{
  "title": "Representative objects, hypotheses, licensed conclusions, and failure boundaries",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "theorem-frameworks-claim-grammar-hypothesis-conclusion-table",
  "caption": "Representative objects, hypotheses, licensed conclusions, and failure boundaries",
  "columns": [
    {
      "text": "Object or framework",
      "html": "Object or framework"
    },
    {
      "text": "Domain and regularity",
      "html": "Domain and regularity"
    },
    {
      "text": "Essential hypotheses",
      "html": "Essential hypotheses"
    },
    {
      "text": "Licensed result and status",
      "html": "Licensed result and status"
    },
    {
      "text": "Excluded converse",
      "html": "Excluded converse"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object or framework": {
        "text": "Cutoff Euclidean φ 4 family",
        "html": "Cutoff Euclidean <var>φ</var><sup>4</sup> family"
      },
      "Domain and regularity": {
        "text": "Finite lattice spacing and volume; correlation functions regarded as distributions after interpolation",
        "html": "Finite lattice spacing and volume; correlation functions regarded as distributions after interpolation"
      },
      "Essential hypotheses": {
        "text": "Specified bare-parameter tuning, uniform moment and tightness estimates, a declared order of ultraviolet and infinite-volume limits, and nontriviality control",
        "html": "Specified bare-parameter tuning, uniform moment and tightness estimates, a declared order of ultraviolet and infinite-volume limits, and nontriviality control"
      },
      "Licensed result and status": {
        "text": "Existence of the stated subsequential or full continuum Schwinger-function limit in the stated topology; nothing stronger",
        "html": "Existence of the stated subsequential or full continuum Schwinger-function limit in the stated topology; nothing stronger"
      },
      "Excluded converse": {
        "text": "Stable simulations or convergent low moments do not prove a unique, non-Gaussian, Wightman-reconstructible continuum theory",
        "html": "Stable simulations or convergent low moments do not prove a unique, non-Gaussian, Wightman-reconstructible continuum theory"
      },
      "Adversarial check": {
        "text": "Reverse the two limits or hold the bare mass fixed and ask which estimate, convergence mode, or nontriviality statement is lost",
        "html": "Reverse the two limits or hold the bare mass fixed and ask which estimate, convergence mode, or nontriviality statement is lost"
      }
    },
    {
      "Object or framework": {
        "text": "Osterwalder–Schrader Schwinger hierarchy",
        "html": "Osterwalder–Schrader Schwinger hierarchy"
      },
      "Domain and regularity": {
        "text": "A complete sequence of Euclidean distributions with the theorem's regularity and growth control",
        "html": "A complete sequence of Euclidean distributions with the theorem's regularity and growth control"
      },
      "Essential hypotheses": {
        "text": "Euclidean covariance, symmetry, reflection positivity, clustering where required, and the corrected analytic hypotheses",
        "html": "Euclidean covariance, symmetry, reflection positivity, clustering where required, and the corrected analytic hypotheses"
      },
      "Licensed result and status": {
        "text": "Reconstruction of a cyclic relativistic theory and Wightman functions, with the theorem's unitary-uniqueness statement",
        "html": "Reconstruction of a cyclic relativistic theory and Wightman functions, with the theorem's unitary-uniqueness statement"
      },
      "Excluded converse": {
        "text": "Euclidean invariance, pointwise positivity, or a positive two-point function alone does not give reconstruction",
        "html": "Euclidean invariance, pointwise positivity, or a positive two-point function alone does not give reconstruction"
      },
      "Adversarial check": {
        "text": "Keep the two-point kernel reflection positive but choose an incompatible four-point function; full polynomial positivity then fails",
        "html": "Keep the two-point kernel reflection positive but choose an incompatible four-point function; full polynomial positivity then fails"
      }
    },
    {
      "Object or framework": {
        "text": "Locally covariant algebraic theory",
        "html": "Locally covariant algebraic theory"
      },
      "Domain and regularity": {
        "text": "A functor from globally hyperbolic spacetimes and admissible embeddings to unital algebras and injective morphisms",
        "html": "A functor from globally hyperbolic spacetimes and admissible embeddings to unital algebras and injective morphisms"
      },
      "Essential hypotheses": {
        "text": "Functorial covariance, Einstein causality, time-slice dynamics, and any declared state-space conditions",
        "html": "Functorial covariance, Einstein causality, time-slice dynamics, and any declared state-space conditions"
      },
      "Licensed result and status": {
        "text": "A theory natural under the specified spacetime embeddings, with locality and dynamics in the stated categorical sense",
        "html": "A theory natural under the specified spacetime embeddings, with locality and dynamics in the stated categorical sense"
      },
      "Excluded converse": {
        "text": "Agreement of one two-point function on one spacetime does not imply natural equivalence to a Wightman, Euclidean, or second algebraic theory",
        "html": "Agreement of one two-point function on one spacetime does not imply natural equivalence to a Wightman, Euclidean, or second algebraic theory"
      },
      "Adversarial check": {
        "text": "Ask for the missing object assignments, morphism compatibility, natural isomorphism, essential-surjectivity statement, and state data",
        "html": "Ask for the missing object assignments, morphism compatibility, natural isomorphism, essential-surjectivity statement, and state data"
      }
    },
    {
      "Object or framework": {
        "text": "Constructive P ( φ ) 2 measure",
        "html": "Constructive <var>P</var>(<var>φ</var>)<sub>2</sub> measure"
      },
      "Domain and regularity": {
        "text": "Cutoff and finite-volume Euclidean measures on distributions, followed by controlled removal of both regulators",
        "html": "Cutoff and finite-volume Euclidean measures on distributions, followed by controlled removal of both regulators"
      },
      "Essential hypotheses": {
        "text": "Stability, uniform bounds, tightness, reflection positivity, clustering, and estimates surviving the chosen limit order",
        "html": "Stability, uniform bounds, tightness, reflection positivity, clustering, and estimates surviving the chosen limit order"
      },
      "Licensed result and status": {
        "text": "A constructed low-dimensional interacting model after the continuum limit and reconstruction steps actually established",
        "html": "A constructed low-dimensional interacting model after the continuum limit and reconstruction steps actually established"
      },
      "Excluded converse": {
        "text": "A nonzero covariance or a finite regulated integral does not prove interaction, uniqueness, universality, or a continuum theory",
        "html": "A nonzero covariance or a finite regulated integral does not prove interaction, uniqueness, universality, or a continuum theory"
      },
      "Adversarial check": {
        "text": "Replace $P$ by a purely quadratic polynomial: the covariance remains nonzero while every higher truncated correlation vanishes",
        "html": "Replace $P$ by a purely quadratic polynomial: the covariance remains nonzero while every higher truncated correlation vanishes"
      }
    },
    {
      "Object or framework": {
        "text": "Quasifree representations of a CCR algebra",
        "html": "Quasifree representations of a CCR algebra"
      },
      "Domain and regularity": {
        "text": "States on one fixed Weyl algebra over a specified real symplectic space; local restrictions when local comparison is claimed",
        "html": "States on one fixed Weyl algebra over a specified real symplectic space; local restrictions when local comparison is claimed"
      },
      "Essential hypotheses": {
        "text": "Comparison of covariance-induced topologies and the relevant Hilbert–Schmidt condition, or the theorem's local Hadamard hypotheses",
        "html": "Comparison of covariance-induced topologies and the relevant Hilbert–Schmidt condition, or the theorem's local Hadamard hypotheses"
      },
      "Licensed result and status": {
        "text": "Unitary equivalence, quasiequivalence, or local quasiequivalence only at the exact strength supplied by the selected criterion",
        "html": "Unitary equivalence, quasiequivalence, or local quasiequivalence only at the exact strength supplied by the selected criterion"
      },
      "Excluded converse": {
        "text": "Isomorphic abstract algebras, matching selected moments, or local quasiequivalence does not imply global unitary equivalence",
        "html": "Isomorphic abstract algebras, matching selected moments, or local quasiequivalence does not imply global unitary equivalence"
      },
      "Adversarial check": {
        "text": "Alter the infrared sector or append a disjoint spectator representation while preserving the selected local two-point data",
        "html": "Alter the infrared sector or append a disjoint spectator representation while preserving the selected local two-point data"
      }
    },
    {
      "Object or framework": {
        "text": "Smeared Klein–Gordon field and stress tensor",
        "html": "Smeared Klein–Gordon field and stress tensor"
      },
      "Domain and regularity": {
        "text": "Test functions for the field; products of distributions constrained by wavefront sets; Wick powers on a common invariant domain",
        "html": "Test functions for the field; products of distributions constrained by wavefront sets; Wick powers on a common invariant domain"
      },
      "Essential hypotheses": {
        "text": "Field equation and propagator choice for φ ( f ); Hadamard singularity, microlocal product criterion, locality, covariance, and renormalization conditions for composites",
        "html": "Field equation and propagator choice for <var>φ</var>(<var>f</var>); Hadamard singularity, microlocal product criterion, locality, covariance, and renormalization conditions for composites"
      },
      "Licensed result and status": {
        "text": "Well-defined smeared fields and locally covariant renormalized composites, modulo the classified finite renormalization freedom",
        "html": "Well-defined smeared fields and locally covariant renormalized composites, modulo the classified finite renormalization freedom"
      },
      "Excluded converse": {
        "text": "The notation $\\phi(x)$ is not automatically a point operator, and the stress tensor is not unique before the allowed counterterms are fixed",
        "html": "The notation $\\phi(x)$ is not automatically a point operator, and the stress tensor is not unique before the allowed counterterms are fixed"
      },
      "Adversarial check": {
        "text": "Substitute a delta distribution or multiply two singular kernels whose wavefront covectors sum to zero",
        "html": "Substitute a delta distribution or multiply two singular kernels whose wavefront covectors sum to zero"
      }
    },
    {
      "Object or framework": {
        "text": "Kernel-checked free Euclidean Gaussian field",
        "html": "Kernel-checked free Euclidean Gaussian field"
      },
      "Domain and regularity": {
        "text": "A pinned Lean environment encoding a Gaussian probability measure on tempered distributions and explicit Euclidean predicates",
        "html": "A pinned Lean environment encoding a Gaussian probability measure on tempered distributions and explicit Euclidean predicates"
      },
      "Essential hypotheses": {
        "text": "Definitions faithful to the intended theorem, imported dependency hashes, a named top-level declaration, and a successful replay without unproved placeholders",
        "html": "Definitions faithful to the intended theorem, imported dependency hashes, a named top-level declaration, and a successful replay without unproved placeholders"
      },
      "Licensed result and status": {
        "text": "A machine-checked proof of the exact encoded free-field existence and properties at that source revision",
        "html": "A machine-checked proof of the exact encoded free-field existence and properties at that source revision"
      },
      "Excluded converse": {
        "text": "Kernel acceptance does not prove that the definitions model an interacting theory, that omitted physics was formalized, or that later revisions preserve the theorem unchanged",
        "html": "Kernel acceptance does not prove that the definitions model an interacting theory, that omitted physics was formalized, or that later revisions preserve the theorem unchanged"
      },
      "Adversarial check": {
        "text": "Change a covariance convention or weaken the encoded positivity predicate while leaving theorem names unchanged, then compare the elaborated statement",
        "html": "Change a covariance convention or weaken the encoded positivity predicate while leaving theorem names unchanged, then compare the elaborated statement"
      }
    }
  ]
}