verified 4 stabilizer terms reproduce |qubit_sqrtT>^{ot 3}, checked symbolically
Implied per-copy exponent γ ≤ 0.6667 against a published 0.5000, no improvement.
1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
|0⟩⊗3, |1⟩⊗3, |+⟩⊗3 and |→^3 span the four-dimensional symmetric subspace (|+⟩⊗3 and |→^3 separate the Dicke states of even and odd weight), so χ ≤ 4 at three copies for every single-qubit state. With the exact Galois lower bound 4 the cell is settled, χ(|sqrtT⟩⊗3) = 4, against 3 for the T state. Coefficients are short polynomials in zeta_16 = e^{i π/8} times powers of √ 2, solved by fit_coeffs inside Q(zeta_16) (the generic sympy route mis-simplifies them), and the identity is checked symbolically. Implied γ 2/3 against the published 1/2. The numerical rank-3 exclusion over the 1,080 three-qubit states (research survey, 2026-10-17) came within 0.0019 of the parallel threshold, under rank_exclusion's margin of 0.01, which is why the exact Galois certificate stands behind the lower bound instead.
Vincent Russo · this repository (stabrank-research, round 31 new-orbit survey) · by hand: four product stabilizer states spanning Sym^3(C^2), coefficients fitted exactly in Q(zeta_16)
{
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"orbit": "qubit_sqrtT",
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"direction": "upper",
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{
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"coeffs": [
"sqrt(2)/4 - sqrt(2)*exp(I*pi/4)/4",
"-sqrt(2)*exp(I*pi/8)/4 + sqrt(2)*exp(3*I*pi/8)/4",
"exp(I*pi/8)/2 + exp(I*pi/4)/2",
"-exp(I*pi/8)/2 + exp(I*pi/4)/2"
]
},
"provenance": {
"author": "Vincent Russo",
"reference": "this repository (stabrank-research, round 31 new-orbit survey)",
"method": "by hand: four product stabilizer states spanning Sym^3(C^2), coefficients fitted exactly in Q(zeta_16)",
"date": "2026-10-17",
"github": [
"vprusso"
],
"compute": {
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"runs": 1,
"hardware": "Apple silicon laptop, one core; direct construction, no search"
}
},
"notes": "|0>^3, |1>^3, |+>^3 and |->^3 span the four-dimensional symmetric subspace (|+>^3 and |->^3 separate the Dicke states of even and odd weight), so chi <= 4 at three copies for every single-qubit state. With the exact Galois lower bound 4 the cell is settled, chi(|sqrtT>^3) = 4, against 3 for the T state. Coefficients are short polynomials in zeta_16 = e^{i pi/8} times powers of sqrt 2, solved by fit_coeffs inside Q(zeta_16) (the generic sympy route mis-simplifies them), and the identity is checked symbolically. Implied gamma 2/3 against the published 1/2. The numerical rank-3 exclusion over the 1,080 three-qubit states (research survey, 2026-10-17) came within 0.0019 of the parallel threshold, under rank_exclusion's margin of 0.01, which is why the exact Galois certificate stands behind the lower bound instead."
}