T-025: $s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate

V3 C3 lower bound confirmed superseded by T-060

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

$s (11) \geq 191 / 50$ = 3.82, by a threshold certificate: 584 point atoms of mass 271052551/31250000 and 320 threshold atoms, every one 2-of-3, of budget 143352577/62500000, on the D4-symmetric site set at shrunken side 9977/10000 and the 181-direction net.

A threshold atom (S, k, w) charges w to every admissible core holding at least k points of S and costs w floor(|S| / k) of the budget, so the family's total budget is 685457679/62500000 = 10.967322864, strictly below 11, while every event cell at every net direction carries charge at least 100000203/100000000. The counting argument then rules out eleven unit squares with pairwise disjoint interiors in a container of side 191/50.

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 0.0033905 beyond T-024 and by 0.0311456 beyond Stromquist, leaving the interval [3.82, 3.877084] and a gap of 0.057084. What earns the score beyond the arithmetic is the architecture: the exact depth-one ceiling family retained at this same side proves that no D4-symmetric point-atom measure of mass below eleven exists here, so no certificate of the earlier one-body form could have reached it, and the threshold atoms carry 2.293641 of this budget that the point method cannot have. It is the first certificate of a new kind in this project, decided from frozen bytes by two routes and reviewed adversarially before registration.
Composition: One certificate, decided twice. The certificate is instantiated at its named side, 191/50; no dilation family is part of T-025.

Conditions 1, 1', 2', 3 and 4 are closed-form consequences of the frozen bytes and were recomputed without sqpack in the review; Condition 5' is decided by the exact event-cell sweep of sqpack.fractional.threshold, whose dense-grid and slab evaluations agree at every direction and whose witness charge is re-evaluated by membership counting, and independently by the interval branch and bound of sqpack.fractional.threshold_interval over the doubled net, which never invokes the D4 argument.

The two share the loader, the conditions and the net, and no part of how Condition 5' is decided; the gate refuses the record unless both accept and agree on the least charge exactly.

The result is C3 on those two machine routes. One adversarial AI review of the theorem is retained and mapped, by an agent of this project; it found no soundness defect, and its open items -- the second route, the controls, and a stranger-facing statement in a case directory -- this registration closes. It is not a human oversight record.
Next rung: Push the side with the same loop: the LP that produced these atoms is run again at a larger side, and the ceiling family says the answer cannot come from point atoms alone on the stated core domain. The finer-net and dilation continuation has now been measured and registered as T-026, proving $s (11) \geq 3.826447410572939744$ by an exact dilation-limit argument. That is a separate corollary. T-025 itself proves the ordinary lower bound $s (11) \geq 191 / 50$ at V3/C3, and its certificate is instantiated at that named side; V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record. The self-contained T-025 claim embeds its standard-library verifier and certificate bytes. T-025 needs no limit argument, so no dilation record belongs at this rung.
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 this result
$s (11) \geq 191 / 50 = 3.82$ , 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-025 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-025 in results.yaml, line 1909
Evidence: E-n011-threshold-certificate · E-n011-threshold-interval-decision
Proofs and certificates: certificate certificate.json · proof t-025-threshold-certificate-proof.md
Sources: Burns–Massaccesi n17
Artifacts: 15 artifacts and controls
cases/n11_threshold_certificate/certificate.json cases/n11_threshold_certificate/verify_claim.py cases/n11_threshold_certificate/t-025-verifiable-claim-191-50.md cases/n11_threshold_certificate/t-025-threshold-certificate-proof.md devtools/decide_threshold_certificate.py src/sqpack/fractional/threshold.py src/sqpack/fractional/threshold_interval.py campaign/series/series-000-smoke-and-calibration/results/agenda-034/lane-b-threshold-atoms-at-191-50.md campaign/series/series-000-smoke-and-calibration/results/agenda-034/ceiling-family-191-50.json tests/test_n11_threshold_certificate.py tests/test_decide_threshold_certificate.py tests/test_fractional_threshold.py tests/test_fractional_threshold_interval.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-025: s(11)≥191/50=3.82s(11)≥191/50=3.82, by a threshold certificate

The case

Results on the case

Links

T-025: $s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate