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V3 C3 case exclusion confirmed superseded in part by T-060

2026-09-08 established · Levy · $n = 11$

At q = 96/25, if four distinct unit squares have selected strict cores of side B = 9977/10000 containing, respectively, the four closed rational patches in arms.endpoint.footprint_union of the retained exp143 receipt, at most five further unit squares fit. Thus the specified four-owner branch contains no eleven-square packing. The five saved rational dots meet every admissible remaining core on the complete 361-direction net; audited containment and boundary arguments transfer this exact check to arbitrary physical angles. This is a conditional exclusion and changes no global bound for $s (11)$ .

Superseded in part by T-060. The conclusion, that the four-owner branch holds no packing of eleven squares at side 96/25, which T-060 gives for every packing of eleven squares, since 96/25 is below T. The count is not implied: that beside four squares holding the four patches at most five further squares fit, nine in all, is a statement about ten squares, on which T-060 says nothing.

Significance, composition and next rung

Significance: A substantive case exclusion at a central open size, demonstrating that forced occupied area can make a five-dot counting argument sufficient. It excludes one owner combination and does not move the global bound.
Composition: Exact full-net coverage and frozen-data identities are machine-checked and replayed here. Universal owner footprint containment, extension to event boundaries, and strict-core selection for physical squares are audited analytic proof steps, so the composed theorem is V3. Exp145 adds an independent inclusion-exclusion computation of the same finite cover. The analytic owner and physical-angle transfer premises remain shared, and the composed theorem is C3; C4 would need two adversarial AI reviews by distinct reviewers and a human oversight record.
Next rung: Exp145 completes the independent finite-union check. Assemble a review packet for the shared analytic premises before revisiting confirmation. V4 for the whole theorem requires mechanizing the analytic transfer, and with C4 two adversarial AI reviews by distinct reviewers and a human oversight record.

Exp146 enlarged twelve wall-aware footprints, but exp147 transferred no additional tuple. Exp149 refuted the selected five-dot extension and exp150 showed its escape survives individual owner-core constraints. Exp151 also refuted a fixed sixth-site extension. Exp152 found a nonempty two-core site region, but exp153 then refuted the entire fixed-D-plus-one-site family with 188 exact support directions.

Next audit a short actual-core obstruction and determine whether it is a disjoint pair or only a higher-order obstruction before drawing weighted-mass conclusions. Wider owner-class coverage remains a new research result, not a rung change.
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 this result
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-023 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-023 in results.yaml, line 1692
Evidence: E-n011-five-dot-full-net · E-n011-five-dot-independent-union · E-n011-five-dot-physical-transfer
Proofs and certificates: certificate exp-144-four-owner-endpoint-full-net-replay.json · certificate exp-145-independent-five-dot-union.json · proof five-dot-transfer-review.md
Artifacts: 11 artifacts and controls
campaign/series/series-000-smoke-and-calibration/results/agenda-032/exp-143-four-owner-footprint-cover.json campaign/series/series-000-smoke-and-calibration/results/agenda-032/exp-144-four-owner-endpoint-full-net-replay.json campaign/series/series-000-smoke-and-calibration/results/agenda-032/proofs/endpoint-footprint-review.md campaign/series/series-000-smoke-and-calibration/results/agenda-032/proofs/five-dot-transfer-review.md campaign/series/series-000-smoke-and-calibration/results/agenda-032/sprint-report.md campaign/series/series-000-smoke-and-calibration/results/agenda-033/exp-145-independent-five-dot-union.json cases/n11_five_dot_cover/union-contract.md tests/test_owner_footprints.py tests/test_multi_owner_domains.py tests/test_replay_owner_footprint_cover.py tests/test_independent_five_dot_union.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-023: Conditional exclusion: no eleven-square packing in the four-owner branch at q=96/25q=96/25

The case

Results on the case

Links

T-023: Conditional exclusion: no eleven-square packing in the four-owner branch at $q = 96 / 25$