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

2026-09-24 established · Levy · $n = 11$

Every packing of eleven unit squares in a square container, six at orientation 0 and five sharing one orientation modulo pi/2 with half-tangent in [91442076901/250000000000, 73154061521/200000000000] (an interval containing [t* - 10^-6, t* + 10^-6] for Trump's exact half-tangent t* = 0.365769307604677...), whose side is at most the rational U_hi of the certificate header (U_hi - U = 2.03e-45), lies, after the quarter turn that puts its tilted centroid in the closed upper-right quadrant and the relabelling that orders each class by x + y/4, strictly within rho = 808514697/200000000000 of Trump's labelled image (rotation 1, labels [3,4,2,5,0,1,8,10,6,9,7]) in every centre coordinate, with every tilted orientation within 2.0e-6 radians of Trump's; every other packing in the family has side greater than U_hi.

The statement is a machine-verified reduction, not optimality: a closed exact cell tree of 139,441,005 records, replayed in full by an independent reader, puts every small packing in the family near Trump's pose. It does not invoke the BC-240 local theorem. T-036 composes it with that theorem's first clause.

Significance, composition and next rung

Significance: A substantive case result and machine audit at the smallest open case: one certificate tree, closed with no unresolved leaf, reduces a whole restricted family to a neighbourhood of Trump's pose. It moves no bound on $s (11)$ , its family is one box of half-tangents 2e-6 wide, and its cost, about 1.7e8 nodes for that one box, argues against S4's reusable technique. The 2026-09-24 closed-tree review scored the composed theorem S3 on the same grounds.
Composition: Not compound. One certificate tree and one reader verdict carry the whole statement, with no local theorem, monotonicity or composition step between them. The reader closes a Trump-degenerate leaf by exact arithmetic, bounding every centre coordinate of the cell's packings within rho of the labelled image, which is why the statement stops at the rho-ball rather than at Trump's pose. The entry is C3 (repository-origin, exact-algebraic, a certificate, a replay and a passing status). One adversarial AI review is retained, the mapped 2026-09-24 closed-tree review: a Fable max W2 review by a different agent than the lanes that produced the tree, inside the same project. It is not an external review and not a human oversight record.
Next rung: V4 and C4 need a second adversarial AI review by a distinct reviewer and a human oversight record; rung 5 would need a proof-assistant formalization of the cell-tree contract and its reader, reviewed by human experts. Three steps would strengthen the record without moving a rung. The certificate could be retained where a gate can reach it; until then the manifest is the record's only hold on it. The reader could compare the declared box with t* +- 10^-6 itself, a comparison only the review has made. A reader that accepts a non-root cell would make the $n = 11$ negative control reader-checked. The mathematics beyond this result is rung 1 of the ladder, H-112, which H-242's pilot (BC-383) prices: a per-node bound that closes this box in far fewer than 1.7e8 nodes comes before more boxes.
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
$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$
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 this result
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-035 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-035 in results.yaml, line 3009
Evidence: E-n011-h236-rung0-reduction
Artifacts: 7 artifacts and controls
campaign/series/series-000-smoke-and-calibration/results/agenda-042/exp-232-h236-reader-final2.json.gz campaign/series/series-000-smoke-and-calibration/results/agenda-042/exp-232-h236-tree-manifest.sha256 campaign/series/series-000-smoke-and-calibration/results/agenda-042/exp-232-h236-frozen.txt cases/trump11/fixed_angle_tree.py cases/trump11/fixed_angle_tree_check.py campaign/series/series-000-smoke-and-calibration/experiments/exp-232-h236-rung-zero-closed.md tests/test_fixed_angle_tree.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-035: Six-plus-five packings near Trump's tilt with side ≤Uhi​≤Uhi lie within rho of his pose

The case

Results on the case

Links

T-035: Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within `rho` of his pose