lean t5_m3_stabRankP_le_fifteen in LeanProofs.T5M3StabRank, machine-checked by Lean
Implied per-copy exponent γ ≤ 0.5609 against a baseline 0.6826 (single-copy product bound; no published exponent), an improvement.
This bound is backed by a machine-checked theorem: t5_m3_stabRankP_le_fifteen in LeanProofs.T5M3StabRank. The recorded lake build confirms it compiles.
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 T5-m2-upper-5.json, T5-m1-upper-3.json: χ(|T5⟩⊗2) ≤ 5 x χ(|T5⟩⊗1) ≤ 3 gives χ(|T5⟩⊗3) ≤ 15, exponent 0.5609. Replaces the rank-24 product of the m=1 and the annealed rank-8 m=2 witnesses (T5-m3-upper-24.json, 2026-09-23). The orbit has no published exponent and is measured against the single-copy product bound log_5(3) = 0.6826; 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. Lean: t5_m3_stabRankP_le_fifteen in LeanProofs.T5M3StabRank proves stabRankP 5 |T5⟩⊗3 ≤ 15 from the m=2 and m=1 theorems and the multiplicativity lemma stabRankP_tensor_le; the fifteen terms here are the products that lemma counts.
this repository · stabrank research/constructions/product_witness.py · structured construction
{
"schema_version": "0.1",
"orbit": "T5",
"m": 3,
"direction": "upper",
"rank": 15,
"provenance": {
"author": "this repository",
"reference": "stabrank research/constructions/product_witness.py",
"method": "structured construction",
"date": "2026-09-24",
"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 T5-m2-upper-5.json, T5-m1-upper-3.json: chi(|T5>^2) <= 5 x chi(|T5>^1) <= 3 gives chi(|T5>^3) <= 15, exponent 0.5609. Replaces the rank-24 product of the m=1 and the annealed rank-8 m=2 witnesses (T5-m3-upper-24.json, 2026-09-23). The orbit has no published exponent and is measured against the single-copy product bound log_5(3) = 0.6826; 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. Lean: t5_m3_stabRankP_le_fifteen in LeanProofs.T5M3StabRank proves stabRankP 5 |T5>^3 <= 15 from the m=2 and m=1 theorems and the multiplicativity lemma stabRankP_tensor_le; the fifteen terms here are the products that lemma counts.",
"witness": {
"terms": [
{
"k": 1,
"x0": [
0,
0,
0
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
0
]
],
"l": [
0
]
},
{
"k": 1,
"x0": [
0,
0,
1
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
0
]
],
"l": [
0
]
},
{
"k": 2,
"x0": [
0,
0,
0
],
"W": [
[
1,
4,
0
],
[
0,
0,
1
]
],
"Q": [
[
0,
0
],
[
0,
4
]
],
"l": [
0,
4
]
},
{
"k": 1,
"x0": [
0,
1,
0
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
3
]
],
"l": [
2
]
},
{
"k": 1,
"x0": [
0,
1,
1
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
3
]
],
"l": [
2
]
},
{
"k": 2,
"x0": [
0,
1,
0
],
"W": [
[
1,
4,
0
],
[
0,
0,
1
]
],
"Q": [
[
3,
0
],
[
0,
4
]
],
"l": [
2,
4
]
},
{
"k": 1,
"x0": [
0,
2,
0
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
1
]
],
"l": [
3
]
},
{
"k": 1,
"x0": [
0,
2,
1
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
1
]
],
"l": [
3
]
},
{
"k": 2,
"x0": [
0,
2,
0
],
"W": [
[
1,
4,
0
],
[
0,
0,
1
]
],
"Q": [
[
1,
0
],
[
0,
4
]
],
"l": [
3,
4
]
},
{
"k": 1,
"x0": [
0,
3,
0
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
4
]
],
"l": [
3
]
},
{
"k": 1,
"x0": [
0,
3,
1
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
4
]
],
"l": [
3
]
},
{
"k": 2,
"x0": [
0,
3,
0
],
"W": [
[
1,
4,
0
],
[
0,
0,
1
]
],
"Q": [
[
4,
0
],
[
0,
4
]
],
"l": [
3,
4
]
},
{
"k": 1,
"x0": [
0,
4,
0
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
2
]
],
"l": [
2
]
},
{
"k": 1,
"x0": [
0,
4,
1
],
"W": [
[
1,
4,
0
]
],
"Q": [
[
2
]
],
"l": [
2
]
},
{
"k": 2,
"x0": [
0,
4,
0
],
"W": [
[
1,
4,
0
],
[
0,
0,
1
]
],
"Q": [
[
2,
0
],
[
0,
4
]
],
"l": [
2,
4
]
}
],
"coeffs": [
"1/4 + sqrt(10)*(-10*sqrt(2) + 4*sqrt(5)*I*sqrt(sqrt(5) + 5))/400",
"sqrt(5)*(10 + 2*sqrt(5)*I*sqrt(5 - 2*sqrt(5)))/100",
"-sqrt(5)/20 + 1/4 - I*sqrt(sqrt(5)/40 + 1/8)",
"sqrt(10)*(4*sqrt(5)*I*sqrt(sqrt(5) + 5)*exp(2*I*pi/5) + 2*sqrt(10)*(-sqrt(5)*exp(2*I*pi/5) + 5*exp(2*I*pi/5)))/400",
"I*sqrt(1/10 - sqrt(5)/50)",
"sqrt(5)/5",
"-I*sqrt(1/10 - sqrt(5)/50)",
"sqrt(5)*(2*sqrt(5)*I*sqrt(5 - 2*sqrt(5))*exp(4*I*pi/5) + 10*exp(4*I*pi/5))*exp(2*I*pi/5)/100",
"-1/4 - sqrt(5)/20 + I*sqrt(1/8 - sqrt(5)/40)",
"sqrt(10)*(4*sqrt(5)*I*sqrt(sqrt(5) + 5)*exp(4*I*pi/5) + 2*sqrt(10)*(-sqrt(5)*exp(4*I*pi/5) + 5*exp(4*I*pi/5)))/400",
"sqrt(5)*(2*sqrt(5)*I*sqrt(5 - 2*sqrt(5))*exp(4*I*pi/5) + 10*exp(4*I*pi/5))/100",
"-sqrt(5)/20 + 1/4 + I*sqrt(sqrt(5)/40 + 1/8)",
"sqrt(10)*(4*sqrt(5)*I*sqrt(sqrt(5) + 5)*exp(2*I*pi/5) + 2*sqrt(10)*(5 - sqrt(5))*exp(2*I*pi/5))*exp(-4*I*pi/5)/400",
"sqrt(5)*(2*sqrt(5)*I*sqrt(5 - 2*sqrt(5))*exp(2*I*pi/5) + 10*exp(2*I*pi/5))*exp(-4*I*pi/5)/100",
"-1/4 - sqrt(5)/20 - I*sqrt(1/8 - sqrt(5)/40)"
]
},
"lean": {
"module": "LeanProofs.T5M3StabRank",
"theorem": "t5_m3_stabRankP_le_fifteen"
}
}