{
  "release": "v1.0.0",
  "external_sources": [
    {
      "id": "HKR2026",
      "authors": "Harrell II, Kennedy, Ramos",
      "title": "Bounds on eigenvalue ratios of quantum graph Laplacians",
      "url": "https://arxiv.org/abs/2603.26172",
      "role": [
        "states the Dirichlet-tree Polya-type bound in the target setting",
        "states the equality-characterization problem as open in the cited version",
        "supplies/uses the generic perturbative nodal framework relied upon by the release"
      ],
      "dependency_type": "external mathematical input"
    },
    {
      "id": "BKKM2017",
      "authors": "Berkolaiko, Kennedy, Kurasov, Mugnolo",
      "title": "Edge connectivity and the spectral gap of combinatorial and quantum graphs",
      "role": ["background spectral graph inequality machinery"],
      "dependency_type": "literature background"
    },
    {
      "id": "BKKS2019",
      "authors": "Berkolaiko, Kennedy, Kurasov, Mugnolo",
      "title": "Surgery principles for the spectral analysis of quantum graphs",
      "role": ["metric-graph spectral surgery and comparison background"],
      "dependency_type": "literature background"
    },
    {
      "id": "BLS2019",
      "authors": "Berkolaiko, Latushkin, Sukhtaiev",
      "title": "Limits of quantum graph operators with shrinking edges",
      "role": ["context for shrinking-edge operator limits; not required for the elementary branch-impedance proof"],
      "dependency_type": "context only"
    },
    {
      "id": "Friedlander2005",
      "authors": "Friedlander",
      "title": "Genericity of simple eigenvalues for a metric graph",
      "role": ["generic spectral background"],
      "dependency_type": "literature background"
    },
    {
      "id": "Kurasov2024",
      "authors": "Kurasov",
      "title": "Spectral Geometry of Graphs",
      "role": ["Dirichlet-tree and metric-graph spectral background, including diameter-type estimates"],
      "dependency_type": "external mathematical input/background"
    }
  ],
  "methodological_provenance": {
    "logical_dependency": false,
    "note": "The author's transducer, defect-conservation, purity-rigidity, conductivity, and Hadamard research suggested proof architecture only; all required mathematics is proved independently in this release.",
    "file": "METHODOLOGICAL_PROVENANCE.md"
  }
}