verified 6 stabilizer terms reproduce |CS>^{ot 3}, checked symbolically
Implied per-copy exponent γ ≤ 0.8617 against a baseline 1.0000 (single-copy product bound; no published exponent), an 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 CS-m2-upper-3.json, CS-m1-upper-2.json: χ(|CS⟩⊗2) ≤ 3 x χ(|CS⟩⊗1) ≤ 2 gives χ(|CS⟩⊗3) ≤ 6, exponent 0.8617. First witness for this cell. The orbit has no published exponent and is measured against the single-copy product bound log_2(2) = 1.0000; this exponent is below that baseline because the m=2 factor already is, so it beats no published exponent and the site does not count it as one. It is the sub-multiplicative value the board already implied. 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": "CS",
"m": 3,
"direction": "upper",
"rank": 6,
"provenance": {
"author": "this repository",
"reference": "stabrank research/constructions/product_witness.py",
"method": "structured construction",
"date": "2026-10-21",
"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 CS-m2-upper-3.json, CS-m1-upper-2.json: chi(|CS>^2) <= 3 x chi(|CS>^1) <= 2 gives chi(|CS>^3) <= 6, exponent 0.8617. First witness for this cell. The orbit has no published exponent and is measured against the single-copy product bound log_2(2) = 1.0000; this exponent is below that baseline because the m=2 factor already is, so it beats no published exponent and the site does not count it as one. It is the sub-multiplicative value the board already implied. 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": 2,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
0
],
[
0,
0
]
],
"l": [
0,
0
]
},
{
"k": 2,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
0,
0,
0,
0,
1,
0
],
[
0,
0,
0,
0,
0,
1
]
],
"Q": [
[
0,
1
],
[
0,
0
]
],
"l": [
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": [
1,
1,
1,
1,
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,
1
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
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,
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,
0
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
3,
3,
3,
3,
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,
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,
1
],
[
0,
0,
0,
0,
0,
0
]
],
"l": [
3,
3,
3,
3,
0,
0
]
}
],
"coeffs": [
"1/4 + I/4",
"1/4 - I/4",
"-I/2",
"-1/2",
"-1/2",
"I/2"
]
}
}