{
  "title": "Objects, hypotheses, licensed conclusions, and failure boundaries for local nets and representations",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "algebraic-locally-covariant-qft-hypothesis-conclusion-table",
  "caption": "Objects, hypotheses, licensed conclusions, and failure boundaries for local nets and representations",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Required hypotheses",
      "html": "Required hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or upgrade",
      "html": "Excluded converse or upgrade"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Regional C*-algebras on a directed family of double cones",
        "html": "Regional C*-algebras on a directed family of double cones"
      },
      "Required hypotheses": {
        "text": "Injective coherent inclusions; spacelike commutation and covariant region maps when those properties are claimed",
        "html": "Injective coherent inclusions; spacelike commutation and covariant region maps when those properties are claimed"
      },
      "Licensed conclusion": {
        "text": "An isotonic local net and its representation-independent quasilocal C*-completion",
        "html": "An isotonic local net and its representation-independent quasilocal C*-completion"
      },
      "Excluded converse or upgrade": {
        "text": "The construction is not a tensor product of sharp regions, and locality does not imply duality",
        "html": "The construction is not a tensor product of sharp regions, and locality does not imply duality"
      },
      "Adversarial check": {
        "text": "Assign independent factors to nested or overlapping regions and find the missing inclusion or inconsistent overlap",
        "html": "Assign independent factors to nested or overlapping regions and find the missing inclusion or inconsistent overlap"
      }
    },
    {
      "Object and domain": {
        "text": "Positive normalized functional on the quasilocal algebra",
        "html": "Positive normalized functional on the quasilocal algebra"
      },
      "Required hypotheses": {
        "text": "C*-positivity and normalization; invariance or KMS analyticity only when separately imposed",
        "html": "C*-positivity and normalization; invariance or KMS analyticity only when separately imposed"
      },
      "Licensed conclusion": {
        "text": "A cyclic GNS representation, with purity equivalent to irreducibility and a representation-dependent von Neumann closure",
        "html": "A cyclic GNS representation, with purity equivalent to irreducibility and a representation-dependent von Neumann closure"
      },
      "Excluded converse or upgrade": {
        "text": "GNS does not supply locality, positive energy, faithfulness, or a density matrix in another representation",
        "html": "GNS does not supply locality, positive energy, faithfulness, or a density matrix in another representation"
      },
      "Adversarial check": {
        "text": "Write an infinite-volume thermal state as a vacuum-Fock Gibbs trace and expose the non-trace-class partition function",
        "html": "Write an infinite-volume thermal state as a vacuum-Fock Gibbs trace and expose the non-trace-class partition function"
      }
    },
    {
      "Object and domain": {
        "text": "Two quasifree representations restricted to bounded local algebras",
        "html": "Two quasifree representations restricted to bounded local algebras"
      },
      "Required hypotheses": {
        "text": "Equivalent induced topologies and regional Hilbert–Schmidt covariance difference; Hadamard hypotheses in the free curved-spacetime theorem",
        "html": "Equivalent induced topologies and regional Hilbert–Schmidt covariance difference; Hadamard hypotheses in the free curved-spacetime theorem"
      },
      "Licensed conclusion": {
        "text": "Equality of local folia and mutual normality on every stated bounded region",
        "html": "Equality of local folia and mutual normality on every stated bounded region"
      },
      "Excluded converse or upgrade": {
        "text": "Local quasiequivalence does not produce a compatible global unitary or erase infrared phase differences",
        "html": "Local quasiequivalence does not produce a compatible global unitary or erase infrared phase differences"
      },
      "Adversarial check": {
        "text": "Let the region exhaust infinite volume and watch the thermal covariance difference fail the global Hilbert–Schmidt test",
        "html": "Let the region exhaust infinite volume and watch the thermal covariance difference fail the global Hilbert–Schmidt test"
      }
    },
    {
      "Object and domain": {
        "text": "Vacuum representation and a nonempty open region",
        "html": "Vacuum representation and a nonempty open region"
      },
      "Required hypotheses": {
        "text": "Positive translation spectrum, global cyclicity, weak additivity, covariance, and locality for the separating conclusion",
        "html": "Positive translation spectrum, global cyclicity, weak additivity, covariance, and locality for the separating conclusion"
      },
      "Licensed conclusion": {
        "text": "The vacuum is cyclic locally and separating when the causal complement has nonempty interior",
        "html": "The vacuum is cyclic locally and separating when the causal complement has nonempty interior"
      },
      "Excluded converse or upgrade": {
        "text": "Orbit density gives no uniform operator norm, energy, success probability, or signaling protocol",
        "html": "Orbit density gives no uniform operator norm, energy, success probability, or signaling protocol"
      },
      "Adversarial check": {
        "text": "Drive the approximation error to zero while recording norm and energy instead of assuming they remain bounded",
        "html": "Drive the approximation error to zero while recording norm and energy instead of assuming they remain bounded"
      }
    },
    {
      "Object and domain": {
        "text": "Additive free-Maxwell net on a causally completed ring-like region",
        "html": "Additive free-Maxwell net on a causally completed ring-like region"
      },
      "Required hypotheses": {
        "text": "Specified contractible generators, region topology, causal complement, and the vacuum representation",
        "html": "Specified contractible generators, region topology, causal complement, and the vacuum representation"
      },
      "Licensed conclusion": {
        "text": "Einstein causality survives while a linked flux gives a strict inclusion into the dual algebra",
        "html": "Einstein causality survives while a linked flux gives a strict inclusion into the dual algebra"
      },
      "Excluded converse or upgrade": {
        "text": "Commutation with the complement does not prove that an operator is generated in the original region",
        "html": "Commutation with the complement does not prove that an operator is generated in the original region"
      },
      "Adversarial check": {
        "text": "Compute the nonzero linking pairing of electric and magnetic fluxes on disjoint thickened tori",
        "html": "Compute the nonzero linking pairing of electric and magnetic fluxes on disjoint thickened tori"
      }
    },
    {
      "Object and domain": {
        "text": "Smeared unbounded fields on a common invariant core",
        "html": "Smeared unbounded fields on a common invariant core"
      },
      "Required hypotheses": {
        "text": "Closability, essential self-adjointness or controlled resolvents, support covariance, and strong spacelike commutativity",
        "html": "Closability, essential self-adjointness or controlled resolvents, support covariance, and strong spacelike commutativity"
      },
      "Licensed conclusion": {
        "text": "Spectral unitaries generate an isotonic local von Neumann net to which the closed fields are affiliated",
        "html": "Spectral unitaries generate an isotonic local von Neumann net to which the closed fields are affiliated"
      },
      "Excluded converse or upgrade": {
        "text": "A vanishing form commutator or an unclosable distribution does not license local exponentials",
        "html": "A vanishing form commutator or an unclosable distribution does not license local exponentials"
      },
      "Adversarial check": {
        "text": "Change the self-adjoint extension or test spectral commutation beyond the proposed core",
        "html": "Change the self-adjoint extension or test spectral commutation beyond the proposed core"
      }
    },
    {
      "Object and domain": {
        "text": "Scalar factorizing amplitude and its wedge algebra in two dimensions",
        "html": "Scalar factorizing amplitude and its wedge algebra in two dimensions"
      },
      "Required hypotheses": {
        "text": "Unitarity, hermitian analyticity, crossing, bounded physical-strip continuation, and modular nuclearity for compact localization",
        "html": "Unitarity, hermitian analyticity, crossing, bounded physical-strip continuation, and modular nuclearity for compact localization"
      },
      "Licensed conclusion": {
        "text": "ZF operators give relatively wedge-local fields and, in the proved regular class, nontrivial local intersections; further scaling and phase-space input can fix factor type",
        "html": "ZF operators give relatively wedge-local fields and, in the proved regular class, nontrivial local intersections; further scaling and phase-space input can fix factor type"
      },
      "Excluded converse or upgrade": {
        "text": "A real-axis unitary phase is not a local QFT, and type III alone is not hyperfinite type III₁",
        "html": "A real-axis unitary phase is not a local QFT, and type III alone is not hyperfinite type III₁"
      },
      "Adversarial check": {
        "text": "Remove strip analyticity so the wedge contour shift fails, or remove scaling control so the Connes type is undetermined",
        "html": "Remove strip analyticity so the wedge contour shift fails, or remove scaling control so the Connes type is undetermined"
      }
    }
  ]
}