{
  "title": "Hypotheses, licensed conclusions, and failure checks for conformal nets",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "conformal-nets-vertex-algebras-hypothesis-conclusion-table",
  "caption": "Hypotheses, licensed conclusions, and failure checks for conformal nets",
  "columns": [
    {
      "text": "Object or domain",
      "html": "Object or domain"
    },
    {
      "text": "Spectral, locality, or index hypotheses",
      "html": "Spectral, locality, or index hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse",
      "html": "Excluded converse"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object or domain": {
        "text": "Interval assignment on the vacuum Hilbert space",
        "html": "Interval assignment on the vacuum Hilbert space"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Isotony, locality, Möbius covariance, nonnegative rotation generator, unique cyclic vacuum",
        "html": "Isotony, locality, Möbius covariance, nonnegative rotation generator, unique cyclic vacuum"
      },
      "Licensed conclusion": {
        "text": "A positive-energy Möbius conformal net",
        "html": "A positive-energy Möbius conformal net"
      },
      "Excluded converse": {
        "text": "One global covariant algebra does not recover localization",
        "html": "One global covariant algebra does not recover localization"
      },
      "Adversarial check": {
        "text": "Delete the interval labels and verify that locality becomes unstated",
        "html": "Delete the interval labels and verify that locality becomes unstated"
      }
    },
    {
      "Object or domain": {
        "text": "Punctured interval and complementary interval",
        "html": "Punctured interval and complementary interval"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Strong additivity and correctly formulated circle Haag duality",
        "html": "Strong additivity and correctly formulated circle Haag duality"
      },
      "Licensed conclusion": {
        "text": "Point deletion preserves the interval algebra; complements have the stated commutants",
        "html": "Point deletion preserves the interval algebra; complements have the stated commutants"
      },
      "Excluded converse": {
        "text": "Ordinary additivity does not imply strong additivity",
        "html": "Ordinary additivity does not imply strong additivity"
      },
      "Adversarial check": {
        "text": "Cover only by overlapping intervals and observe that the deleted point is never tested",
        "html": "Cover only by overlapping intervals and observe that the deleted point is never tested"
      }
    },
    {
      "Object or domain": {
        "text": "Two disjoint intervals",
        "html": "Two disjoint intervals"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Split, strong additivity, finite Jones index of the dual inclusion",
        "html": "Split, strong additivity, finite Jones index of the dual inclusion"
      },
      "Licensed conclusion": {
        "text": "Complete rationality and finite total squared sector dimension",
        "html": "Complete rationality and finite total squared sector dimension"
      },
      "Excluded converse": {
        "text": "Split and additivity alone do not force finite index",
        "html": "Split and additivity alone do not force finite index"
      },
      "Adversarial check": {
        "text": "Use the U(1) current net with uncountably many charge sectors",
        "html": "Use the U(1) current net with uncountably many charge sectors"
      }
    },
    {
      "Object or domain": {
        "text": "DHR endomorphism category",
        "html": "DHR endomorphism category"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Complete rationality, finite dimensions, conjugates, nondegenerate braiding",
        "html": "Complete rationality, finite dimensions, conjugates, nondegenerate braiding"
      },
      "Licensed conclusion": {
        "text": "A unitary modular tensor category",
        "html": "A unitary modular tensor category"
      },
      "Excluded converse": {
        "text": "An abstract modular category need not determine or realize a net",
        "html": "An abstract modular category need not determine or realize a net"
      },
      "Adversarial check": {
        "text": "Add a transparent nonvacuum sector and watch the S matrix become degenerate",
        "html": "Add a transparent nonvacuum sector and watch the S matrix become degenerate"
      }
    },
    {
      "Object or domain": {
        "text": "Finite-index inclusion",
        "html": "Finite-index inclusion"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Positive special Q-system; braided commutativity for locality",
        "html": "Positive special Q-system; braided commutativity for locality"
      },
      "Licensed conclusion": {
        "text": "A finite-index extension; a local extension in the commutative case",
        "html": "A finite-index extension; a local extension in the commutative case"
      },
      "Excluded converse": {
        "text": "A sector sum or Jones index does not supply multiplication and positivity",
        "html": "A sector sum or Jones index does not supply multiplication and positivity"
      },
      "Adversarial check": {
        "text": "Use the chiral Ising fermion sum and detect its nontrivial exchange sign",
        "html": "Use the chiral Ising fermion sum and detect its nontrivial exchange sign"
      }
    },
    {
      "Object or domain": {
        "text": "Alpha-induced coupling matrix",
        "html": "Alpha-induced coupling matrix"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Finite braided system, nondegenerate braiding, chiral locality",
        "html": "Finite braided system, nondegenerate braiding, chiral locality"
      },
      "Licensed conclusion": {
        "text": "Nonnegative integral matrix commuting with S and T",
        "html": "Nonnegative integral matrix commuting with S and T"
      },
      "Excluded converse": {
        "text": "Modular commutation alone does not produce a local Q-system",
        "html": "Modular commutation alone does not produce a local Q-system"
      },
      "Adversarial check": {
        "text": "Supply a matrix but no associative commutative multiplication",
        "html": "Supply a matrix but no associative commutative multiplication"
      }
    },
    {
      "Object or domain": {
        "text": "Left–right chiral product",
        "html": "Left–right chiral product"
      },
      "Spectral, locality, or index hypotheses": {
        "text": "Completely rational chiral nets and a positive commutative full-center Q-system",
        "html": "Completely rational chiral nets and a positive commutative full-center Q-system"
      },
      "Licensed conclusion": {
        "text": "A local finite-index full two-dimensional net",
        "html": "A local finite-index full two-dimensional net"
      },
      "Excluded converse": {
        "text": "A multiplicity matrix commuting with one modular generator is insufficient",
        "html": "A multiplicity matrix commuting with one modular generator is insufficient"
      },
      "Adversarial check": {
        "text": "For Ising, test diag(1,0,1): it commutes with T but not S",
        "html": "For Ising, test diag(1,0,1): it commutes with T but not S"
      }
    }
  ]
}