{
  "abstract": "We apply the Constructive Cavity Method (CCM) to the Ising perceptron. The process starts from positive soft channels admitting an auxiliary representation $$ U(x)=\\int_{\\mathbb{R}} e^{tx}ρ(\\mathrm{d}t),\\qquad ρ\\ge0, $$ with an auxiliary measure having a finite exponential quadratic moment. This rewrites the perceptron as a dense, bipartite Gaussian model with an Ising layer and a continuous auxiliary layer. The CCM blanket geometry then gives two constructive variance profiles, $Q_\\ell$ on the Ising side and $Γ_\\ell$ on the auxiliary side. The bipartite shell scale is $$ Q_{\\ell+1}Γ_{\\ell+1}-Q_\\ellΓ_\\ell, $$ and Gaussian core comparison removes the core contribution $ΔQ_\\ellΔΓ_\\ell$ under blanket refinement. The scalar cavity functional has three terms, $$ \\mathcal{A}_{\\mathrm{perc}}(ν)=\\mathbb{E}\\left[\\log\\langle2\\cosh x\\rangle_ν+α\\log\\langle U(s)\\rangle_ν-\\log\\langle e^y\\rangle_ν\\right], $$ where $x$ is the new-spin field, $s$ is the new-constraint field, and $y$ is the cavity correction. CCM identifies the constructive cavity variables with the Aizenman--Sims--Starr random overlap structure (ROSt) variables. Parisi's recursive averaging then gives $$ \\mathcal{G}_{\\mathrm{perc}}^{\\mathrm{CCM}}(ζ,Q,Γ)=X_0(0)+αY_0(0)-\\frac{1}{2}\\sum_{\\ell=0}^{R}ζ_\\ell\\left(Q_{\\ell+1}Γ_{\\ell+1}-Q_\\ellΓ_\\ell\\right). $$ The replica-symmetric specialization gives Gardner's formula for the generalized Ising perceptron. Specializing the constraint terminal to the hard channel gives the Krauth--Mézard functional for the binary perceptron.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39141",
  "author_total": 3,
  "authors": [
    {
      "name": "Simone Franchini"
    },
    {
      "name": "Francesco Concetti"
    },
    {
      "name": "Giampiero Bardella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12b6p04000brg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39141",
  "fulltext_url": null,
  "id": "12b6p04000brg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39141",
  "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",
      "cond-mat.stat-mech"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39141",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12b6p04000brg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.39141",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39141"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "The Constructive Cavity Method for the Ising Perceptron",
  "title_text": "The Constructive Cavity Method for the Ising Perceptron",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:45Z",
  "url": "https://research.draconitic.com/paper/12b6p04000brg",
  "venue": "arXiv",
  "versions": []
}