The Spin Coefficient Is a Prescription: What a Wigner-Function Derivation of ⟨S²⟩ Actually Returns

The pin memorandum deposited with paper #6 states a kill condition for the Aldred Limit's pinned band: "a microphysical (Wigner-function) derivation of ⟨S²⟩ with coef outside [1/16, 1/2] rescales the band as 1/coef," where ⟨S²⟩ = coef · ħ²n² is the spin average on which the whole Einstein–Cartan bounce balance rests. This paper runs that derivation — the Wigner function of an unpolarized spin-½ gas, computed exactly at the one-particle level — and reports what it returns. It does not return a coefficient. The one-particle Wigner function fixes the spin two-point structure completely: for the unpolarized state, ⟨S_i⟩ = 0 and ⟨S_iS_j⟩ = (ħ²/4)δ_ij. But the Einstein–Cartan term needs a quadratic of the macroscopic spin tensor, and passing from the one-particle structure to that macroscopic quadratic is a coarse-graining step the Wigner function does not contain. Under the canonical readings, the coefficient comes out as: exactly 0 (mean-field reading — an unpolarized gas has no mean spin to square); 1/(4N) or 3/(4N), for a fluid element of N particles, under the two fluctuation readings (spin-vector and operator-Casimir), which differ from each other by a factor 3, sweep the memo's whole kill band as N runs from 1 to 4, and vanish in the thermodynamic limit; and f²/4 under net-spin readings, which reproduce the pin memo's own prescription table exactly and serve as this paper's calibration. The kill condition therefore cannot fire in either direction: no Wigner-faithful computation outputs a unique coefficient inside the band to confirm it, or outside it to rescale it. The band does not move. What the computation names, exactly, is the missing input: a coarse-graining prescription — a stated rule fixing the macroscopic spin tensor's norm from the one-particle Wigner function, equivalently a spin-correlation assumption at the fluid-element scale. The incoherent value coef = 1/8 of the Kopczyński–Trautman / Böhmer et al. spin-fluid model stands where it always stood: as that model's macroscopic prescription, model-labeled, neither confirmed nor refuted by microphysics. Quantum gravity is NOT SOLVED. The vacuum catastrophe is NOT SOLVED.