{
  "title": "Defects Generalized Symmetry Categorical Qft Failure Map",
  "question": "Which concrete missing hypotheses break defect composition, categorical coherence, complete charge claims, gauging or condensation, and promotion of categorical equivalence to equivalence of physical QFTs?",
  "takeaway": "A mismatched link makes composition undefined; fusion coefficients without a pentagon leave only a based ring; one action spectrum omits module coherence; anomaly, nonseparability, or failed locality blocks theory change; and a center or Morita class omits enrichment and local dynamics.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checks",
      "label": "typed stratified labels\\\\defined defect composition"
    },
    {
      "id": "a1",
      "row": "claim checks",
      "label": "fusion and junction data\\\\coherent associators, units, and duals"
    },
    {
      "id": "a2",
      "row": "claim checks",
      "label": "full module action\\\\categorical sectors and selection rules"
    },
    {
      "id": "a3",
      "row": "claim checks",
      "label": "vanishing obstruction and admissible algebra\\\\consistent finite gauging or condensation"
    },
    {
      "id": "a4",
      "row": "claim checks",
      "label": "named equivalence and realization hypotheses\\\\exactly the proved classification"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "mismatch incident phase labels at a junction\\\\the composite hom-space is undefined"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "keep fusion coefficients but violate the pentagon\\\\only a based ring survives"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "diagonalize one fusion matrix and call it complete charge data\\\\module associators and physical realization remain unknown"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "gauge an anomalous action or condense a nonseparable nonlocal algebra\\\\the claimed fusion or modular output fails"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "promote matching centers or Morita classes to QFT equivalence\\\\symmetry enrichment, local observables, positivity, and dynamics remain distinct"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "cohere",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct categorical settings"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "action",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct categorical settings"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "test",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct categorical settings"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "theorem",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct categorical settings"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "wrong link",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "omit coherence",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "decategorify action",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "ignore obstruction",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "overstate equivalence",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}