[
  {
    "id": "ARR-2026-53CTRKDSP685PT51-v4",
    "type": "article",
    "title": "The Schwarzian Bridge in a Single Sigmoid Neuron: Oracle Gram Resolution and Spherical Saturation Phases",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          30
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-53CTRKDSP685PT51",
    "version": "v4",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-53CTRKDSP685PT51/versions/v4/",
    "note": "Research preprint; ARR-2026-53CTRKDSP685PT51 v4. Not peer reviewed by AIRR.",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 saturation laws and sharp pointwise bilateral oracle-Gram complexity Theta_p(r[d+log(1/delta)]/epsilon^2), now with an autonomous Gaussian-product lower tail and explicit constants. For the weaker bottom-eigenspace target at fixed d and n, the extreme-saturation projector converges to the normal of the d-1 samples nearest the transition hyperplane; its exact angle law gives matching Theta(d^(3/2)/epsilon) complexity at fixed nondegenerate confidence. For spherical X=RU with f_R(rho) asymptotic to c rho^(a-1) at zero, an autonomous Abelian kernel lemma yields exact leading constants in three radial phases: r^(-(a+2)) for 0<a<1, r^(-3) log r for a=1, and r^(-3) for a>1. Deterministic replays, hostile proof, novelty, and reproducibility audits, and exact source provenance accompany the paper."
  }
]