The Spin Coefficient of the Einstein-Cartan Bound: Displaying the Coefficient 1/8 and Computing Its Relativistic Form for a Free Dirac Gas

The Einstein-Cartan spin-density bound rho_A = m^2/(kappa^2 coef hbar^2) = 9.72e54 g/cm^3 for neutron matter rests on coef = 1/8 in = coef hbar^2 n^2, previously quoted without a displayed derivation and accompanied by an argument from G_F E_F^2 that no perturbative coefficient exists at bounce conditions. This paper displays the coefficient in full (Frenkel rest frame S^2 = |s|^2; aligned population 1/4; vanishing square of the cell average; two spin populations summed incoherently giving 1/8) and identifies the spent quantity as the spin covariance, quantum mechanically the exchange (Fock) part of the Hehl-Datta axial-axial interaction after the Hartree part cancels. It then computes that exchange term for a free Dirac gas at T = 0, normal-ordered with respect to the vacuum: coef(u) = (1/24)[1 + 2 A(u)^2] with A(u) = 3[u sqrt(1+u^2) - arsinh u]/(2 u^3), u = p_F/(mc), giving coef(0) = 1/8, coef(1) = 0.0949, and the ultra-relativistic limit coef = 1/24. At the bound's density u = 1.17e13, so coef = 1/24 and the density rescales by a factor of 3, to 2.92e55 g/cm^3 within the same balance equation. The G_F E_F^2 argument is withdrawn: the expansion parameters at p_F are the Standard Model gauge couplings (alpha_1 = alpha_2 = 0.0236, alpha_s = 0.0271 at one loop) and (p_F/M_Pl)^2 = 8.07e-13. Disclosure: this paper was prepared with the assistance of Dave, an AI system; all computations were performed by the archived script compute.py deposited with the paper, and every numerical value reported is an output of that script.