{
  "abstract": "We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (τ_\\varepsilon=Θ(Δ_1^{-γ}d^ρ\\log(Ψ_0/\\varepsilon))), where (Δ_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12996",
  "author_total": 1,
  "authors": [
    {
      "name": "Zongmin Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fzgmvsd6t832.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12996",
  "fulltext_url": null,
  "id": "9fzgmvsd6t832",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12996",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7213065705",
  "pdf_url": "https://arxiv.org/pdf/2609.12996",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fzgmvsd6t832.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12996",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12996"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Full Adam Theorem for Spectral Heavy-Tail Onset",
  "title_text": "A Full Adam Theorem for Spectral Heavy-Tail Onset",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:17Z",
  "url": "https://research.draconitic.com/paper/9fzgmvsd6t832",
  "venue": "arXiv",
  "versions": []
}