T-018: $s (11) \geq 381 / 100 = 3.81$

V3 C3 lower bound confirmed superseded by T-060

2026-09-04 established · Levy after Burns, Massaccesi · $n = 11$

$s (11) \geq 381 / 100$ , by this project's weighted fractional unavoidable-set certificate at container side 381/100 = 3.81. This improves Stromquist's 2 + 4/sqrt(5) = 3.788854..., stated in 1984 (Memo III, p. 10) and published in 2003.

No intervening improvement was found by the recorded search, and $n = 11$ was then the smallest open case. The movement was +0.021146, and the case stayed open against Trump's 1979 packing at 3.877084: the interval narrowed from 0.088230 to 0.067084. Two rungs are retained below it -- 19/5, the value that first passed Stromquist, and the 189/50 calibration rung below him.

Significance, composition and next rung

Significance: Scored against the rubric's own anchor for S5, "movement on a central open case". $s (11)$ is the smallest open case and a central one for this project; the recorded public search found no movement beyond Stromquist's value, stated in 1984 and published in 2003, and this displaces a bound from the refereed literature rather than an inherited or closed-form one.

What S5 does not claim: the weighted-resource lineage predates this project and the pure-atomic direction-net architecture follows Burns and Massaccesi, so what is new is an instance and the generator that found it; apparent novelty is not absolute priority; and the remaining gap to Trump's 1979 packing is 0.067, which this approach does not close. The calibration rung at 189/50, below Stromquist's bound, was run first by design and is retained.
Composition: Primary: one certificate, one verifier, one accepted verdict. The bound is the certificate's container side directly, with no monotonicity or composition step. It raises no other case: every n > 11 already carries a larger verified bound.
Next rung: A second machine method decides it, the interval-certified decision, which is independent in method and, deciding coverage on the doubled net, never invokes the D4 reflection and so does not need Condition 1 at all. One adversarial AI review is retained, PR 78's, ported here as a dated record on 2026-09-05; V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record. An outside mathematical review and an archival release would strengthen the priority record without changing this rung.

The bound itself: 3.81 converged with margin -- objective 10.8603 against eleven, rationalised to 434547/40000 = 10.863675 -- so the next rung is a search question rather than a limit of the method. Getting there took two attempts at the same side, and the difference between them is worth keeping. The first stopped with its row loop still finding violated placements (final least 0.9764) and was refused by the exact sweep at directions 55 to 63, least cell 199531/200000. The second ran the loop to convergence -- three row rounds, 25318 rows, final oracle least mass exactly 1 -- and was accepted at every direction. An objective below n proves nothing on its own; the loop's stopping condition is the thing to read.

Where it stops: 3.82 was then attacked from both sides and neither route closes. The covering LP was run on two independent site sets, and the two stop at exactly eleven for different reasons, which is worth separating because only one of them converged.

The grid-built set, 6637 sites and 14820 rows at the end, ran its row loop to convergence and finished at an objective of exactly 11.000000, descending to it from 11.6 over twelve rounds and never crossing below.

The set seeded from this certificate's own 1121 atoms, 24069 sites, has not converged and does not need to. Its objective has stood at 11.000000 through twenty-four row rounds and 30240 rows while the loop's least covered mass climbed from 0.8490 towards 1, reaching 0.9997 at the round the run was stopped -- so violated placements remained and the loop had not exhausted. Adding rows can only raise a restricted optimum, so the optimum for that site set is at least eleven whatever the loop does next, and no certificate exists on it. Reading the objective alone would have understated this: the loop's stopping state is what makes the conclusion independent of finishing.

A certificate needs mass strictly below eleven, so neither site set yields one. The rejection route does not close either, and it is much further from closing than the search's own progress figures suggest. H-061's object was built and decided exactly: the converged dual is 76 squares, 608 after the D4 images, with a raw total of exactly 11, and sqpack.fractional.ceiling checked all 1650944 vertices of their arrangement, 272244 of them in exact arithmetic. The maximum pointwise depth is 1925/1152 = 1.671007, so the feasible total -- the raw total scaled to make the family a packing -- is 1152/175 = 6.5829 against the eleven a ceiling needs.

The gap between that and the search's running estimate is the thing to carry forward. Column generation prices against a grid sample and reported a depth of 12/11 = 1.0909, which would have made the feasible total 121/12 = 10.08. The exact maximum is 53 per cent higher, because depth peaks at vertices of the arrangement that no grid samples. A ceiling must therefore be judged on the exact check and never on the sampled depth, which flatters it.

One thing 3.82 is not: the method's ceiling. A certificate for n cannot exist above ceil(sqrt(n)) * B, because a wider container holds ceil(sqrt(n))^2 pairwise disjoint axis-parallel B-squares whose masses Condition 5 forces past n; for $n = 11$ that is 4B = 3.9908, leaving 0.1808 of runway above the retained 381/100. The wall at 3.82 is a covering wall, met far below where the method stops working.

If tau*(3.82) is exactly eleven then this is the one configuration where neither pre-registered route can close, since a certificate needs mass below n and a ceiling needs the scaled dual to reach n, and both fail by an infinitesimal at exactly n. That is a limit on the reach of the rejection route rather than a stalled search. It is recorded as measurement: two LP runs agreeing to six decimals is evidence about the restricted optima, not a proof that tau* equals eleven, and nothing here rules out a site set neither run found.
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 this result
$s (11) \geq 381 / 100 = 3.81$
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
$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$
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-018 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-018 in results.yaml, line 1041
Evidence: E-n011-fractional-certificate · E-fractional-interval-decision
Proofs and certificates: certificate certificate.json
Sources: Burns–Massaccesi n17
Artifacts: 11 artifacts and controls
cases/n11_fractional_certificate/certificate.json cases/n11_fractional_certificate/certificate-19-5.json cases/n11_fractional_certificate/certificate-189-50.json cases/n11_fractional_certificate/t-018-proof-visual.svg cases/n11_fractional_certificate/t-018-proof-card.md cases/n11_fractional_certificate/minimal_verify.py cases/n11_fractional_certificate/verify_claim.py cases/n11_fractional_certificate/t-018-verifiable-claim-19-5.md cases/n11_fractional_certificate/t-018-verifiable-claim-381-100.md src/sqpack/fractional/colgen.py tests/test_fractional_certificate.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-018: s(11)≥381/100=3.81s(11)≥381/100=3.81

The case

Results on the case

Links

T-018: $s (11) \geq 381 / 100 = 3.81$