{
  "title": "Domains, hypotheses, directional conclusions, and failure tests in locally covariant QFT",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "locally-covariant-dynamical-locality-hypothesis-conclusion-table",
  "caption": "Domains, hypotheses, directional conclusions, and failure tests in locally covariant QFT",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Required hypotheses",
      "html": "Required hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or upgrade",
      "html": "Excluded converse or upgrade"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Embedding between oriented, time-oriented globally hyperbolic spacetimes",
        "html": "Embedding between oriented, time-oriented globally hyperbolic spacetimes"
      },
      "Required hypotheses": {
        "text": "Same dimension; isometric; orientation- and time-orientation-preserving; causally convex open image",
        "html": "Same dimension; isometric; orientation- and time-orientation-preserving; causally convex open image"
      },
      "Licensed conclusion": {
        "text": "A Loc morphism that preserves causal propagation and composes with other admissible embeddings",
        "html": "A Loc morphism that preserves causal propagation and composes with other admissible embeddings"
      },
      "Excluded converse or upgrade": {
        "text": "An arbitrary isometric open embedding is not admissible, and a Loc morphism need not be Cauchy",
        "html": "An arbitrary isometric open embedding is not admissible, and a Loc morphism need not be Cauchy"
      },
      "Adversarial check": {
        "text": "Let a target causal curve leave and re-enter the image and test Green-operator intertwining",
        "html": "Let a target causal curve leave and re-enter the image and test Green-operator intertwining"
      }
    },
    {
      "Object and domain": {
        "text": "Functor $\\mathcal A:\\mathsf{Loc}\\to\\mathsf{Phys}$",
        "html": "Functor $\\mathcal A:\\mathsf{Loc}\\to\\mathsf{Phys}$"
      },
      "Required hypotheses": {
        "text": "Identity and composition; injective $*$-maps; Einstein causality and time-slice imposed separately",
        "html": "Identity and composition; injective $*$-maps; Einstein causality and time-slice imposed separately"
      },
      "Licensed conclusion": {
        "text": "Coherent observable transport, spacelike commutation, and isomorphisms for Cauchy morphisms",
        "html": "Coherent observable transport, spacelike commutation, and isomorphisms for Cauchy morphisms"
      },
      "Excluded converse or upgrade": {
        "text": "Functoriality alone supplies neither causality nor time-slice, and no preferred state follows",
        "html": "Functoriality alone supplies neither causality nor time-slice, and no preferred state follows"
      },
      "Adversarial check": {
        "text": "Use a non-causally-convex embedding and compare the transported Klein–Gordon commutator",
        "html": "Use a non-causally-convex embedding and compare the transported Klein–Gordon commutator"
      }
    },
    {
      "Object and domain": {
        "text": "Field or theory transformation",
        "html": "Field or theory transformation"
      },
      "Required hypotheses": {
        "text": "A component on every object and a commuting naturality square for every Loc morphism",
        "html": "A component on every object and a commuting naturality square for every Loc morphism"
      },
      "Licensed conclusion": {
        "text": "A locally covariant field, subtheory embedding, or natural equivalence with a fixed direction",
        "html": "A locally covariant field, subtheory embedding, or natural equivalence with a fixed direction"
      },
      "Excluded converse or upgrade": {
        "text": "Objectwise formulas or one component isomorphism do not form a natural transformation or equivalence",
        "html": "Objectwise formulas or one component isomorphism do not form a natural transformation or equivalence"
      },
      "Adversarial check": {
        "text": "Choose curvature-renormalization constants independently on embedded spacetimes",
        "html": "Choose curvature-renormalization constants independently on embedded spacetimes"
      }
    },
    {
      "Object and domain": {
        "text": "Compact metric perturbation and its represented derivative",
        "html": "Compact metric perturbation and its represented derivative"
      },
      "Required hypotheses": {
        "text": "Both metrics globally hyperbolic; four Cauchy maps; time-slice; for the derivative, a Hadamard representation, common domain, and differentiability",
        "html": "Both metrics globally hyperbolic; four Cauchy maps; time-slice; for the derivative, a Hadamard representation, common domain, and differentiability"
      },
      "Licensed conclusion": {
        "text": "A relative-Cauchy automorphism; in the free scalar theorem, weak generation by the smeared renormalized stress tensor",
        "html": "A relative-Cauchy automorphism; in the free scalar theorem, weak generation by the smeared renormalized stress tensor"
      },
      "Excluded converse or upgrade": {
        "text": "Finite rce does not imply a norm derivative or an abstract inner derivation",
        "html": "Finite rce does not imply a norm derivative or an abstract inner derivation"
      },
      "Adversarial check": {
        "text": "Pass through a nonglobally hyperbolic metric, or add a nonconserved curvature term and test a compact diffeomorphism",
        "html": "Pass through a nonglobally hyperbolic metric, or add a nonconserved curvature term and test a compact diffeomorphism"
      }
    },
    {
      "Object and domain": {
        "text": "Kinematic and dynamical regional subobjects",
        "html": "Kinematic and dynamical regional subobjects"
      },
      "Required hypotheses": {
        "text": "Time-slice, admissible exterior metric perturbations, and target equalizers/unions; model-specific support theorem",
        "html": "Time-slice, admissible exterior metric perturbations, and target equalizers/unions; model-specific support theorem"
      },
      "Licensed conclusion": {
        "text": "Equality for the massive scalar and stated reduced gauge theories; regional localization is recovered dynamically",
        "html": "Equality for the massive scalar and stated reduced gauge theories; regional localization is recovered dynamically"
      },
      "Excluded converse or upgrade": {
        "text": "Exterior invariance alone does not prove kinematic localization",
        "html": "Exterior invariance alone does not prove kinematic localization"
      },
      "Adversarial check": {
        "text": "Retain a locally constant massless mode or a universal Maxwell flux observable",
        "html": "Retain a locally constant massless mode or a universal Maxwell flux observable"
      }
    },
    {
      "Object and domain": {
        "text": "Natural transformation between dynamically local theories",
        "html": "Natural transformation between dynamically local theories"
      },
      "Required hypotheses": {
        "text": "Dynamical locality for both theories and one component isomorphism; categorical subobject hypotheses",
        "html": "Dynamical locality for both theories and one component isomorphism; categorical subobject hypotheses"
      },
      "Licensed conclusion": {
        "text": "The transformation is a natural isomorphism on all spacetimes",
        "html": "The transformation is a natural isomorphism on all spacetimes"
      },
      "Excluded converse or upgrade": {
        "text": "One-spacetime agreement between unrestricted LCQFTs does not imply same physics everywhere",
        "html": "One-spacetime agreement between unrestricted LCQFTs does not imply same physics everywhere"
      },
      "Adversarial check": {
        "text": "Construct a diagonal theory whose species number changes with the spacetime class",
        "html": "Construct a diagonal theory whose species number changes with the spacetime class"
      }
    },
    {
      "Object and domain": {
        "text": "Natural state or boundary-enriched theory",
        "html": "Natural state or boundary-enriched theory"
      },
      "Required hypotheses": {
        "text": "For the no-go theorem: dynamical locality, faithful Reeh–Schlieder GNS on a noncompact-Cauchy spacetime, and extended locality; for boundaries: fixed well-posed boundary data preserved by arrows",
        "html": "For the no-go theorem: dynamical locality, faithful Reeh–Schlieder GNS on a noncompact-Cauchy spacetime, and extended locality; for boundaries: fixed well-posed boundary data preserved by arrows"
      },
      "Licensed conclusion": {
        "text": "The natural state forces trivial rce and, with the full hypotheses, a trivial theory; a fixed-boundary propagator may define a restricted functor",
        "html": "The natural state forces trivial rce and, with the full hypotheses, a trivial theory; a fixed-boundary propagator may define a restricted functor"
      },
      "Excluded converse or upgrade": {
        "text": "No-natural-state does not mean no Hadamard states, and a fixed Robin construction is not generic boundary covariance",
        "html": "No-natural-state does not mean no Hadamard states, and a fixed Robin construction is not generic boundary covariance"
      },
      "Adversarial check": {
        "text": "Confuse a covariant state space with one natural section, or change the Robin parameter while keeping the same propagator map",
        "html": "Confuse a covariant state space with one natural section, or change the Robin parameter while keeping the same propagator map"
      }
    }
  ]
}