{
  "title": "Microlocal Qft Dependency Map",
  "question": "Which one-way theorem dependencies carry directed distributional singularities to Hadamard local fields and then to a finite classification of renormalization freedom, and where do positivity and state existence enter separately?",
  "takeaway": "Wavefront orientation first controls products and restrictions; Green hyperbolicity and real-principal-type propagation carry the null relation; the Hadamard cone licenses local subtraction; covariance, causal factorization, and scaling degree then restrict Wick, time-ordered, and stress-tensor ambiguities to finite local curvature terms, while positivity remains additional state data.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "directed singular data\\\\$\\operatorname{WF}(u)\\subset T^*M\\setminus0$"
    },
    {
      "id": "n1",
      "label": "transverse operations and hyperbolic transport\\\\products, pullbacks, proper pushforwards\\\\$G_\\pm$ and null Hamilton flow"
    },
    {
      "id": "n2",
      "label": "Hadamard two-point cone\\\\$(x,k;x',-k')$, $k$ future null\\\\smooth difference of admissible two-point functions"
    },
    {
      "id": "n3",
      "label": "local covariant composites\\\\Hadamard subtraction, diagonal extension\\\\naturality and almost homogeneous scaling"
    },
    {
      "id": "n4",
      "label": "time ordering, stress tensor, and finite freedom\\\\causal factorization plus scaling degree\\\\local curvature counterterms"
    },
    {
      "id": "branch",
      "label": "field equation and commutator\\\\plus positivity and normalization\\\\existence requires a separate construction"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "operate",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "propagate",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "subtract",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "classify",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "branch",
      "label": "additional state conditions",
      "logical_status": "licensed implication"
    }
  ]
}