{
  "title": "Objects, hypotheses, licensed conclusions, and failure boundaries in Euclidean reconstruction",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "euclidean-os-reconstruction-hypothesis-conclusion-table",
  "caption": "Objects, hypotheses, licensed conclusions, and failure boundaries in Euclidean reconstruction",
  "columns": [
    {
      "text": "Euclidean object and domain",
      "html": "Euclidean object and domain"
    },
    {
      "text": "OS hypotheses",
      "html": "OS hypotheses"
    },
    {
      "text": "Reconstruction or result",
      "html": "Reconstruction or result"
    },
    {
      "text": "Excluded converse",
      "html": "Excluded converse"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Euclidean object and domain": {
        "text": "Random distribution or full Schwinger hierarchy on Schwartz test functions",
        "html": "Random distribution or full Schwinger hierarchy on Schwartz test functions"
      },
      "OS hypotheses": {
        "text": "Consistent cylinder laws or a continuous positive-definite characteristic functional; finite, jointly continuous moments when the hierarchy is used",
        "html": "Consistent cylinder laws or a continuous positive-definite characteristic functional; finite, jointly continuous moments when the hierarchy is used"
      },
      "Reconstruction or result": {
        "text": "A Euclidean probability law determines compatible moments, while Minlos extension constructs a law from a valid characteristic functional",
        "html": "A Euclidean probability law determines compatible moments, while Minlos extension constructs a law from a valid characteristic functional"
      },
      "Excluded converse": {
        "text": "A symmetric Euclidean-covariant moment list need not exist as, or uniquely determine, a probability law",
        "html": "A symmetric Euclidean-covariant moment list need not exist as, or uniquely determine, a probability law"
      },
      "Adversarial check": {
        "text": "Test every finite moment matrix; a negative variance or an indeterminate moment problem stops the claimed inference",
        "html": "Test every finite moment matrix; a negative variance or an indeterminate moment problem stops the claimed inference"
      }
    },
    {
      "Euclidean object and domain": {
        "text": "Positive-time algebra with a fixed reflection across the time-zero plane",
        "html": "Positive-time algebra with a fixed reflection across the time-zero plane"
      },
      "OS hypotheses": {
        "text": "Euclidean covariance and symmetry, suitable regularity, and nonnegativity of the reflected form for every positive-time polynomial",
        "html": "Euclidean covariance and symmetry, suitable regularity, and nonnegativity of the reflected form for every positive-time polynomial"
      },
      "Reconstruction or result": {
        "text": "Quotienting its null space and completing gives the physical Hilbert space and a contraction semigroup with nonnegative generator",
        "html": "Quotienting its null space and completing gives the physical Hilbert space and a contraction semigroup with nonnegative generator"
      },
      "Excluded converse": {
        "text": "Ordinary covariance positivity does not imply reflection positivity, locality, or a relativistic field theory",
        "html": "Ordinary covariance positivity does not imply reflection positivity, locality, or a relativistic field theory"
      },
      "Adversarial check": {
        "text": "Use a positive rational Fourier multiplier with one negative partial-fraction residue and find a negative reflected direction",
        "html": "Use a positive rational Fourier multiplier with one negative partial-fraction residue and find a negative reflected direction"
      }
    },
    {
      "Euclidean object and domain": {
        "text": "Complete hierarchy at all orders on Euclidean configuration space",
        "html": "Complete hierarchy at all orders on Euclidean configuration space"
      },
      "OS hypotheses": {
        "text": "Corrected E0′ or E0″ growth together with Euclidean invariance, reflection positivity, permutation symmetry, and clustering",
        "html": "Corrected E0′ or E0″ growth together with Euclidean invariance, reflection positivity, permutation symmetry, and clustering"
      },
      "Reconstruction or result": {
        "text": "The corrected reconstruction theorem yields a tempered local positive-energy Wightman hierarchy and its cyclic theory, unique up to unitary equivalence",
        "html": "The corrected reconstruction theorem yields a tempered local positive-energy Wightman hierarchy and its cyclic theory, unique up to unitary equivalence"
      },
      "Excluded converse": {
        "text": "A valid two-point kernel or fixed-order temperedness does not establish a complete reconstruction",
        "html": "A valid two-point kernel or fixed-order temperedness does not establish a complete reconstruction"
      },
      "Adversarial check": {
        "text": "Keep the free two-point function but set the four-point function to zero; polynomial positivity then becomes negative",
        "html": "Keep the free two-point function but set the four-point function to zero; polynomial positivity then becomes negative"
      }
    },
    {
      "Euclidean object and domain": {
        "text": "Ordered imaginary-time regions and their permuted complex tubes",
        "html": "Ordered imaginary-time regions and their permuted complex tubes"
      },
      "OS hypotheses": {
        "text": "Spectral support, Euclidean symmetry, uniform growth, an unobstructed path inside the named tube, and a specified boundary approach",
        "html": "Spectral support, Euclidean symmetry, uniform growth, an unobstructed path inside the named tube, and a specified boundary approach"
      },
      "Reconstruction or result": {
        "text": "Analytic functions acquire Wightman, opposite-order, or time-ordered distributional boundary values according to the approach domain",
        "html": "Analytic functions acquire Wightman, opposite-order, or time-ordered distributional boundary values according to the approach domain"
      },
      "Excluded converse": {
        "text": "The substitution t = − iτ does not choose an ordering, cross a singularity, or prove numerical stability",
        "html": "The substitution <var>t</var> = −<var>iτ</var> does not choose an ordering, cross a singularity, or prove numerical stability"
      },
      "Adversarial check": {
        "text": "Move an energy contour through a pinch and verify that residues or a cut obstruct the proposed continuation",
        "html": "Move an energy contour through a pinch and verify that residues or a cut obstruct the proposed continuation"
      }
    },
    {
      "Euclidean object and domain": {
        "text": "Clustered Euclidean state and connected semigroup correlations",
        "html": "Clustered Euclidean state and connected semigroup correlations"
      },
      "OS hypotheses": {
        "text": "Qualitative clustering on a dense observable set; for a gap, a uniform exponential rate in a sector that detects the full vacuum complement",
        "html": "Qualitative clustering on a dense observable set; for a gap, a uniform exponential rate in a sector that detects the full vacuum complement"
      },
      "Reconstruction or result": {
        "text": "Qualitative clustering selects a unique vacuum under the stated density assumptions; exponential decay locates a positive spectral threshold",
        "html": "Qualitative clustering selects a unique vacuum under the stated density assumptions; exponential decay locates a positive spectral threshold"
      },
      "Excluded converse": {
        "text": "A Hamiltonian gap need not contain an isolated one-particle mass shell, and one channel need not see every low-energy state",
        "html": "A Hamiltonian gap need not contain an isolated one-particle mass shell, and one channel need not see every low-energy state"
      },
      "Adversarial check": {
        "text": "Choose a continuous positive spectral measure supported above a positive threshold; decay is exponential but there is no particle pole",
        "html": "Choose a continuous positive spectral measure supported above a positive threshold; decay is exponential but there is no particle pole"
      }
    },
    {
      "Euclidean object and domain": {
        "text": "Fermionic or gauge lattice algebra on one side of a site- or link-centered reflection",
        "html": "Fermionic or gauge lattice algebra on one side of a site- or link-centered reflection"
      },
      "OS hypotheses": {
        "text": "Antilinear graded order reversal, orientation reversal of crossing links, a gauge-invariant positive algebra, and a nonnegative crossing-action expansion",
        "html": "Antilinear graded order reversal, orientation reversal of crossing links, a gauge-invariant positive algebra, and a nonnegative crossing-action expansion"
      },
      "Reconstruction or result": {
        "text": "The null quotient gives the positive physical sector and, for the stated Wilson construction, a positive transfer operator at fixed lattice spacing",
        "html": "The null quotient gives the positive physical sector and, for the stated Wilson construction, a positive transfer operator at fixed lattice spacing"
      },
      "Excluded converse": {
        "text": "Gauge-fixed propagator positivity and continuum existence do not follow; an improved action is not automatically reflection positive",
        "html": "Gauge-fixed propagator positivity and continuum existence do not follow; an improved action is not automatically reflection positive"
      },
      "Adversarial check": {
        "text": "Omit fermionic order reversal or replace a reflected link adjoint by the original orientation and watch the square acquire the wrong sign",
        "html": "Omit fermionic order reversal or replace a reflected link adjoint by the original orientation and watch the square acquire the wrong sign"
      }
    }
  ]
}