{
  "title": "Objects, hypotheses, licensed conclusions, and boundary tests",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "causal-perturbation-theory-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": "Equicausal functionals and Peierls kernel",
        "html": "Equicausal functionals and Peierls kernel"
      },
      "Essential hypotheses": {
        "text": "Normally hyperbolic linearization; compact functional support; Hörmander-compatible microcausal cones plus equicontinuity for every derivative family.",
        "html": "Normally hyperbolic linearization; compact functional support; Hörmander-compatible microcausal cones plus equicontinuity for every derivative family."
      },
      "Licensed conclusion": {
        "text": "The retarded and advanced variations and the Peierls bracket are defined and remain smooth on the declared functional domain.",
        "html": "The retarded and advanced variations and the Peierls bracket are defined and remain smooth on the declared functional domain."
      },
      "Excluded converse or extension": {
        "text": "Pointwise microcausality alone need not give smooth closure, and locality does not make arbitrary distributional smearings admissible.",
        "html": "Pointwise microcausality alone need not give smooth closure, and locality does not make arbitrary distributional smearings admissible."
      },
      "Adversarial check": {
        "text": "Use the regular-functional counterexample to smooth closure, or smear on a characteristic surface so covectors sum to zero.",
        "html": "Use the regular-functional counterexample to smooth closure, or smear on a characteristic surface so covectors sum to zero."
      }
    },
    {
      "Object and domain": {
        "text": "Local S-matrix $S(F)$ and coefficients $T_n$",
        "html": "Local S-matrix $S(F)$ and coefficients $T_n$"
      },
      "Essential hypotheses": {
        "text": "Compact supports, a specified causal order, normalized unit, and causal factorization.",
        "html": "Compact supports, a specified causal order, normalized unit, and causal factorization."
      },
      "Licensed conclusion": {
        "text": "Factorization in ordered configurations and inductive determination of $T_n$ away from coincidence diagonals.",
        "html": "Factorization in ordered configurations and inductive determination of $T_n$ away from coincidence diagonals."
      },
      "Excluded converse or extension": {
        "text": "No factorization formula is licensed for overlapping causally unordered supports.",
        "html": "No factorization formula is licensed for overlapping causally unordered supports."
      },
      "Adversarial check": {
        "text": "Choose two overlapping supports with neither later than the other and attempt to swap them.",
        "html": "Choose two overlapping supports with neither later than the other and attempt to swap them."
      }
    },
    {
      "Object and domain": {
        "text": "Off-diagonal distribution $t^0$",
        "html": "Off-diagonal distribution $t^0$"
      },
      "Essential hypotheses": {
        "text": "Finite scaling degree at the diagonal and lower-order extensions already compatible on partial diagonals.",
        "html": "Finite scaling degree at the diagonal and lower-order extensions already compatible on partial diagonals."
      },
      "Licensed conclusion": {
        "text": "An extension exists with controlled local ambiguity; it is unique only below the codimension threshold.",
        "html": "An extension exists with controlled local ambiguity; it is unique only below the codimension threshold."
      },
      "Excluded converse or extension": {
        "text": "Existence does not imply uniqueness at $\\operatorname{sd}(t^0)\\ge d$, nor does choosing extensions independently preserve causal induction.",
        "html": "Existence does not imply uniqueness at $\\operatorname{sd}(t^0)\\ge d$, nor does choosing extensions independently preserve causal induction."
      },
      "Adversarial check": {
        "text": "Extend $(\\Delta_F)^2$ in four dimensions and exhibit its delta-function ambiguity.",
        "html": "Extend $(\\Delta_F)^2$ in four dimensions and exhibit its delta-function ambiguity."
      }
    },
    {
      "Object and domain": {
        "text": "Renormalized $T_n$ and comparison map $Z$",
        "html": "Renormalized $T_n$ and comparison map $Z$"
      },
      "Essential hypotheses": {
        "text": "Causal Wick expansion, locality, additivity, field independence, covariance, scaling, and the same declared basic axioms for two prescriptions.",
        "html": "Causal Wick expansion, locality, additivity, field independence, covariance, scaling, and the same declared basic axioms for two prescriptions."
      },
      "Licensed conclusion": {
        "text": "Finite differences are local diagonal counterterms organized by a Stückelberg–Petermann transformation.",
        "html": "Finite differences are local diagonal counterterms organized by a Stückelberg–Petermann transformation."
      },
      "Excluded converse or extension": {
        "text": "A momentum-dependent nonlocal redefinition is not licensed, and a change of Hadamard normal ordering must be re-expanded consistently.",
        "html": "A momentum-dependent nonlocal redefinition is not licensed, and a change of Hadamard normal ordering must be re-expanded consistently."
      },
      "Adversarial check": {
        "text": "Insert a nonlocal kernel into $Z$ and test additivity for disjointly supported interactions.",
        "html": "Insert a nonlocal kernel into $Z$ and test additivity for disjointly supported interactions."
      }
    },
    {
      "Object and domain": {
        "text": "Bogoliubov map, quadratic background shift, or Ward differential",
        "html": "Bogoliubov map, quadratic background shift, or Ward differential"
      },
      "Essential hypotheses": {
        "text": "Compact interaction; controlled Møller maps for perturbative agreement; local anomaly consistency and an admissible counterterm class for Ward identities.",
        "html": "Compact interaction; controlled Møller maps for perturbative agreement; local anomaly consistency and an admissible counterterm class for Ward identities."
      },
      "Licensed conclusion": {
        "text": "Formal retarded observables, equivalence for the proved background split, and restoration of a Ward identity exactly when its local class is trivial.",
        "html": "Formal retarded observables, equivalence for the proved background split, and restoration of a Ward identity exactly when its local class is trivial."
      },
      "Excluded converse or extension": {
        "text": "The formal series need not converge; arbitrary noncompact metric changes are not covered; consistency does not force an anomaly class to vanish.",
        "html": "The formal series need not converge; arbitrary noncompact metric changes are not covered; consistency does not force an anomaly class to vanish."
      },
      "Adversarial check": {
        "text": "Treat the series as norm convergent, use an uncontrolled kinetic shift, or cancel one current divergence while breaking consistency.",
        "html": "Treat the series as norm convergent, use an uncontrolled kinetic shift, or cancel one current divergence while breaking consistency."
      }
    },
    {
      "Object and domain": {
        "text": "Local interacting algebra $\\mathfrak A_g(O)$",
        "html": "Local interacting algebra $\\mathfrak A_g(O)$"
      },
      "Essential hypotheses": {
        "text": "Switchings agree on a causal neighborhood of $O$ and relative-S intertwiners obey the cocycle relation.",
        "html": "Switchings agree on a causal neighborhood of $O$ and relative-S intertwiners obey the cocycle relation."
      },
      "Licensed conclusion": {
        "text": "The local algebra is independent of switching up to canonical isomorphism and fits into an isotonic local net.",
        "html": "The local algebra is independent of switching up to canonical isomorphism and fits into an isotonic local net."
      },
      "Excluded converse or extension": {
        "text": "Local stabilization does not imply a global constant-coupling S-matrix, vacuum, KMS state, or massless adiabatic limit.",
        "html": "Local stabilization does not imply a global constant-coupling S-matrix, vacuum, KMS state, or massless adiabatic limit."
      },
      "Adversarial check": {
        "text": "Let a massless switching sequence grow and observe infrared-dependent coefficients while fixed bounded regions remain equivalent.",
        "html": "Let a massless switching sequence grow and observe infrared-dependent coefficients while fixed bounded regions remain equivalent."
      }
    },
    {
      "Object and domain": {
        "text": "Formal state or separately constructed local operator algebra",
        "html": "Formal state or separately constructed local operator algebra"
      },
      "Essential hypotheses": {
        "text": "For a formal state, ordered-series positivity and coefficientwise microlocal conditions; for a factorizing model, regular-strip analyticity and modular nuclearity; for a dynamical C*-algebra, its exact defining relations.",
        "html": "For a formal state, ordered-series positivity and coefficientwise microlocal conditions; for a factorizing model, regular-strip analyticity and modular nuclearity; for a dynamical C*-algebra, its exact defining relations."
      },
      "Licensed conclusion": {
        "text": "A normalized positive formal functional, or the particular nonperturbative local-net conclusion proved by the separate theorem.",
        "html": "A normalized positive formal functional, or the particular nonperturbative local-net conclusion proved by the separate theorem."
      },
      "Excluded converse or extension": {
        "text": "Negative numerical truncations do not refute formal positivity; wedge locality alone does not prove compact intersections; neither construction proves general four-dimensional interacting existence.",
        "html": "Negative numerical truncations do not refute formal positivity; wedge locality alone does not prove compact intersections; neither construction proves general four-dimensional interacting existence."
      },
      "Adversarial check": {
        "text": "Evaluate $1-2\\lambda$ at $\\lambda=1$, remove the regularity strip, or demand an unspecified vacuum representation.",
        "html": "Evaluate $1-2\\lambda$ at $\\lambda=1$, remove the regularity strip, or demand an unspecified vacuum representation."
      }
    }
  ]
}