The Schwarzian Bridge in a Single Sigmoid Neuron: Spherical Inputs, All-Power Laws, and Empirical Resolution

For a spherically symmetric isotropic input X, we identify the local coefficient linking the radial-tangential anisotropy of a single-neuron sensitivity matrix to the two-point Loewner determinant of its activation: the multiplier is q_X=E||X||^4/[d(d+2)]. Explicit derivative-supremum bounds turn this limiting identity into a finite-scale inequality. For every real p>0 and h_p=sigma'^p, generalized-logistic Gamma moments yield strict global monotonicity, a small-signal expansion through r^6, and a refined large-saturation law. For finite Gaussian samples we prove an explicit pointwise, block-resolved (1+/-epsilon) relative Loewner theorem with radial-eigenvalue and eigenspace-angle bounds and sufficient n proportional to r[d+log(1/delta)]/epsilon^2. A rank obstruction and an empty-transition-slab argument give separate necessary factors; no matching Omega(rd) or minimax estimator lower bound is claimed. Classical transform, Schwarzian, Fisher-splitting, and prior empirical-Hessian results are explicitly delimited.