{
  "title": "Algebraic Locally Covariant Qft Failure Map",
  "question": "Which concrete failures block the passage from regional algebras to a faithful quasilocal net, from local folia to global equivalence, from locality to duality or compact localization, and from type III to the hyperfinite type-III_1 conclusion?",
  "takeaway": "Independent tensor factors or noninjective bonding maps destroy the claimed net, an infinite-volume KMS state need not be a vacuum density matrix, local quasiequivalence need not patch globally, Maxwell flux defeats locality-to-duality, a real-axis unitary phase need not be wedge-local, and type III alone fixes neither the Connes subtype nor hyperfiniteness.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "licensed implication",
      "label": "coherent injective local maps\\\\faithful quasilocal C*-net"
    },
    {
      "id": "a1",
      "row": "licensed implication",
      "label": "state in its own GNS representation\\\\regional folia tested separately"
    },
    {
      "id": "a2",
      "row": "licensed implication",
      "label": "named locality or model theorem\\\\duality, affiliation, or wedge intersection"
    },
    {
      "id": "a3",
      "row": "licensed implication",
      "label": "factoriality plus scaling and nuclearity\\\\hyperfinite type $\\mathrm{III}_1$"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "independent factors on overlaps or $q:\\mathbb C^2\\to\\mathbb C$\\\\no faithful original net"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "$e^{-\\beta H_0}$ not trace class; local intertwiners incompatible\\\\KMS state survives, global unitary does not"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "linked Maxwell flux or nonanalytic unitary $S_2$ phase\\\\causality or ZF algebra survives, duality or wedge locality does not"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "only type III proved, or a type-I lattice cutoff\\\\Connes subtype and hyperfiniteness remain open"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "GNS",
      "logical_status": "licensed implication"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "regional theorem",
      "logical_status": "licensed implication"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "type theorem",
      "logical_status": "licensed implication"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "break maps",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "false converse",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "drop theorem input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "drop phase-space input",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}