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V3 C3 case exclusion confirmed superseded by T-060

2026-09-20 established · Levy · $n = 11$

At L = 96/25 and B = 9977/10000 on the 181-direction net (half-tangents k*207107/90000000, k = 0..180), the D4-symmetric point measure of total mass 10868617/1000000 = 10.868617 in cases/n11_corner_class_certificate/certificate.json, the retained exp-220 covering, charges at least 2000013/2000000 to every closed B-square at a net direction whose minimum of x + y is at least 1/2 in each of the four corner frames, decided by the exact event-cell sweep and by the interval branch and bound, which agree at that value.

Consequently every packing of eleven unit squares in [0, 96/25]^2 contains a unit square whose minimum of x + y in some corner frame is strictly below 1/2, that is, one that meets the open corner triangle x + y < 1/2 at that corner; equivalently, the all-free (octagon) class of lane-a Theorem B at threshold 1/2 contains no eleven-square packing at side 96/25.

This is a conditional exclusion of one corner-bin class and changes no bound on $s (11)$ ; the all-deep class is separately known to be outside this language's reach (BC-366), so the corner tree cannot close at 96/25 by clipping alone.

Superseded by T-060. The exclusion, that every packing of eleven unit squares in [0, 96/25]^2 has a square meeting an open corner triangle, so that the octagon class holds none at side 96/25. T-060 gives every packing of eleven squares a side of at least T, which is above 96/25, so there is no such packing.

Significance, composition and next rung

Significance: The first decided item of the corner-conditioned point language at a central open size: one D4 class of the corner tree is excluded at 3.84 by a certificate on a clipped domain, with the structural consequence that every eleven-square packing in a square of side 96/25 has a square meeting an open corner triangle of depth 1/2. It moves no bound, and the registration review shows the tree it belongs to cannot close at this side in the point language alone, since the all-deep class is a Lemma-D theorem for every site set (BC-366); the mixed classes are a bounded follow-up.
Composition: The certificate is a weighted fractional unavoidable-set certificate of the retained form whose row domain is the D4-symmetric convex clip of the admissible core domain by the four corner triangles at depth 1/2, admitted as a domain predicate on the row generator and threaded through both gate routes and the ceiling readers in Session 145 with a residual-7 control on the transported 88-family.

The declared C3 is what the two cited atoms derive. The exact event-cell sweep and the interval branch and bound are the two internal routes of one decide_certificate --corner-clip 1/2 invocation, which refuses the file unless both accept: they share the Certificate parser and the closed-form conditions, carry a byte-identical replay string, and the exact leg records relationship_to_generator: same-implementation, so the repository counts the pair as one confirmation and not as two method-distinct derivations. The Session 146 registration review says exactly that; a genuinely separate derivation of this clipped decision would be a second machine method, and none exists.

The verification rests on the two machine decisions of the frozen bytes, which fail differently: the exact event-cell sweep with the clip as extra half-planes on the per-slab centre range, and the interval branch and bound whose boxes are dropped only when provably inside the excluded set; the two decide the same set only because the clip is D4-invariant and folded, which the instrument's tests check. Conditions 1 to 4 are decided exactly before either route runs.

The conclusion transfers from the shrunken cores to the unit squares without loss in the direction that matters, because the kept domain is closed and a core lies strictly inside its square by Condition 4. The retained bytes declare variant class and the clip, and the gate refuses them without the flag, so the file cannot be read as a bound.
Next rung: The complementary class, in which some core meets a corner triangle, is the hard one: the all-deep class is a Lemma-D theorem for every site set (the transported 88-family keeps residual 7 against a requirement of 7, BC-366), so no point covering excludes it and the corner tree cannot make 96/25 unconditional by clipping. The four mixed D4 classes (fourteen bin vectors) are a bounded follow-up under BC-367 and need a box cut on the row domain. The instrument that could matter is a 2-of-3 threshold atom on ring-centre overlaps inside the all-deep class, which needs the clip wired into the threshold routes.

V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained: the Session 146 registration review is not mapped under docs/project/reviews. The case directory and the gate control test in the T-022 pattern are retained.
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 this result
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-031 in the results table · The $n = 11$ optimality paper · The $n = 11$ explainer

On GitHub, at main

Register: T-031 in results.yaml, line 2471
Evidence: E-n011-corner-class-96-25-exact-decision · E-n011-corner-class-96-25-interval-decision
Proofs and certificates: certificate certificate.json
Artifacts: 12 artifacts and controls
cases/n11_corner_class_certificate/certificate.json campaign/series/series-000-smoke-and-calibration/results/agenda-040/exp-220-n11-96-25-class-covering.json campaign/series/series-000-smoke-and-calibration/results/agenda-040/exp-220-n11-96-25-class-decide.stdout campaign/series/series-000-smoke-and-calibration/results/agenda-040/exp-220-n11-96-25-class-decide-noflag.stdout campaign/series/series-000-smoke-and-calibration/results/agenda-040/exp-220-n11-96-25-class-receipt.md campaign/series/series-000-smoke-and-calibration/results/agenda-040/exp-219-n11-96-25-clip-receipt.md campaign/series/series-000-smoke-and-calibration/results/agenda-040/h222-registration-review.md src/sqpack/fractional/corner_clip.py devtools/decide_certificate.py tests/test_n11_corner_class_certificate.py tests/test_fractional_corner_clip.py tests/test_decide_certificate.py
Case file: frontier/n-011.md (verified lower, verified upper, reported lower, reported upper)

T-031: The octagon corner class (threshold 1/21/2) holds no eleven-square packing at side 96/2596/25

The case

Results on the case

Links

T-031: The octagon corner class (threshold $1 / 2$ ) holds no eleven-square packing at side $96 / 25$