The Schwarzian Bridge in a Single Sigmoid Neuron: Sharp Empirical Complexity and Global Spherical Saturation

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 all-power saturation laws and a pointwise bilateral relative Loewner theorem. A conditional-Gaussian lower bound closes its sample complexity at Theta_p(r[d+log(1/delta)]/epsilon^2); a full-dyadic-shell extension gives tunable uniform precision, and explicit sufficient constants are C_1=22929 and C_2=294162. For isotropic spherical X=RU with E[R^-1]<infinity, the global anisotropy constant is Q_R=E[R]/[(d-1)E[R^-1]]. Gaussian, fixed-sphere, and isotropic Student inputs yield distinct closed constants. Within radial laws regularly varying at zero, the inverse-radius condition has a sharp three-regime phase transition. An isotropic Rademacher counterexample shows covariance isotropy alone is insufficient. Deterministic replays, fixed-seed diagnostics, proof audits, and exact source provenance accompany the paper.