{
  "title": "Modular Nuclearity Split Failure Map",
  "question": "Which dropped hypothesis first invalidates a modular, split, localization, or entropy conclusion?",
  "takeaway": "Separation is needed to define Tomita's map, geometry needs a dedicated modular-covariance theorem, splitness needs quantitative phase-space control, and type III forbids an intrinsic sharp-region density matrix.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "licensed checkpoints",
      "label": "Tomita map on $\\mathcal M\\Omega$\\\\cyclic and separating vector"
    },
    {
      "id": "a1",
      "row": "licensed checkpoints",
      "label": "geometric or expected subalgebra\\\\BW or modular invariance input"
    },
    {
      "id": "a2",
      "row": "licensed checkpoints",
      "label": "nuclear map and split inclusion\\\\summable bound, positive collar"
    },
    {
      "id": "a3",
      "row": "licensed checkpoints",
      "label": "wedge-local reconstruction\\\\quarter-modular nuclearity"
    },
    {
      "id": "a4",
      "row": "licensed checkpoints",
      "label": "entropy object\\\\relative modular or regulated type I"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "cyclic but not separating\\\\$S_0(A\\Omega)$ is ambiguous"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "generic massive double cone\\\\no boost flow or invariant subalgebra"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "species proliferation or zero collar\\\\nuclear norm or split bound diverges"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "wedge locality alone\\\\double-cone intersection may be trivial"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "exact type-III wedge\\\\no intrinsic trace-one reduced state"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "symmetry",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "phase space",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "intersection",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "entropy domain",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "no separation",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "assume geometry",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "lose bound",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "false localization",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "force a trace",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}