{
  "title": "Domains, required hypotheses, directional results, and failure tests in curved-spacetime AQFT",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "curved-spacetime-aqft-stress-qei-hypothesis-conclusion-table",
  "caption": "Domains, required hypotheses, directional results, and failure tests in curved-spacetime AQFT",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Required hypotheses",
      "html": "Required hypotheses"
    },
    {
      "text": "Licensed result",
      "html": "Licensed result"
    },
    {
      "text": "Excluded converse or upgrade",
      "html": "Excluded converse or upgrade"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Real free field on a boundaryless spacetime",
        "html": "Real free field on a boundaryless spacetime"
      },
      "Required hypotheses": {
        "text": "Global hyperbolicity; formally self-adjoint Green-hyperbolic operator; quotient by the field-equation image",
        "html": "Global hyperbolicity; formally self-adjoint Green-hyperbolic operator; quotient by the field-equation image"
      },
      "Licensed result": {
        "text": "Nondegenerate symplectic space, CCR algebra, causal covariance, and time-slice",
        "html": "Nondegenerate symplectic space, CCR algebra, causal covariance, and time-slice"
      },
      "Excluded converse or upgrade": {
        "text": "The algebra does not select a representation, vacuum, or Hadamard state",
        "html": "The algebra does not select a representation, vacuum, or Hadamard state"
      },
      "Adversarial check": {
        "text": "Retain $P C_0^\\infty$ and exhibit the radical of the proposed symplectic form",
        "html": "Retain $P C_0^\\infty$ and exhibit the radical of the proposed symplectic form"
      }
    },
    {
      "Object and domain": {
        "text": "Quasifree Klein–Gordon state",
        "html": "Quasifree Klein–Gordon state"
      },
      "Required hypotheses": {
        "text": "Globally hyperbolic metric; positive ultrastatic endpoint covariance or equivalent construction; Cauchy transport; propagation of singularities",
        "html": "Globally hyperbolic metric; positive ultrastatic endpoint covariance or equivalent construction; Cauchy transport; propagation of singularities"
      },
      "Licensed result": {
        "text": "Existence of Hadamard states on the target spacetime",
        "html": "Existence of Hadamard states on the target spacetime"
      },
      "Excluded converse or upgrade": {
        "text": "Existence supplies neither stationarity nor a preferred natural state",
        "html": "Existence supplies neither stationarity nor a preferred natural state"
      },
      "Adversarial check": {
        "text": "Demand invariance on a future metric with no timelike Killing field",
        "html": "Demand invariance on a future metric with no timelike Killing field"
      }
    },
    {
      "Object and domain": {
        "text": "Renormalized scalar stress tensor in four dimensions",
        "html": "Renormalized scalar stress tensor in four dimensions"
      },
      "Required hypotheses": {
        "text": "Hadamard state; local covariance; conservation; scaling and regularity; fixed subtraction scale and boundaryless domain",
        "html": "Hadamard state; local covariance; conservation; scaling and regularity; fixed subtraction scale and boundaryless domain"
      },
      "Licensed result": {
        "text": "Finite state-dependent expectation plus conserved local curvature freedom; conformal trace anomaly",
        "html": "Finite state-dependent expectation plus conserved local curvature freedom; conformal trace anomaly"
      },
      "Excluded converse or upgrade": {
        "text": "Conservation alone does not give uniqueness, and the $\\Box R$ anomaly coefficient is not universal",
        "html": "Conservation alone does not give uniqueness, and the $\\Box R$ anomaly coefficient is not universal"
      },
      "Adversarial check": {
        "text": "Add a finite $R^2$ counterterm and track the shifted trace and gravitational coupling",
        "html": "Add a finite $R^2$ counterterm and track the shifted trace and gravitational coupling"
      }
    },
    {
      "Object and domain": {
        "text": "Sampled energy or null contraction",
        "html": "Sampled energy or null contraction"
      },
      "Required hypotheses": {
        "text": "Declared field and dimension; Hadamard state class; exact timelike or null domain; smooth sampler and normalization",
        "html": "Declared field and dimension; Hadamard state class; exact timelike or null domain; smooth sampler and normalization"
      },
      "Licensed result": {
        "text": "Timelike free-field QEI; two-dimensional chiral null bound; theorem-specific volume bounds",
        "html": "Timelike free-field QEI; two-dimensional chiral null bound; theorem-specific volume bounds"
      },
      "Excluded converse or upgrade": {
        "text": "No state-independent pointwise bound or general compact null-geodesic QEI in four dimensions",
        "html": "No state-independent pointwise bound or general compact null-geodesic QEI in four dimensions"
      },
      "Adversarial check": {
        "text": "Narrow a timelike sampler, or take an infinite boost while holding the wrong normalization fixed",
        "html": "Narrow a timelike sampler, or take an infinite boost while holding the wrong normalization fixed"
      }
    },
    {
      "Object and domain": {
        "text": "Complete null average or focusing argument",
        "html": "Complete null average or focusing argument"
      },
      "Required hypotheses": {
        "text": "Affine completeness and limiting prescription; theorem-specific achronality, causal tube, field, state, initial contraction, and global causality",
        "html": "Affine completeness and limiting prescription; theorem-specific achronality, causal tube, field, state, initial contraction, and global causality"
      },
      "Licensed result": {
        "text": "ANEC or a focal/incompleteness conclusion only under the named theorem’s full inputs",
        "html": "ANEC or a focal/incompleteness conclusion only under the named theorem’s full inputs"
      },
      "Excluded converse or upgrade": {
        "text": "ANEC does not bound a finite negative pulse or imply a singularity without geometry",
        "html": "ANEC does not bound a finite negative pulse or imply a singularity without geometry"
      },
      "Adversarial check": {
        "text": "Truncate the geodesic, or use a noncontracting Minkowski congruence",
        "html": "Truncate the geodesic, or use a noncontracting Minkowski congruence"
      }
    },
    {
      "Object and domain": {
        "text": "Semiclassical equation or interacting perturbation",
        "html": "Semiclassical equation or interacting perturbation"
      },
      "Required hypotheses": {
        "text": "Matched renormalized couplings; conserved Hadamard source; state–geometry data; gauge and causal response; or compact interaction and formal series",
        "html": "Matched renormalized couplings; conserved Hadamard source; state–geometry data; gauge and causal response; or compact interaction and formal series"
      },
      "Licensed result": {
        "text": "Special FLRW existence results, controlled linear stability, or local pAQFT algebras order by order",
        "html": "Special FLRW existence results, controlled linear stability, or local pAQFT algebras order by order"
      },
      "Excluded converse or upgrade": {
        "text": "No generic nonlinear well-posedness, convergent perturbation series, or global interacting vacuum follows",
        "html": "No generic nonlinear well-posedness, convergent perturbation series, or global interacting vacuum follows"
      },
      "Adversarial check": {
        "text": "Count Planckian runaway roots outside the approximation, or set the coupling constant globally without an infrared theorem",
        "html": "Count Planckian runaway roots outside the approximation, or set the coupling constant globally without an infrared theorem"
      }
    },
    {
      "Object and domain": {
        "text": "Scalar field with timelike boundary",
        "html": "Scalar field with timelike boundary"
      },
      "Required hypotheses": {
        "text": "Fixed positive self-adjoint extension; boundary-preserving maps; boundary-adapted microlocal state condition",
        "html": "Fixed positive self-adjoint extension; boundary-preserving maps; boundary-adapted microlocal state condition"
      },
      "Licensed result": {
        "text": "Extension-dependent dynamics, propagator, algebra, and ground state in the proved static model",
        "html": "Extension-dependent dynamics, propagator, algebra, and ground state in the proved static model"
      },
      "Excluded converse or upgrade": {
        "text": "The bulk differential expression does not determine a unique theory, and one Robin model is not generic boundary covariance",
        "html": "The bulk differential expression does not determine a unique theory, and one Robin model is not generic boundary covariance"
      },
      "Adversarial check": {
        "text": "Compare two admissible extensions and their different spectra",
        "html": "Compare two admissible extensions and their different spectra"
      }
    }
  ]
}