{
  "abstract": "In a slowly varying aperiodic potential $h_i=h\\cos(2πβi^n+ϕ)$ with random power-law density interactions $V_{ij}/|i-j|^α$, the resonant-object ensemble changes across $α=2n$, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling $x_{\\rm pair}^{4-2n-α}\\ln x_{\\rm pair}\\propto h^2/(Vt_0)$ is unchanged~\\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics --- the supply $N_0\\propto L^{2-n}/h$, the correlated common-phase comb, and the closed-form Hartree variance --- proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at $L\\le18$ shows that the order-one forward-branching scale $h_{\\rm FB}\\propto L^{2-n}$ carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged $q\\ln(1/q)$ law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold $V_*(α)$ and the boundary-healing recursion; and we treat the marginal case $α=2$, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings --- $α=1/2$ (the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate), $α=2n$ (change of the local resonant objects), and $α=2$ (marginality of the long-range shell sum) --- none of which is, by itself, a localization transition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30761",
  "author_total": 1,
  "authors": [
    {
      "name": "Yogeshwar Prasad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e0q34b9yjj75w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30761",
  "fulltext_url": null,
  "id": "e0q34b9yjj75w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30761",
  "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.dis-nn"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "cond-mat.str-el"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7204894875",
  "pdf_url": "https://arxiv.org/pdf/2608.30761",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e0q34b9yjj75w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.30761",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30761"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains",
  "title_text": "Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:41Z",
  "url": "https://research.draconitic.com/paper/e0q34b9yjj75w",
  "venue": "arXiv",
  "versions": []
}