verified 15 stabilizer terms reproduce |qubit_T>^{ot 9}, checked to 60 significant digits
Implied per-copy exponent γ ≤ 0.4341 against a published 0.3962, no improvement.
1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction from the m=10 witness, no search.
Slice of the fifteen-term decomposition of |T⟩⊗10 (S. Bevins, arXiv:2610.17500, filed as qubit_T-m10-upper-15.json): applying (I ⊗ ⟨0|) to the last qubit on both sides gives ⟨0|T⟩ |T⟩⊗9 as a combination of the fifteen projected terms, each a nine-qubit stabilizer state, all nonzero and distinct. Each coefficient is Bevins' coefficient times the scalar by which the projected term differs from the stabilizer state written here, divided by ⟨0|T⟩ = √((3 + √ 3)/6); every step is exact and nothing is fitted. χ(|T⟩⊗9) ≤ 15 gives exponent 0.4341 against the m=10 record 0.3907 and the product bound 18 (m=4 times m=5). It fills a cell that had no bound and moves no exponent; the lower bound of the same cell is 6 by projection monotonicity.
this repository · arxiv:2610.17500 (the m=10 identity) and stabrank bounds/qubit_T-m10-upper-15.json · projection of a verified witness
{
"schema_version": "0.1",
"orbit": "qubit_T",
"direction": "upper",
"m": 9,
"rank": 15,
"witness": {
"terms": [
{
"k": 9,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
0,
0,
0,
0,
0,
0,
0
]
},
{
"k": 4,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
1,
0
]
],
"Q": [
[
0,
0,
0,
0
],
[
0,
0,
0,
0
],
[
0,
0,
0,
0
],
[
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1
]
},
{
"k": 9,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1,
1,
1,
1,
1
]
},
{
"k": 7,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
0,
0,
1,
1,
0
]
},
{
"k": 7,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
0,
0,
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
1,
1,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
0,
0,
0,
0,
1
]
},
{
"k": 9,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
1,
1,
0,
0,
1,
1,
0
]
},
{
"k": 9,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
0,
0,
1,
1,
0,
0,
0
]
},
{
"k": 9,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
1,
1,
1,
1,
0,
0,
1
]
},
{
"k": 9,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
0,
0,
0,
0,
1,
1,
1
]
},
{
"k": 7,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1,
1,
1
]
},
{
"k": 7,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
1,
0
],
[
0,
0,
0,
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1,
1,
1
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1,
1
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0,
0,
0,
0
],
[
0,
0,
1,
1,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0,
1,
1,
0
]
],
"Q": [
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1,
1
]
}
],
"coeffs": [
"(-sqrt(2)*sqrt(3 - sqrt(3)) + sqrt(6)*sqrt(3 - sqrt(3)) + sqrt(2)*I*sqrt(3 - sqrt(3)) + sqrt(6)*I*sqrt(3 - sqrt(3)))/9",
"4*sqrt(3 - sqrt(3))/9",
"(-sqrt(2)*sqrt(3 - sqrt(3)) + sqrt(6)*sqrt(3 - sqrt(3)) - sqrt(6)*I*sqrt(3 - sqrt(3)) - sqrt(2)*I*sqrt(3 - sqrt(3)))/9",
"(sqrt(6)*sqrt(3 - sqrt(3)) + 3*sqrt(2)*sqrt(3 - sqrt(3)) - sqrt(6)*I*sqrt(3 - sqrt(3)) + 3*sqrt(2)*I*sqrt(3 - sqrt(3)))/27",
"(sqrt(6)*sqrt(3 - sqrt(3)) + 3*sqrt(2)*sqrt(3 - sqrt(3)) - sqrt(6)*I*sqrt(3 - sqrt(3)) + 3*sqrt(2)*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(3)*sqrt(3 - sqrt(3)) + 6*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(3)*sqrt(3 - sqrt(3)) + 6*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(6)*sqrt(3 - sqrt(3)) + 2*sqrt(6)*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(6)*sqrt(3 - sqrt(3)) + 2*sqrt(6)*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(6)*sqrt(3 - sqrt(3)) - 2*sqrt(6)*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(6)*sqrt(3 - sqrt(3)) - 2*sqrt(6)*I*sqrt(3 - sqrt(3)))/27",
"(sqrt(6)*sqrt(3 - sqrt(3)) + 3*sqrt(2)*sqrt(3 - sqrt(3)) - 3*sqrt(2)*I*sqrt(3 - sqrt(3)) + sqrt(6)*I*sqrt(3 - sqrt(3)))/27",
"(sqrt(6)*sqrt(3 - sqrt(3)) + 3*sqrt(2)*sqrt(3 - sqrt(3)) - 3*sqrt(2)*I*sqrt(3 - sqrt(3)) + sqrt(6)*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(3)*sqrt(3 - sqrt(3)) - 6*I*sqrt(3 - sqrt(3)))/27",
"(2*sqrt(3)*sqrt(3 - sqrt(3)) - 6*I*sqrt(3 - sqrt(3)))/27"
]
},
"provenance": {
"author": "this repository",
"reference": "arxiv:2610.17500 (the m=10 identity) and stabrank bounds/qubit_T-m10-upper-15.json",
"method": "projection of a verified witness",
"date": "2026-10-18",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.001,
"wall_clock_hours": 0.001,
"runs": 1,
"hardware": "Apple silicon laptop, one core; direct construction from the m=10 witness, no search"
}
},
"notes": "Slice of the fifteen-term decomposition of |T>^10 (S. Bevins, arXiv:2610.17500, filed as qubit_T-m10-upper-15.json): applying (I (x) <0|) to the last qubit on both sides gives <0|T> |T>^9 as a combination of the fifteen projected terms, each a nine-qubit stabilizer state, all nonzero and distinct. Each coefficient is Bevins' coefficient times the scalar by which the projected term differs from the stabilizer state written here, divided by <0|T> = sqrt((3 + sqrt 3)/6); every step is exact and nothing is fitted. chi(|T>^9) <= 15 gives exponent 0.4341 against the m=10 record 0.3907 and the product bound 18 (m=4 times m=5). It fills a cell that had no bound and moves no exponent; the lower bound of the same cell is 6 by projection monotonicity."
}