{
  "title": "Objects, hypotheses, licensed conclusions, and boundary tests",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "modular-nuclearity-split-hypothesis-conclusion-table",
  "caption": "Objects, hypotheses, licensed conclusions, and boundary tests",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Essential hypotheses",
      "html": "Essential hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or extension",
      "html": "Excluded converse or extension"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Tomita map $S_0$ on $\\mathcal M\\Omega$",
        "html": "Tomita map $S_0$ on $\\mathcal M\\Omega$"
      },
      "Essential hypotheses": {
        "text": "$\\Omega$ is both cyclic and separating for $\\mathcal M$.",
        "html": "$\\Omega$ is both cyclic and separating for $\\mathcal M$."
      },
      "Licensed conclusion": {
        "text": "$S_0$ is well defined and closable; $S=J\\Delta^{1/2}$ gives $J\\mathcal M J=\\mathcal M'$ and modular automorphisms of $\\mathcal M$.",
        "html": "$S_0$ is well defined and closable; $S=J\\Delta^{1/2}$ gives $J\\mathcal M J=\\mathcal M'$ and modular automorphisms of $\\mathcal M$."
      },
      "Excluded converse or extension": {
        "text": "Cyclicity alone does not define $S_0$, and abstract modular flow is not automatically spacetime flow.",
        "html": "Cyclicity alone does not define $S_0$, and abstract modular flow is not automatically spacetime flow."
      },
      "Adversarial check": {
        "text": "Choose $A\\Omega=0$ with $A^*\\Omega\\neq0$: the proposed rule is ambiguous.",
        "html": "Choose $A\\Omega=0$ with $A^*\\Omega\\neq0$: the proposed rule is ambiguous."
      }
    },
    {
      "Object and domain": {
        "text": "Standard form, faithful state $\\varphi$, and subalgebra $\\mathcal N\\subset\\mathcal M$",
        "html": "Standard form, faithful state $\\varphi$, and subalgebra $\\mathcal N\\subset\\mathcal M$"
      },
      "Essential hypotheses": {
        "text": "Natural-cone representatives exist; for an expectation, $\\mathcal N$ is invariant under $\\sigma^\\varphi$.",
        "html": "Natural-cone representatives exist; for an expectation, $\\mathcal N$ is invariant under $\\sigma^\\varphi$."
      },
      "Licensed conclusion": {
        "text": "Relative modular data compare states, and a unique $\\varphi$-preserving normal conditional expectation onto $\\mathcal N$ exists.",
        "html": "Relative modular data compare states, and a unique $\\varphi$-preserving normal conditional expectation onto $\\mathcal N$ exists."
      },
      "Excluded converse or extension": {
        "text": "An arbitrary inclusion need not admit an expectation preserving the chosen state; $J$ is not ordinary charge conjugation.",
        "html": "An arbitrary inclusion need not admit an expectation preserving the chosen state; $J$ is not ordinary charge conjugation."
      },
      "Adversarial check": {
        "text": "Pick a subalgebra moved by $\\sigma^\\varphi$: no $\\varphi$-preserving expectation can satisfy the claimed theorem.",
        "html": "Pick a subalgebra moved by $\\sigma^\\varphi$: no $\\varphi$-preserving expectation can satisfy the claimed theorem."
      }
    },
    {
      "Object and domain": {
        "text": "Vacuum wedge algebra or half-sided modular inclusion",
        "html": "Vacuum wedge algebra or half-sided modular inclusion"
      },
      "Essential hypotheses": {
        "text": "For wedges, the Bisognano–Wichmann field or net hypotheses; for inclusions, a common standard vector and the stated half-line modular compression.",
        "html": "For wedges, the Bisognano–Wichmann field or net hypotheses; for inclusions, a common standard vector and the stated half-line modular compression."
      },
      "Licensed conclusion": {
        "text": "Geometric boosts and wedge reflection, or a positive translation group satisfying the affine commutation law.",
        "html": "Geometric boosts and wedge reflection, or a positive translation group satisfying the affine commutation law."
      },
      "Excluded converse or extension": {
        "text": "Generic double-cone modular flow need not be geometric; two unrelated standard factors need not reconstruct translations.",
        "html": "Generic double-cone modular flow need not be geometric; two unrelated standard factors need not reconstruct translations."
      },
      "Adversarial check": {
        "text": "Replace the wedge by a massive double cone, or drop the half-sided condition: the geometric identification stops.",
        "html": "Replace the wedge by a massive double cone, or drop the half-sided condition: the geometric identification stops."
      }
    },
    {
      "Object and domain": {
        "text": "Energy-damped map $\\Theta_{\\beta,\\mathcal O}$ from the local operator unit ball",
        "html": "Energy-damped map $\\Theta_{\\beta,\\mathcal O}$ from the local operator unit ball"
      },
      "Essential hypotheses": {
        "text": "Positive energy, localization, and a summable nuclear decomposition with controlled dependence on $\\beta$ and $\\mathcal O$.",
        "html": "Positive energy, localization, and a summable nuclear decomposition with controlled dependence on $\\beta$ and $\\mathcal O$."
      },
      "Licensed conclusion": {
        "text": "A quantitative phase-space bound; with the theorem-specific net assumptions, thermal and split consequences.",
        "html": "A quantitative phase-space bound; with the theorem-specific net assumptions, thermal and split consequences."
      },
      "Excluded converse or extension": {
        "text": "Locality and positive energy alone do not imply nuclearity; compactness alone gives no nuclear norm bound.",
        "html": "Locality and positive energy alone do not imply nuclearity; compactness alone gives no nuclear norm bound."
      },
      "Adversarial check": {
        "text": "Add infinitely many species with rapidly increasing multiplicity: the nuclear sum diverges.",
        "html": "Add infinitely many species with rapidly increasing multiplicity: the nuclear sum diverges."
      }
    },
    {
      "Object and domain": {
        "text": "Strict inclusion $\\mathcal N\\subset\\mathcal M$ with a nonzero localization collar",
        "html": "Strict inclusion $\\mathcal N\\subset\\mathcal M$ with a nonzero localization collar"
      },
      "Essential hypotheses": {
        "text": "A nuclearity or equivalent split criterion with the required factoriality and standardness assumptions.",
        "html": "A nuclearity or equivalent split criterion with the required factoriality and standardness assumptions."
      },
      "Licensed conclusion": {
        "text": "A type-I factor $\\mathcal F$ between the endpoints and normal product states for $\\mathcal N$ and $\\mathcal M'$.",
        "html": "A type-I factor $\\mathcal F$ between the endpoints and normal product states for $\\mathcal N$ and $\\mathcal M'$."
      },
      "Excluded converse or extension": {
        "text": "The endpoint algebras do not become type I, the factor is not canonical, and zero collar is not covered.",
        "html": "The endpoint algebras do not become type I, the factor is not canonical, and zero collar is not covered."
      },
      "Adversarial check": {
        "text": "Send the collar width to zero while holding the same bound fixed: the estimate generally becomes singular.",
        "html": "Send the collar width to zero while holding the same bound fixed: the estimate generally becomes singular."
      }
    },
    {
      "Object and domain": {
        "text": "Translated wedge inclusion in a Borchers triple",
        "html": "Translated wedge inclusion in a Borchers triple"
      },
      "Essential hypotheses": {
        "text": "Wedge covariance and locality plus nuclearity of the quarter-modular map for the specified translation.",
        "html": "Wedge covariance and locality plus nuclearity of the quarter-modular map for the specified translation."
      },
      "Licensed conclusion": {
        "text": "Splitness and nontrivial relative commutants; in the constructed factorizing models, compactly localized observables.",
        "html": "Splitness and nontrivial relative commutants; in the constructed factorizing models, compactly localized observables."
      },
      "Excluded converse or extension": {
        "text": "Wedge-local generators or an input scattering function alone do not prove nontrivial double-cone intersections.",
        "html": "Wedge-local generators or an input scattering function alone do not prove nontrivial double-cone intersections."
      },
      "Adversarial check": {
        "text": "Remove the modular nuclearity estimate: the candidate intersection may be trivial.",
        "html": "Remove the modular nuclearity estimate: the candidate intersection may be trivial."
      }
    },
    {
      "Object and domain": {
        "text": "Sharp type-III local algebra and a normal state",
        "html": "Sharp type-III local algebra and a normal state"
      },
      "Essential hypotheses": {
        "text": "No intrinsic semifinite trace; relative modular data are formed in standard form, or a regulator or split factor is specified explicitly.",
        "html": "No intrinsic semifinite trace; relative modular data are formed in standard form, or a regulator or split factor is specified explicitly."
      },
      "Licensed conclusion": {
        "text": "Algebraic relative entropy, and regulated or collar-dependent type-I entropies when the extra structure is present.",
        "html": "Algebraic relative entropy, and regulated or collar-dependent type-I entropies when the extra structure is present."
      },
      "Excluded converse or extension": {
        "text": "No canonical sharp-region density matrix or von Neumann entropy follows from the local algebra.",
        "html": "No canonical sharp-region density matrix or von Neumann entropy follows from the local algebra."
      },
      "Adversarial check": {
        "text": "Demand a trace-one density operator for an exact continuum wedge without a regulator or split factor.",
        "html": "Demand a trace-one density operator for an exact continuum wedge without a regulator or split factor."
      }
    }
  ]
}