{
  "title": "Domains, assumptions, directional conclusions, and failure tests for equilibrium and nonequilibrium claims",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "thermal-aqft-kms-nonequilibrium-hypothesis-conclusion-table",
  "caption": "Domains, assumptions, directional conclusions, and failure tests for equilibrium and nonequilibrium claims",
  "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": "State on $(\\mathcal A,\\alpha)$",
        "html": "State on $(\\mathcal A,\\alpha)$"
      },
      "Required hypotheses": {
        "text": "Strongly continuous automorphisms; analytic elements; positive normalized functional; KMS boundary identity at fixed $\\beta$",
        "html": "Strongly continuous automorphisms; analytic elements; positive normalized functional; KMS boundary identity at fixed $\\beta$"
      },
      "Licensed conclusion": {
        "text": "Equilibrium relative to the chosen time flow, including infinite systems without a density matrix",
        "html": "Equilibrium relative to the chosen time flow, including infinite systems without a density matrix"
      },
      "Excluded converse or upgrade": {
        "text": "Stationarity alone is not KMS, and a finite-volume Gibbs formula is not an infinite-volume construction",
        "html": "Stationarity alone is not KMS, and a finite-volume Gibbs formula is not an infinite-volume construction"
      },
      "Adversarial check": {
        "text": "Test a stationary diagonal state whose level populations are not Gibbs weights",
        "html": "Test a stationary diagonal state whose level populations are not Gibbs weights"
      }
    },
    {
      "Object and domain": {
        "text": "Cyclic process and tensor powers",
        "html": "Cyclic process and tensor powers"
      },
      "Required hypotheses": {
        "text": "Well-defined generator and differentiable cyclic perturbations; passivity for every finite tensor power",
        "html": "Well-defined generator and differentiable cyclic perturbations; passivity for every finite tensor power"
      },
      "Licensed conclusion": {
        "text": "Under the Pusz–Woronowicz hypotheses, a completely passive state is KMS or a ground state",
        "html": "Under the Pusz–Woronowicz hypotheses, a completely passive state is KMS or a ground state"
      },
      "Excluded converse or upgrade": {
        "text": "Single-copy passivity does not imply complete passivity",
        "html": "Single-copy passivity does not imply complete passivity"
      },
      "Adversarial check": {
        "text": "Use a passive non-Gibbs three-level state and activate work extraction on several copies",
        "html": "Use a passive non-Gibbs three-level state and activate work extraction on several copies"
      }
    },
    {
      "Object and domain": {
        "text": "KMS representation and phase decomposition",
        "html": "KMS representation and phase decomposition"
      },
      "Required hypotheses": {
        "text": "GNS representation, normal extension, faithful support reduction; extremality taken inside the KMS simplex",
        "html": "GNS representation, normal extension, faithful support reduction; extremality taken inside the KMS simplex"
      },
      "Licensed conclusion": {
        "text": "Modular implementation of equilibrium and factorial characterization of a pure thermodynamic phase",
        "html": "Modular implementation of equilibrium and factorial characterization of a pure thermodynamic phase"
      },
      "Excluded converse or upgrade": {
        "text": "Factorial does not mean type I or vector-pure, and a convex mixture is not one phase",
        "html": "Factorial does not mean type I or vector-pure, and a convex mixture is not one phase"
      },
      "Adversarial check": {
        "text": "Decompose a central mixture of two disjoint factorial KMS states",
        "html": "Decompose a central mixture of two disjoint factorial KMS states"
      }
    },
    {
      "Object and domain": {
        "text": "Thermal construction and relaxation",
        "html": "Thermal construction and relaxation"
      },
      "Required hypotheses": {
        "text": "For existence, thermal/phase-space nuclearity and controlled volume limits; for mixing, Liouvillean spectral and perturbative estimates",
        "html": "For existence, thermal/phase-space nuclearity and controlled volume limits; for mixing, Liouvillean spectral and perturbative estimates"
      },
      "Licensed conclusion": {
        "text": "Existence of locally normal KMS states, or return to equilibrium in the stated folium and topology",
        "html": "Existence of locally normal KMS states, or return to equilibrium in the stated folium and topology"
      },
      "Excluded converse or upgrade": {
        "text": "Nuclearity does not imply uniqueness or mixing; KMS analyticity does not remove nonzero Liouvillean resonances",
        "html": "Nuclearity does not imply uniqueness or mixing; KMS analyticity does not remove nonzero Liouvillean resonances"
      },
      "Adversarial check": {
        "text": "Add a conserved quantity or phase multiplicity and test whether correlations decay",
        "html": "Add a conserved quantity or phase multiplicity and test whether correlations decay"
      }
    },
    {
      "Object and domain": {
        "text": "Reservoir scattering state",
        "html": "Reservoir scattering state"
      },
      "Required hypotheses": {
        "text": "Thermodynamic reservoirs, Møller or Cesàro limit, current domains, relative-entropy balance, and steady conservation",
        "html": "Thermodynamic reservoirs, Møller or Cesàro limit, current domains, relative-entropy balance, and steady conservation"
      },
      "Licensed conclusion": {
        "text": "A stationary NESS and nonnegative weighted entropy production with a fixed current convention",
        "html": "A stationary NESS and nonnegative weighted entropy production with a fixed current convention"
      },
      "Excluded converse or upgrade": {
        "text": "A finite recurrent system does not establish a unique NESS; zero production need not imply equal temperatures",
        "html": "A finite recurrent system does not establish a unique NESS; zero production need not imply equal temperatures"
      },
      "Adversarial check": {
        "text": "Set the transmission to zero between reservoirs at unequal temperatures",
        "html": "Set the transmission to zero between reservoirs at unequal temperatures"
      }
    },
    {
      "Object and domain": {
        "text": "Local or horizon temperature claim",
        "html": "Local or horizon temperature claim"
      },
      "Required hypotheses": {
        "text": "For LTE, a selected local-observable space and positive reference measure; for horizons, geometric modular action or a regular invariant state plus normalized Killing flow",
        "html": "For LTE, a selected local-observable space and positive reference measure; for horizons, geometric modular action or a regular invariant state plus normalized Killing flow"
      },
      "Licensed conclusion": {
        "text": "Thermal compatibility at stated resolution, or a KMS period for the specified boost/Killing parameter",
        "html": "Thermal compatibility at stated resolution, or a KMS period for the specified boost/Killing parameter"
      },
      "Excluded converse or upgrade": {
        "text": "One thermometer does not determine a universal temperature; a local Rindler argument does not prove Hawking flux",
        "html": "One thermometer does not determine a universal temperature; a local Rindler argument does not prove Hawking flux"
      },
      "Adversarial check": {
        "text": "Match the Wick-square moment with two distinct mixtures, or change the global state while keeping the same local horizon geometry",
        "html": "Match the Wick-square moment with two distinct mixtures, or change the global state while keeping the same local horizon geometry"
      }
    }
  ]
}