T-033: $s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$

V3 C3 lower bound confirmed superseded by T-060

2026-09-22 established · Levy · $n = 11$

$s (11) \geq 955000$ *sqrt(2073600042893309449)/359341754646249 = 3.826997548829543624, proved by an exact dilation-limit corollary of T-025's threshold certificate re-certified on the 2880-step 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 2880-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.

The shrink is T-026's and does not move under refinement; what this net changes is the largest half-gap tangent, 207107/1440000000 over the 2881 directions of the net, exactly half the 1440-step value, and the bound rises with it.

For every positive rational q with q^2 below 129600002680831840562500000000/129126496632245014779129770001, 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.

What that conclusion does not carry is stated with it. No individual certificate is supplied at the supremum, because the sharpened containment test is equality there; no strict inequality is established there; and 191/50 remains the largest side at which the retained T-025/T-026/T-033 fixed-core family holds an individual-side certificate, which is T-025's and not this result's.

Significance, composition and next rung

Significance: This is a substantive controlled case result and machine audit of this project's retained threshold family. It moves T-026 from 3.826447410572939744 to 3.826997548829543624 by halving D inside T-022's sharpened containment factor sqrt(1 + D^2)/(B(1 + D)), at T-026's own crossing shrink, which does not rise under refinement; Condition 4's own coarse test B(1 + D) < 1 has the lower ceiling 152800000000000/39926861627361 = 3.826997509248, which this value exceeds. The sharpened factor is what reaches; 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.

The certificate is the same measure T-026 registered: the rescaled budget is unchanged at 5483661432/498684619 = 10.996251384, still strictly below the eleven a certificate may not reach, and only the net and the half-gap tangent differ.

The unchanged family cannot reach the stronger current verified bound 31/8: its refinement ceiling 955000/249507 is about 3.82755, already below 3.875. The result therefore records controlled method and calibration evidence, rather than a current public-bound advance. Changed weights, sites, parent domains, and richer charge atoms remain unresolved.
Composition: Derived from a full replay of the 2880-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 comparison of squares; rational density and upward embedding supply the limit step.

The source certificate's own acceptance rests on two machine decisions of one frozen file that 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.

The derived bound does not inherit that. The dilation step on top of them is decided by one exact-algebraic entry and by nothing else, and epistemics.md takes a derived claim to the minimum over its inputs and the derivation itself, and the derivation is machine-replayed here like its inputs, so the rung is C3. T-022 read its own case the same way. This reconciliation applies the same rule to T-024, whose earlier declaration counted only the source certificate's two routes.

A method-distinct decision of the corollary itself, an interval enclosure of the strict factor test and of c squared recorded as its own interval-certified entry, not the source's two routes, would give the derivation a second machine method. No review has been mapped, so rung 4, which needs two adversarial AI reviews by distinct reviewers and a human oversight record, is not in reach.

The step is T-022's argument reused unchanged, at the same B and a halved D, whose proof note is cited here rather than rewritten. T-026's 1440-step rung is the control at the coarser net -- the same atoms, the same weights and the same crossing shrink, the two records differing in direction_steps, in their id and provenance, and in no other field -- and the refinement run reproduced its registered surd exactly, so the 0.000550138 between the two rungs is the halving of D alone.
Next rung: V4 and C4 need two adversarial AI reviews by distinct reviewers, of this corollary and the frozen 2880-step certificate, each a document under docs/project/reviews/ entered in docs/project/document-map.yaml with role review, and a human oversight record. A method-distinct decision of the corollary itself would be shown beside the rung. T-026's mapped review is scoped to the 1440-step claim and reviewing that one is not reviewing this one, so it is not cited here.

No self-contained verifiable-claim document is rendered for this rung either: devtools.render_verifiable_claim carries one branch per retained threshold claim and its T-026 branch is bound to that rung's witness direction, surd and proof note, so a third rung needs the branch parameterised and a proof note of its own.

The rescaled budget is 5483661432/498684619, leaving 1869377/498684619 below eleven, and the side this rung reaches is within about 0.000550 of 955000/249507, which is all that any further refinement of the net can buy while the shrink stands where it does. Refinement is therefore near exhausted on these atoms, and re-optimization at a finer net, changed site generation and richer charge atoms remain distinct hypotheses rather than a continuation of this one. A further assurance step would require proof-assistant formalization, external review, or an independently reproduced full decision.
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
$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$
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 this result
$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$
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-033 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-033 in results.yaml, line 2732
Evidence: E-n011-threshold-net2880-certificate · E-n011-threshold-net2880-interval-decision · E-n011-threshold-net2880-dilation-limit · E-n011-threshold-net1440-certificate · E-n011-threshold-net1440-dilation-limit
Proofs and certificates: certificate certificate-191-50-net2880.json · certificate t-033-dilation-limit-corollary.json · proof t-026-dilation-limit-proof.md · certificate certificate-191-50-net1440.json · certificate t-026-dilation-limit-corollary.json
Sources: Burns–Massaccesi n17
Artifacts: 12 artifacts and controls
cases/n11_threshold_certificate/certificate-191-50-net2880.json cases/n11_threshold_certificate/t-033-dilation-limit-corollary.json cases/n11_threshold_certificate/t-026-dilation-limit-proof.md devtools/measure_threshold_net_refinement.py devtools/dilation_corollary.py devtools/decide_threshold_certificate.py campaign/series/series-000-smoke-and-calibration/results/agenda-041/exp-226-n11-net2880-receipt.md tests/test_dilation_corollary.py tests/test_measure_threshold_net_refinement.py tests/test_decide_threshold_certificate.py tests/test_fractional_threshold.py tests/test_fractional_threshold_interval.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-033: s(11)≥9550002073600042893309449​/359341754646249=3.8269975…s(11)≥9550002073600042893309449/359341754646249=3.8269975…

The case

Results on the case

Links

T-033: $s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$