{
  "title": "Modular Nuclearity Split Dependency Map",
  "question": "Which standardness, phase-space, and localization hypotheses lead to modular data, nuclearity bounds, and split inclusions?",
  "takeaway": "A standard local pair gives Tomita data, but a separately defined damped localization map needs a quantitative nuclearity estimate before a strict inclusion is split; geometric flow and type-I factorization each require additional hypotheses.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "standard local pair $(\\mathcal M,\\Omega)$\\\\$\\Omega$ cyclic and separating"
    },
    {
      "id": "n1",
      "label": "damped localization map $\\Theta_{\\beta,\\mathcal O}$\\\\$\\mathcal A(\\mathcal O)_1\\to\\mathcal H$"
    },
    {
      "id": "n2",
      "label": "nuclear decomposition\\\\$\\sum_k\\|\\phi_k\\|\\,\\|\\xi_k\\|<\\infty$"
    },
    {
      "id": "n3",
      "label": "strict inclusion $\\mathcal N\\subset\\mathcal M$\\\\nonzero localization collar"
    },
    {
      "id": "n4",
      "label": "type-I intermediate $\\mathcal F$\\\\normal product states"
    },
    {
      "id": "n5",
      "label": "scope boundary\\\\sharp endpoints remain type III"
    },
    {
      "id": "branch",
      "label": "$S=J\\Delta^{1/2}$ and KMS strip\\\\geometry needs BW or half-sided input"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "define",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "nuclearity",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "split",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "factorize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "qualify",
      "logical_status": "licensed implication"
    },
    {
      "from": "n0",
      "to": "branch",
      "label": "modular path",
      "logical_status": "licensed implication"
    }
  ]
}