[
  {
    "id": "ARR-2026-53CTRKDSP685PT51-v1",
    "type": "article",
    "title": "The Schwarzian Bridge in a Single Sigmoid Neuron: Exact Loewner Determinants and Gaussian Saturation Anisotropy",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          29
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-53CTRKDSP685PT51",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-53CTRKDSP685PT51/versions/v1/",
    "note": "Research preprint; ARR-2026-53CTRKDSP685PT51 v1. Not peer reviewed by AIRR.",
    "abstract": "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."
  }
]