{
  "title": "Algebraic Locally Covariant Qft Dependency Map",
  "question": "How do a region system, coherent local C*-algebras, a state, and theorem-specific analytic hypotheses lead to a represented local net, local folia, cyclicity and separation, affiliated generators, wedge-local models, and a qualified factor classification?",
  "takeaway": "The abstract net and its inductive limit precede every representation-dependent statement: a positive state gives GNS data, after which local restriction, commutants, positive-energy analyticity, strong field commutativity, wedge intersections, scaling, and phase-space bounds license different one-way conclusions rather than a single equivalence.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "region system $\\mathcal K$\\\\$O_1\\subset O_2$, causal complement $O'$"
    },
    {
      "id": "n1",
      "label": "local C*-net $\\mathcal A(O)$\\\\isotony, locality, covariance\\\\additivity and time-slice named separately"
    },
    {
      "id": "n2",
      "label": "coherent injective embeddings\\\\$\\mathcal A_{\\rm ql}=\\overline{\\bigcup_O\\mathcal A(O)}^{\\|\\cdot\\|}$"
    },
    {
      "id": "n3",
      "label": "state $\\omega\\geq0$, $\\omega(1)=1$\\\\GNS $(\\mathcal H_\\omega,\\pi_\\omega,\\Omega_\\omega)$\\\\$\\mathfrak M_\\omega(O)=\\pi_\\omega(\\mathcal A(O))''$"
    },
    {
      "id": "n4",
      "label": "regional tests after GNS\\\\folia and local quasiequivalence\\\\$\\mathfrak A^d(O)=\\mathfrak M(O')'$; Reeh from spectrum plus weak additivity"
    },
    {
      "id": "n5",
      "label": "strong generators and local structure\\\\affiliation or wedge intersections\\\\factor type from factoriality, scaling, and phase-space control"
    },
    {
      "id": "branch",
      "label": "regular scalar $S_2$\\\\ZF wedge fields and Borchers triple\\\\modular nuclearity $\\Rightarrow\\mathfrak A(O)\\neq\\mathbb C1$"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "assign",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "complete",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "GNS",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "local tests",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "named theorem",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "branch",
      "label": "integrable branch",
      "logical_status": "licensed implication"
    }
  ]
}