[
  {
    "id": "ARR-2026-53CTRKDSP685PT51-v5",
    "type": "article",
    "title": "The Schwarzian Bridge in a Single Sigmoid Neuron: Oracle Gram Resolution, Joint Spectral Lexicography, 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": "v5",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-53CTRKDSP685PT51/versions/v5/",
    "note": "Research preprint; ARR-2026-53CTRKDSP685PT51 v5. Not peer reviewed by AIRR.",
    "abstract": "A sigmoid neuron induces two distinct spectral geometries: a population radial-tangential anisotropy controlled locally by the Schwarzian derivative, and an empirical extreme-saturation hierarchy controlled by samples nearest the transition hyperplane. For logistic powers h_p=sigma'^p, p>0, we prove exact saturation constants, sharp pointwise bilateral oracle-Gram complexity, and regularly varying spherical radius phases. We then make the empirical hierarchy quantitative. A deterministic exterior-power theorem resolves every leading eigenspace; an exact inverse-Gaussian angle law and a finite hierarchy perturbation lemma yield an explicit joint finite-radius, finite-sample guarantee for the bottom eigenspace and teacher direction. Finally, a two-dimensional tangent Gaussian process proves that uniform control over a full angular shell intrinsically incurs a square-root log-radius factor in the iterated high-probability limit. Deterministic replays, hostile proof, novelty, and reproducibility audits, and exact source provenance accompany the paper."
  }
]