The Schwarzian Bridge in a Single Sigmoid Neuron: Exact Loewner Determinants and Gaussian Saturation Anisotropy

We identify a local coefficient identity linking two geometries of an increasing activation g at a centered inflection: for h_p=(g')^p under Gaussian input, the initial radial-tangential anisotropy is 1-p Sg(0)r^2+O(r^4), while the adjacent-point Loewner determinant has coefficient g'(0)^2 Sg(0)/6. For the logistic sigmoid we give a closed determinant formula, a dimension-minimal 2 by 2 PSD-order witness certified with 256-bit interval arithmetic, profile-generic finite-radius curvature bounds, exact large-saturation constants, and strict anisotropy monotonicity for Bernoulli and squared-output profiles. Classical Schwarzian/composition results and prior one-unit Fisher decompositions are explicitly separated from the new bridge and exact refinements. The optimization consequence is restricted to stationary fixed-step local descent; no global, deep-network, or non-Gaussian claim is made.