[
  {
    "id": "ARR-2026-53CTRKDSP685PT51-v2",
    "type": "article",
    "title": "The Schwarzian Bridge in a Single Sigmoid Neuron: Spherical Inputs, All-Power Laws, and Empirical Resolution",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          29
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-53CTRKDSP685PT51",
    "version": "v2",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-53CTRKDSP685PT51/versions/v2/",
    "note": "Research preprint; ARR-2026-53CTRKDSP685PT51 v2. Not peer reviewed by AIRR.",
    "abstract": "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."
  }
]