verified 16 stabilizer terms reproduce |H3>^{ot 6}, checked symbolically
Implied per-copy exponent γ ≤ 0.4206 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_m3_rank4.json (decomposition 0 of 1, 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⟩⊗3) ≤ 4 x χ(|H3⟩⊗3) ≤ 4 gives χ(|H3⟩⊗6) ≤ 16, exponent 0.4206. 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": 6,
"direction": "upper",
"rank": 16,
"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_m3_rank4.json (decomposition 0 of 1, 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>^3) <= 4 x chi(|H3>^3) <= 4 gives chi(|H3>^6) <= 16, exponent 0.4206. 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,
0
],
"W": [],
"Q": [],
"l": []
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
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,
0
],
"W": [
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
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,
0
],
"W": [
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
2,
0,
0
],
[
0,
2,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
]
],
"Q": [
[
0,
0,
0
],
[
0,
0,
0
],
[
0,
0,
0
]
],
"l": [
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
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,
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
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
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,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"l": [
0,
0,
0,
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
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,
0
],
[
0,
0,
0,
2,
0,
0
],
[
0,
0,
0,
0,
2,
0
],
[
0,
0,
0,
0,
0,
2
]
],
"l": [
0,
0,
0,
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
]
],
"Q": [
[
1,
0,
0
],
[
0,
1,
0
],
[
0,
0,
1
]
],
"l": [
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
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
]
],
"l": [
0,
0,
0,
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"l": [
0,
0,
0,
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
2,
0,
0
],
[
0,
0,
0,
0,
2,
0
],
[
0,
0,
0,
0,
0,
2
]
],
"l": [
0,
0,
0,
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
]
],
"Q": [
[
2,
0,
0
],
[
0,
2,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
2,
0,
0,
0,
0,
0
],
[
0,
2,
0,
0,
0,
0
],
[
0,
0,
2,
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,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
2,
0,
0,
0,
0,
0
],
[
0,
2,
0,
0,
0,
0
],
[
0,
0,
2,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"l": [
0,
0,
0,
0,
0,
0
]
},
{
"k": 6,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
2,
0,
0,
0,
0,
0
],
[
0,
2,
0,
0,
0,
0
],
[
0,
0,
2,
0,
0,
0
],
[
0,
0,
0,
2,
0,
0
],
[
0,
0,
0,
0,
2,
0
],
[
0,
0,
0,
0,
0,
2
]
],
"l": [
0,
0,
0,
0,
0,
0
]
}
],
"coeffs": [
"3*sqrt(3)/32 + 9/32",
"3*sqrt(3)/32 + 9/32",
"-3*sqrt(3)/32 - 3*sqrt(3)*I/32",
"-3*sqrt(3)/32 + 3*sqrt(3)*I/32",
"3*sqrt(3)/32 + 9/32",
"3*sqrt(3)/32 + 9/32",
"-3*sqrt(3)/32 - 3*sqrt(3)*I/32",
"-3*sqrt(3)/32 + 3*sqrt(3)*I/32",
"-3*sqrt(3)/32 - 3*sqrt(3)*I/32",
"-3*sqrt(3)/32 - 3*sqrt(3)*I/32",
"I*(9/32 - 3*sqrt(3)/32)",
"9/32 - 3*sqrt(3)/32",
"-3*sqrt(3)/32 + 3*sqrt(3)*I/32",
"-3*sqrt(3)/32 + 3*sqrt(3)*I/32",
"9/32 - 3*sqrt(3)/32",
"I*(-9/32 + 3*sqrt(3)/32)"
]
}
}