-limit step has one exact-algebraic decision. Both are machine-replayed here, so the whole result is C3 under the derived-claim minimum rule. No review has been mapped for this result.

The 720-step certificate retained beside it is the rung the standalone reader also decides; its own limit, 38100000*sqrt(129600042893309449)/3594594251080001, is registered as evidence and is weaker.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers, of the corollary and the two frozen certificates, and a human oversight record. A method-distinct decision of the corollary itself, rather than a second decision of only the source certificate, would be shown beside the rung; T-022's review covered the argument at the 181-step net and this result reuses it unchanged with new B and D.

The standalone reader verify_claim.py decides the 720-step rung and refuses the 1440-step rung by its declared 1000-direction ceiling; a new reader release that lifts the ceiling would let a stranger decide the registered rung without trusting this project, and that is a change to a retained verifier rather than to the mathematics. The shrink is a 10^-7 grid crossing, not a minimum, so a smaller passing shrink off the grid would move the endpoint by less than the grid step.

Two limits are measured on the frozen atoms: at the original shrink the same atoms fail every finer net at an intermediate direction, so the gain is entirely the shrink the finer net admits, and the exact ceiling family at 191/50 caps every one-body point certificate at unit side 3.8288 on any net containing its six directions.

The next movement is therefore a re-optimised certificate at a finer net (H-153), which the covering LP on the frozen sites already shows to be below eleven at 720 steps, and past the cap the threshold atoms of H-152. The displayed lower bound is proved by a dilation-limit argument rather than one individual-side certificate at the algebraic value.
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 this result
$s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$
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
$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$
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-024 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-024 in results.yaml, line 1788
Evidence: E-n011-fractional-net1440-certificate · E-n011-fractional-net1440-interval-decision · E-n011-fractional-net1440-dilation-limit · E-n011-fractional-net720-certificate · E-n011-fractional-net720-dilation-limit
Proofs and certificates: certificate certificate-381-100-net1440.json · certificate t-024-net1440-dilation-limit-corollary.json · proof t-024-dilation-limit-proof.md · certificate certificate-381-100-net720.json · certificate t-024-dilation-limit-corollary.json
Sources: Burns–Massaccesi n17
Artifacts: 12 artifacts and controls
cases/n11_fractional_certificate/certificate-381-100-net1440.json cases/n11_fractional_certificate/t-024-net1440-dilation-limit-corollary.json cases/n11_fractional_certificate/certificate-381-100-net720.json cases/n11_fractional_certificate/t-024-dilation-limit-corollary.json cases/n11_fractional_certificate/t-024-dilation-limit-proof.md devtools/measure_net_refinement.py devtools/dilation_corollary.py devtools/decide_certificate.py campaign/explorations/X-023-three-losses-and-a-new-atom.md tests/test_dilation_corollary.py tests/test_fractional_certificate.py tests/test_fractional_interval.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-024: s(11)≥3175000518400042893309449​/598960960743657=3.8166095…s(11)≥3175000518400042893309449/598960960743657=3.8166095…

The case

Results on the case

Links

T-024: $s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$