[
  {
    "id": "ARR-2026-53CTRKDSP685PT51-v3",
    "type": "article",
    "title": "The Schwarzian Bridge in a Single Sigmoid Neuron: Sharp Empirical Complexity and Global Spherical Saturation",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          29
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-53CTRKDSP685PT51",
    "version": "v3",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-53CTRKDSP685PT51/versions/v3/",
    "note": "Research preprint; ARR-2026-53CTRKDSP685PT51 v3. Not peer reviewed by AIRR.",
    "abstract": "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."
  }
]