The Schwarzian Bridge in a Single Sigmoid Neuron: Oracle Gram Resolution and Spherical Saturation Phases
Corrected version of a screened paper. Not peer reviewed. Screening record
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.
Preview PDF on this page v4
Uses your browser's PDF viewer. Open the PDF directly if a preview is unavailable.
Cite this paper
Lluis Eriksson (2026). The Schwarzian Bridge in a Single Sigmoid Neuron: Oracle Gram Resolution and Spherical Saturation Phases. AIRR.SCIENCE, ARR-2026-53CTRKDSP685PT51, v4. https://airr.science/papers/ARR-2026-53CTRKDSP685PT51/versions/v4/
Use this version link to cite the manuscript you read. A Working paper label means the version has not completed AIRR screening. AIRR screening is not journal peer review. Import BibTeX, RIS or CSL JSON into your reference manager to apply a citation style.
Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
Revision statement
Major revision. Adds spectral lexicography with an exact limiting angle law, exact three-phase spherical constants, explicit confidence lower-tail constants, and new deterministic replays.
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
OpenAI Codex assisted with public-repository and primary-literature triage, theorem exploration, proof development, adversarial proof/algebra/priority audits, numerical replay, drafting, LaTeX production, visual QA, and deposit preparation. The retained AI audits are not human peer review, formal verification, independent reproduction, or a guarantee of priority; the author remains responsible for every claim.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
No supported exact-version report is published for this exact version. Missing evidence is not scored as zero.
No model reports are published for this version.
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Version v4 supersedes v3. Declared major revision: Adds spectral lexicography with an exact limiting angle law, exact three-phase spherical constants, explicit confidence lower-tail constants, and new deterministic replays. All assessments apply only when explicitly rerun and recorded for this exact version.