{
  "title": "Vertex Operator Algebras Net Bridges Dependency Map",
  "question": "Which formal-algebraic, module-analytic, finiteness, unitary, energy-bound, strong-locality, and comparison hypotheses license a VOA-to-net or net-to-VOA conclusion?",
  "takeaway": "Formal VOA locality must first be supplemented by analytic module and finiteness theorems, then by a positive Hilbert completion, polynomial energy bounds, and strong locality before interval von Neumann algebras form a conformal net; reverse and extension comparisons require additional direction-specific hypotheses.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "formal VOA object\\\\vacuum, translation, grading\\\\Jacobi or uniform locality"
    },
    {
      "id": "n1",
      "label": "module analytic theory\\\\intertwining operators and convergence\\\\associativity and braiding"
    },
    {
      "id": "n2",
      "label": "finiteness regime\\\\$C_2$-cofinite plus rational where required\\\\regular or modular conclusions"
    },
    {
      "id": "n3",
      "label": "unitary completion\\\\PCT adjoints and polynomial energy bounds\\\\closable smeared fields"
    },
    {
      "id": "n4",
      "label": "strong locality\\\\commuting spectral data on disjoint intervals\\\\conformal net $\\mathcal A_V$"
    },
    {
      "id": "n5",
      "label": "direction-specific comparison\\\\extension or point-field hypotheses\\\\only the proved functor or recovery"
    },
    {
      "id": "branch",
      "label": "$C_2$-cofinite but nonrational\\\\generalized modules, logarithms, and pseudotraces\\\\no semisimple modular category"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "modules",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "finiteness",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "Hilbert",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "locality",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "compare",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "branch",
      "label": "nonsemisimple branch",
      "logical_status": "licensed implication"
    }
  ]
}