verified 12 stabilizer terms reproduce |H3>^{ot 5}, checked symbolically
Implied per-copy exponent γ ≤ 0.4524 against a published 0.4206, no improvement.
1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
Tensor product of the exact decompositions research/constructions/data/H3_m2_rank3.json (decomposition 0 of 9, coefficients fitted exactly by fit_coeffs.fit), research/constructions/data/H3_m3_rank4.json (decomposition 0 of 1, coefficients fitted exactly by fit_coeffs.fit): χ(|H3⟩⊗2) ≤ 3 x χ(|H3⟩⊗3) ≤ 4 gives χ(|H3⟩⊗5) ≤ 12, exponent 0.4524. First witness for this cell; it is the sub-multiplicative value the board already implied and does not move the exponent. Terms are the exact products (x0 and l concatenated, W and Q block diagonal) and coefficients the exact products, built by research/constructions/product_witness.py.
this repository · stabrank research/constructions/product_witness.py · structured construction
{
"schema_version": "0.1",
"orbit": "H3",
"m": 5,
"direction": "upper",
"rank": 12,
"provenance": {
"author": "this repository",
"reference": "stabrank research/constructions/product_witness.py",
"method": "structured construction",
"date": "2026-09-23",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.001,
"wall_clock_hours": 0.001,
"runs": 1,
"hardware": "Apple silicon laptop, one core; direct construction, no search"
}
},
"notes": "Tensor product of the exact decompositions research/constructions/data/H3_m2_rank3.json (decomposition 0 of 9, coefficients fitted exactly by fit_coeffs.fit), research/constructions/data/H3_m3_rank4.json (decomposition 0 of 1, coefficients fitted exactly by fit_coeffs.fit): chi(|H3>^2) <= 3 x chi(|H3>^3) <= 4 gives chi(|H3>^5) <= 12, exponent 0.4524. First witness for this cell; it is the sub-multiplicative value the board already implied and does not move the exponent. Terms are the exact products (x0 and l concatenated, W and Q block diagonal) and coefficients the exact products, built by research/constructions/product_witness.py.",
"witness": {
"terms": [
{
"k": 0,
"x0": [
0,
0,
0,
0,
0
],
"W": [],
"Q": [],
"l": []
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
0,
0
],
[
0,
0,
0
],
[
0,
0,
0
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0
],
[
0,
1,
0
],
[
0,
0,
1
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
2,
0,
0
],
[
0,
2,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
0
]
},
{
"k": 2,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
]
],
"Q": [
[
0,
0
],
[
0,
0
]
],
"l": [
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
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,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"l": [
0,
0,
0,
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
2,
0,
0
],
[
0,
0,
0,
2,
0
],
[
0,
0,
0,
0,
2
]
],
"l": [
0,
0,
0,
0,
0
]
},
{
"k": 2,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
]
],
"Q": [
[
1,
0
],
[
0,
1
]
],
"l": [
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
0,
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"l": [
0,
0,
0,
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
1,
0,
0
],
[
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0
],
[
0,
0,
2,
0,
0
],
[
0,
0,
0,
2,
0
],
[
0,
0,
0,
0,
2
]
],
"l": [
0,
0,
0,
0,
0
]
}
],
"coeffs": [
"sqrt(2)*(sqrt(3)*sqrt(3*sqrt(3) + 9) + 3*sqrt(3*sqrt(3) + 9) - sqrt(3)*I*sqrt(3*sqrt(3) + 9) + 3*I*sqrt(3*sqrt(3) + 9))/64",
"sqrt(2)*(sqrt(3)*sqrt(3*sqrt(3) + 9) + 3*sqrt(3*sqrt(3) + 9) - sqrt(3)*I*sqrt(3*sqrt(3) + 9) + 3*I*sqrt(3*sqrt(3) + 9))/64",
"(-5*sqrt(3)*sqrt(9 - 3*sqrt(3)) - 3*sqrt(6*sqrt(3) + 18) + 3*sqrt(3)*I*sqrt(-6 - 2*sqrt(3)) - 3*sqrt(3)*sqrt(3 - sqrt(3)) + 3*sqrt(9 - 3*sqrt(3)) + 9*sqrt(3 - sqrt(3)) + 2*sqrt(3)*sqrt(6*sqrt(3) + 18) - 15*I*sqrt(9 - 3*sqrt(3)) - 9*sqrt(-6 - 2*sqrt(3)) - 9*I*sqrt(3 - sqrt(3)) - 3*sqrt(3)*I*sqrt(3 - sqrt(3)) - sqrt(3)*I*sqrt(9 - 3*sqrt(3)) + sqrt(3)*I*sqrt(6*sqrt(3) + 18) + 6*I*sqrt(6*sqrt(3) + 18))/192",
"(-15*sqrt(9 - 3*sqrt(3)) - 9*sqrt(2*sqrt(3) + 6) - 9*sqrt(3 - sqrt(3)) - sqrt(3)*sqrt(6*sqrt(3) + 18) + sqrt(3)*sqrt(9 - 3*sqrt(3)) + 3*sqrt(3)*sqrt(3 - sqrt(3)) + 6*sqrt(6*sqrt(3) + 18) - 2*sqrt(3)*I*sqrt(6*sqrt(3) + 18) - 3*I*sqrt(6*sqrt(3) + 18) + 3*I*sqrt(9 - 3*sqrt(3)) + 3*sqrt(3)*I*sqrt(3 - sqrt(3)) + 9*I*sqrt(3 - sqrt(3)) + 3*sqrt(3)*I*sqrt(2*sqrt(3) + 6) + 5*sqrt(3)*I*sqrt(9 - 3*sqrt(3)))/192",
"sqrt(2)*(sqrt(3)*sqrt(3*sqrt(3) + 9) + 3*sqrt(3*sqrt(3) + 9) - 3*I*sqrt(3*sqrt(3) + 9) + sqrt(3)*I*sqrt(3*sqrt(3) + 9))/64",
"sqrt(2)*(sqrt(3)*sqrt(3*sqrt(3) + 9) + 3*sqrt(3*sqrt(3) + 9) - 3*I*sqrt(3*sqrt(3) + 9) + sqrt(3)*I*sqrt(3*sqrt(3) + 9))/64",
"(-15*sqrt(9 - 3*sqrt(3)) + 9*I*sqrt(-6 - 2*sqrt(3)) - 9*sqrt(3 - sqrt(3)) - sqrt(3)*sqrt(6*sqrt(3) + 18) + sqrt(3)*sqrt(9 - 3*sqrt(3)) + 3*sqrt(3)*sqrt(3 - sqrt(3)) + 6*sqrt(6*sqrt(3) + 18) - 5*sqrt(3)*I*sqrt(9 - 3*sqrt(3)) - 3*sqrt(3)*sqrt(-6 - 2*sqrt(3)) - 9*I*sqrt(3 - sqrt(3)) - 3*sqrt(3)*I*sqrt(3 - sqrt(3)) - 3*I*sqrt(9 - 3*sqrt(3)) + 3*I*sqrt(6*sqrt(3) + 18) + 2*sqrt(3)*I*sqrt(6*sqrt(3) + 18))/192",
"(-5*sqrt(3)*sqrt(9 - 3*sqrt(3)) - 3*sqrt(6*sqrt(3) + 18) - 3*sqrt(3)*sqrt(2*sqrt(3) + 6) - 3*sqrt(3)*sqrt(3 - sqrt(3)) + 3*sqrt(9 - 3*sqrt(3)) + 9*sqrt(3 - sqrt(3)) + 2*sqrt(3)*sqrt(6*sqrt(3) + 18) - 6*I*sqrt(6*sqrt(3) + 18) - sqrt(3)*I*sqrt(6*sqrt(3) + 18) + sqrt(3)*I*sqrt(9 - 3*sqrt(3)) + 3*sqrt(3)*I*sqrt(3 - sqrt(3)) + 9*I*sqrt(3 - sqrt(3)) + 9*I*sqrt(2*sqrt(3) + 6) + 15*I*sqrt(9 - 3*sqrt(3)))/192",
"sqrt(2)*(-3*I*sqrt(3*sqrt(3) + 9) + sqrt(3)*I*sqrt(3*sqrt(3) + 9))/32",
"sqrt(2)*(-3*I*sqrt(3*sqrt(3) + 9) + sqrt(3)*I*sqrt(3*sqrt(3) + 9))/32",
"(-18*sqrt(9 - 3*sqrt(3)) - 3*sqrt(3)*sqrt(6*sqrt(3) + 18) + 9*I*sqrt(-6 - 2*sqrt(3)) - 18*sqrt(3 - sqrt(3)) + 6*sqrt(3)*sqrt(3 - sqrt(3)) - 3*sqrt(3)*I*sqrt(-6 - 2*sqrt(3)) + 6*sqrt(3)*sqrt(9 - 3*sqrt(3)) + 9*sqrt(6*sqrt(3) + 18) - 3*I*sqrt(6*sqrt(3) + 18) - 3*sqrt(3)*sqrt(-6 - 2*sqrt(3)) - 4*sqrt(3)*I*sqrt(9 - 3*sqrt(3)) + sqrt(3)*I*sqrt(6*sqrt(3) + 18) + 12*I*sqrt(9 - 3*sqrt(3)) + 9*sqrt(-6 - 2*sqrt(3)))/192",
"(-9*sqrt(6*sqrt(3) + 18) - 6*sqrt(3)*sqrt(9 - 3*sqrt(3)) - 3*sqrt(3)*sqrt(2*sqrt(3) + 6) - 6*sqrt(3)*sqrt(3 - sqrt(3)) + 18*sqrt(3 - sqrt(3)) + 9*sqrt(2*sqrt(3) + 6) + 3*sqrt(3)*sqrt(6*sqrt(3) + 18) + 18*sqrt(9 - 3*sqrt(3)) - 3*I*sqrt(6*sqrt(3) + 18) - 3*sqrt(3)*I*sqrt(2*sqrt(3) + 6) - 4*sqrt(3)*I*sqrt(9 - 3*sqrt(3)) + sqrt(3)*I*sqrt(6*sqrt(3) + 18) + 12*I*sqrt(9 - 3*sqrt(3)) + 9*I*sqrt(2*sqrt(3) + 6))/192"
]
}
}