The Schwarzian Bridge in a Single Sigmoid Neuron: Oracle Gram Resolution and Spherical Saturation Phases

A negative-Schwarzian sigmoid is not matrix-monotone of order two, and the same local invariant determines the quadratic onset of radial-tangential anisotropy in a single neuron. For Gaussian inputs and every h_p=sigma'^p, p>0, we prove closed saturation laws and sharp pointwise bilateral oracle-Gram complexity Theta_p(r[d+log(1/delta)]/epsilon^2), now with an autonomous Gaussian-product lower tail and explicit constants. For the weaker bottom-eigenspace target at fixed d and n, the extreme-saturation projector converges to the normal of the d-1 samples nearest the transition hyperplane; its exact angle law gives matching Theta(d^(3/2)/epsilon) complexity at fixed nondegenerate confidence. For spherical X=RU with f_R(rho) asymptotic to c rho^(a-1) at zero, an autonomous Abelian kernel lemma yields exact leading constants in three radial phases: r^(-(a+2)) for 0<a<1, r^(-3) log r for a=1, and r^(-3) for a>1. Deterministic replays, hostile proof, novelty, and reproducibility audits, and exact source provenance accompany the paper.