{
  "title": "Constructive hypotheses, licensed conclusions, and excluded converses",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "constructive-qft-hypothesis-conclusion-table",
  "caption": "Constructive hypotheses, licensed conclusions, and excluded converses",
  "columns": [
    {
      "text": "Object or stage",
      "html": "Object or stage"
    },
    {
      "text": "Essential hypotheses",
      "html": "Essential hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "What does not follow",
      "html": "What does not follow"
    }
  ],
  "rows": [
    {
      "Object or stage": {
        "text": "Gaussian random distribution",
        "html": "Gaussian random distribution"
      },
      "Essential hypotheses": {
        "text": "Continuous positive covariance on the declared test space; reflection-positive kernel when OS use is intended",
        "html": "Continuous positive covariance on the declared test space; reflection-positive kernel when OS use is intended"
      },
      "Licensed conclusion": {
        "text": "A unique Gaussian law and Wick moments; free OS theory under the full conditions",
        "html": "A unique Gaussian law and Wick moments; free OS theory under the full conditions"
      },
      "What does not follow": {
        "text": "Ordinary point values, an interaction, or reflection positivity from positive definiteness alone",
        "html": "Ordinary point values, an interaction, or reflection positivity from positive definiteness alone"
      }
    },
    {
      "Object or stage": {
        "text": "Interacting finite-volume law",
        "html": "Interacting finite-volume law"
      },
      "Essential hypotheses": {
        "text": "Defined Wick powers or determinant; lower stability bound; finite normalization",
        "html": "Defined Wick powers or determinant; lower stability bound; finite normalization"
      },
      "Licensed conclusion": {
        "text": "A probability law at the stated cutoff and volume",
        "html": "A probability law at the stated cutoff and volume"
      },
      "What does not follow": {
        "text": "Uniform ultraviolet removal, thermodynamic uniqueness, or non-Gaussian continuum behavior",
        "html": "Uniform ultraviolet removal, thermodynamic uniqueness, or non-Gaussian continuum behavior"
      }
    },
    {
      "Object or stage": {
        "text": "Cluster or correlation control",
        "html": "Cluster or correlation control"
      },
      "Essential hypotheses": {
        "text": "Convergent polymer norm, or a named order inequality with uniform moment bounds",
        "html": "Convergent polymer norm, or a named order inequality with uniform moment bounds"
      },
      "Licensed conclusion": {
        "text": "Analyticity or monotonicity and controlled connected correlations within that domain",
        "html": "Analyticity or monotonicity and controlled connected correlations within that domain"
      },
      "What does not follow": {
        "text": "Behavior outside the norm, absence of phase transitions, or a critical limit",
        "html": "Behavior outside the norm, absence of phase transitions, or a critical limit"
      }
    },
    {
      "Object or stage": {
        "text": "Regulator limit",
        "html": "Regulator limit"
      },
      "Essential hypotheses": {
        "text": "Tightness plus uniqueness or a Cauchy estimate; counterterm tuning; uniform integrability",
        "html": "Tightness plus uniqueness or a Cauchy estimate; counterterm tuning; uniform integrability"
      },
      "Licensed conclusion": {
        "text": "A limiting measure or Schwinger hierarchy in the stated topology",
        "html": "A limiting measure or Schwinger hierarchy in the stated topology"
      },
      "What does not follow": {
        "text": "Independence of a different regulator path, reflection positivity, or interaction unless checked",
        "html": "Independence of a different regulator path, reflection positivity, or interaction unless checked"
      }
    },
    {
      "Object or stage": {
        "text": "Relativistic reconstruction",
        "html": "Relativistic reconstruction"
      },
      "Essential hypotheses": {
        "text": "Symmetry, Euclidean covariance, OS positivity, growth and regularity; clustering for a unique vacuum",
        "html": "Symmetry, Euclidean covariance, OS positivity, growth and regularity; clustering for a unique vacuum"
      },
      "Licensed conclusion": {
        "text": "Positive-energy local Wightman theory, with vacuum properties covered by the inputs",
        "html": "Positive-energy local Wightman theory, with vacuum properties covered by the inputs"
      },
      "What does not follow": {
        "text": "A mass gap, isolated particle, scattering, or asymptotic completeness",
        "html": "A mass gap, isolated particle, scattering, or asymptotic completeness"
      }
    },
    {
      "Object or stage": {
        "text": "Particle scattering",
        "html": "Particle scattering"
      },
      "Essential hypotheses": {
        "text": "Reconstructed local theory, isolated mass shell, interpolating operators, separated velocity supports",
        "html": "Reconstructed local theory, isolated mass shell, interpolating operators, separated velocity supports"
      },
      "Licensed conclusion": {
        "text": "Haag–Ruelle in/out states and wave operators for the covered particle species",
        "html": "Haag–Ruelle in/out states and wave operators for the covered particle species"
      },
      "What does not follow": {
        "text": "Completeness of those states or absence of additional sectors and bound states",
        "html": "Completeness of those states or absence of additional sectors and bound states"
      }
    }
  ]
}