{
  "title": "Microlocal Qft Failure Map",
  "question": "Which concrete countertests expose the first invalid step when frequency orientation, transversality, global hyperbolicity, Hadamard regularity, or local covariance is omitted?",
  "takeaway": "An unoriented singular support cannot certify positive frequency, a null-curve conormal obstructs pullback, an unconditioned timelike boundary destroys Green-operator uniqueness, a non-Hadamard remainder obstructs coincidence, and a Green-operator or coordinate counterterm violates locality even when its dimension is correct.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "oriented wavefront relation\\\\positive-frequency singularity class"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "cone transversality plus proper support\\\\canonical product, pullback, or pushforward"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "global hyperbolicity plus two-sided Green identities\\\\unique causal propagator and exact sequence"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "Hadamard subtraction plus local covariance and scaling\\\\local Wick and time-ordered fields"
    },
    {
      "id": "a4",
      "row": "claim checkpoints",
      "label": "diagonal support plus finite-jet curvature dependence\\\\admissible finite renormalization"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "$W_2$, $W_2^{\\mathrm{op}}$, and $E$ share singular support\\\\location survives, frequency orientation does not"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "Hadamard null covector lies in a null curve's conormal\\\\transverse smearing may survive, canonical restriction does not"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "Minkowski space with a removed timelike tube and no boundary condition\\\\local field equation survives, unique $G_\\pm$ does not"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "wrong-oriented image singularity remains after point splitting\\\\separated-point expression may survive, diagonal Wick field does not"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "$R\\Box^{-1}R$ or a chart function depends on remote or coordinate data\\\\dimension may match, local-covariant ambiguity does not"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "operate",
      "logical_status": "ordered diagnostic sequence; each missing hypothesis blocks only its named implication"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "propagate",
      "logical_status": "ordered diagnostic sequence; each missing hypothesis blocks only its named implication"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "subtract",
      "logical_status": "ordered diagnostic sequence; each missing hypothesis blocks only its named implication"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "classify",
      "logical_status": "ordered diagnostic sequence; each missing hypothesis blocks only its named implication"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "orientation erased",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "conormal intersection",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "boundary data missing",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "non-Hadamard remainder",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "nonlocal dependence",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}