{
  "title": "Conformal Nets Vertex Algebras Failure Map",
  "question": "Which dropped hypothesis first invalidates localization, complete rationality, modularity, local extension, or full-CFT realization?",
  "takeaway": "A global algebra, infinite-index current net, transparent sector, noncommutative Q-system, and bare modular matrix fail at different checkpoints and leave different weaker conclusions intact.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "licensed checkpoints",
      "label": "interval vacuum net\\\\local algebras and $L_0\\ge0$"
    },
    {
      "id": "a1",
      "row": "licensed checkpoints",
      "label": "complete-rationality package\\\\split, strong additivity, finite $\\mu$"
    },
    {
      "id": "a2",
      "row": "licensed checkpoints",
      "label": "modular DHR category\\\\finite dimensions and trivial center"
    },
    {
      "id": "a3",
      "row": "licensed checkpoints",
      "label": "local extension\\\\positive commutative Q-system"
    },
    {
      "id": "a4",
      "row": "licensed checkpoints",
      "label": "full two-dimensional net\\\\full center and unique vacuum"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "one global algebra only\\\\isotony and locality cannot be stated"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "U(1) current charges $q\\in\\mathbb R$\\\\split may survive but $\\mu=\\infty$"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "nontrivial transparent sector\\\\braiding is degenerate and $S$ singular"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "associative but noncommutative Q-system\\\\finite-index, only relatively local"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "integer matrix commuting with $S,T$ only\\\\no multiplication, positivity, or net"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "finite $\\mu$",
      "logical_status": "ordered theorem checks, not reversible implications"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "sectors",
      "logical_status": "ordered theorem checks, not reversible implications"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "Q-system",
      "logical_status": "ordered theorem checks, not reversible implications"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "full center",
      "logical_status": "ordered theorem checks, not reversible implications"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "forget localization",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "drop finite index",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "ignore Müger center",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "omit braided commutativity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "reverse a necessary condition",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}