{
  "title": "Microlocal objects, essential hypotheses, licensed conclusions, and decisive failure tests.",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "microlocal-qft-hypothesis-conclusion-table",
  "caption": "Microlocal objects, essential hypotheses, licensed conclusions, and decisive failure tests.",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Essential hypotheses",
      "html": "Essential hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or adversarial check",
      "html": "Excluded converse or adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Distribution on a smooth manifold",
        "html": "Distribution on a smooth manifold"
      },
      "Essential hypotheses": {
        "text": "Localized Fourier estimates in conic cotangent neighborhoods; nonzero covectors",
        "html": "Localized Fourier estimates in conic cotangent neighborhoods; nonzero covectors"
      },
      "Licensed conclusion": {
        "text": "A closed conic wavefront set with coordinate-invariant position and direction data",
        "html": "A closed conic wavefront set with coordinate-invariant position and direction data"
      },
      "Excluded converse or adversarial check": {
        "text": "Equal singular supports do not distinguish a positive-frequency kernel from its transpose",
        "html": "Equal singular supports do not distinguish a positive-frequency kernel from its transpose"
      }
    },
    {
      "Object and domain": {
        "text": "Products, restrictions, and kernel compositions",
        "html": "Products, restrictions, and kernel compositions"
      },
      "Essential hypotheses": {
        "text": "No opposing covector sum for products; conormal transversality for pullback; proper support for pushforward",
        "html": "No opposing covector sum for products; conormal transversality for pullback; proper support for pushforward"
      },
      "Licensed conclusion": {
        "text": "A canonical distributional operation with a calculable wavefront-set bound",
        "html": "A canonical distributional operation with a calculable wavefront-set bound"
      },
      "Excluded converse or adversarial check": {
        "text": "A null worldline can have a conormal Hadamard covector; local cone compatibility does not imply proper fiber support",
        "html": "A null worldline can have a conormal Hadamard covector; local cone compatibility does not imply proper fiber support"
      }
    },
    {
      "Object and domain": {
        "text": "Green-hyperbolic solution and propagation",
        "html": "Green-hyperbolic solution and propagation"
      },
      "Essential hypotheses": {
        "text": "Globally hyperbolic spacetime; two-sided advanced and retarded inverses; real principal type in the source-free region",
        "html": "Globally hyperbolic spacetime; two-sided advanced and retarded inverses; real principal type in the source-free region"
      },
      "Licensed conclusion": {
        "text": "An exact source–solution sequence and transport of singularities along null Hamilton orbits",
        "html": "An exact source–solution sequence and transport of singularities along null Hamilton orbits"
      },
      "Excluded converse or adversarial check": {
        "text": "A timelike boundary without boundary conditions destroys uniqueness; a caustic does not terminate a wavefront",
        "html": "A timelike boundary without boundary conditions destroys uniqueness; a caustic does not terminate a wavefront"
      }
    },
    {
      "Object and domain": {
        "text": "Higher-point distributions",
        "html": "Higher-point distributions"
      },
      "Essential hypotheses": {
        "text": "Directed causal graph cones; compatible products; for the simple derivation, quasifree Wick factorization and a Hadamard two-point function",
        "html": "Directed causal graph cones; compatible products; for the simple derivation, quasifree Wick factorization and a Hadamard two-point function"
      },
      "Licensed conclusion": {
        "text": "The microlocal spectrum bound for the full quasifree hierarchy",
        "html": "The microlocal spectrum bound for the full quasifree hierarchy"
      },
      "Excluded converse or adversarial check": {
        "text": "Reversing one graph edge violates orientation; two-point data do not determine nonquasifree truncated functions",
        "html": "Reversing one graph edge violates orientation; two-point data do not determine nonquasifree truncated functions"
      }
    },
    {
      "Object and domain": {
        "text": "Free-field Hadamard two-point function",
        "html": "Free-field Hadamard two-point function"
      },
      "Essential hypotheses": {
        "text": "Field equation in both variables; fixed commutator; global hyperbolicity; local Hadamard form; positivity for state status",
        "html": "Field equation in both variables; fixed commutator; global hyperbolicity; local Hadamard form; positivity for state status"
      },
      "Licensed conclusion": {
        "text": "Equivalence of the Hadamard expansion and the future-directed null wavefront relation",
        "html": "Equivalence of the Hadamard expansion and the future-directed null wavefront relation"
      },
      "Excluded converse or adversarial check": {
        "text": "The right cone and field equation alone do not prove positive type or construct a state",
        "html": "The right cone and field equation alone do not prove positive type or construct a state"
      }
    },
    {
      "Object and domain": {
        "text": "Wick powers and local covariant fields",
        "html": "Wick powers and local covariant fields"
      },
      "Essential hypotheses": {
        "text": "Hadamard subtraction and diagonal transversality; common operator domain; covariance, scaling, and analytic background dependence for classification",
        "html": "Hadamard subtraction and diagonal transversality; common operator domain; covariance, scaling, and analytic background dependence for classification"
      },
      "Licensed conclusion": {
        "text": "Well-defined local composites whose prescription changes are finite curvature-polynomial mixings",
        "html": "Well-defined local composites whose prescription changes are finite curvature-polynomial mixings"
      },
      "Excluded converse or adversarial check": {
        "text": "A non-Hadamard subtraction obstructs the diagonal; a coordinate-dependent finite term violates naturality",
        "html": "A non-Hadamard subtraction obstructs the diagonal; a coordinate-dependent finite term violates naturality"
      }
    },
    {
      "Object and domain": {
        "text": "Time-ordered products and stress tensor",
        "html": "Time-ordered products and stress tensor"
      },
      "Essential hypotheses": {
        "text": "Causal factorization; scaling-degree extension; microlocal regularity; local covariance; field identities and conservation normalization",
        "html": "Causal factorization; scaling-degree extension; microlocal regularity; local covariance; field identities and conservation normalization"
      },
      "Licensed conclusion": {
        "text": "Formal local time-ordering and a conserved renormalized stress tensor, unique up to finite local covariant terms",
        "html": "Formal local time-ordering and a conserved renormalized stress tensor, unique up to finite local covariant terms"
      },
      "Excluded converse or adversarial check": {
        "text": "Flat-space subtraction misses curvature singularities; nonlocal inverse operators are not admissible counterterms",
        "html": "Flat-space subtraction misses curvature singularities; nonlocal inverse operators are not admissible counterterms"
      }
    }
  ]
}