Geometric Derivation of the Universal Threshold ACF = 0.65

The Axial Coherence Factor (ACF = (S × U × R)1/3) threshold of 0.65 was previously identified as an empirically calibrated parameter through piecewise regression on 100 Solar System bodies. Its independent confirmation in biological and mechanical domains established convergent empirical support, but not derivation from first principles. Here we demonstrate that ACF = 0.65 is geometrically necessary: it is the unique value simultaneously satisfying the minimum conditions of three-dimensional structural connectivity and rotational stability. Two independent inputs are used: (1) the percolation threshold for three-dimensional random close-packed networks, p_c = 0.4120 (Bernal 1960; Torquato and Stillinger 2002), and (2) the Jacobi ellipsoid stability limit for self-gravitating rotating bodies, R_min = 2/3 (Chandrasekhar 1969). Neither value was calibrated within the ACF framework. The derived threshold is ACF_derived = (p_c × R_min)1/3 = (0.4120 × 0.6667)1/3 = 0.650033, deviating 0.0051% from the empirical value. Formal uncertainty propagation from the literature range of p_c yields σ(ACF) = 0.00263, placing the agreement at z = 0.013σ — within numerical precision. Alternative derivation paths via the fine-structure constant α and the Bohr orbit velocity ratio are tested against NIST CODATA 2018 and definitively closed. The geometric derivation elevates the threshold from calibrated parameter to derived geometric constant, resolving the primary epistemological objection to universality claims based on ACF.