{
  "title": "Localization and reconstruction claims, their essential hypotheses, and the first decisive countertest.",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "superselection-statistics-gauge-reconstruction-hypothesis-conclusion-table",
  "caption": "Localization and reconstruction claims, their essential hypotheses, and the first decisive countertest.",
  "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 adversarial check",
      "html": "Excluded converse or adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "DHR representation on a Minkowski vacuum net",
        "html": "DHR representation on a Minkowski vacuum net"
      },
      "Essential hypotheses": {
        "text": "Local normality; vacuum equivalence on every double-cone exterior; transportability; Haag duality for endomorphism form",
        "html": "Local normality; vacuum equivalence on every double-cone exterior; transportability; Haag duality for endomorphism form"
      },
      "Licensed conclusion": {
        "text": "Transportable compactly localized endomorphism, up to unitary intertwiner",
        "html": "Transportable compactly localized endomorphism, up to unitary intertwiner"
      },
      "Excluded converse or adversarial check": {
        "text": "An infinite-volume KMS representation differs from the vacuum in arbitrarily remote regions",
        "html": "An infinite-volume KMS representation differs from the vacuum in arbitrarily remote regions"
      }
    },
    {
      "Object and domain": {
        "text": "Concrete endomorphism category",
        "html": "Concrete endomorphism category"
      },
      "Essential hypotheses": {
        "text": "Localized endomorphisms; coherent transporters; arrows satisfying the full intertwiner equation; closure under composition",
        "html": "Localized endomorphisms; coherent transporters; arrows satisfying the full intertwiner equation; closure under composition"
      },
      "Licensed conclusion": {
        "text": "Strict C*-tensor product by endomorphism composition and $S\\otimes T=S\\rho(T)$",
        "html": "Strict C*-tensor product by endomorphism composition and $S\\otimes T=S\\rho(T)$"
      },
      "Excluded converse or adversarial check": {
        "text": "Fusion labels without transporters or local arrow spaces do not define a category acting on the net",
        "html": "Fusion labels without transporters or local arrow spaces do not define a category acting on the net"
      }
    },
    {
      "Object and domain": {
        "text": "Finite-statistics DHR sector",
        "html": "Finite-statistics DHR sector"
      },
      "Essential hypotheses": {
        "text": "Conjugate equations; positive standard solution; sufficiently high dimension for symmetric exchange",
        "html": "Conjugate equations; positive standard solution; sufficiently high dimension for symmetric exchange"
      },
      "Licensed conclusion": {
        "text": "Statistical dimension, canonical left inverse, and permutation statistics",
        "html": "Statistical dimension, canonical left inverse, and permutation statistics"
      },
      "Excluded converse or adversarial check": {
        "text": "A dimension-one anyon can have nontrivial monodromy, so dimension does not fix statistics",
        "html": "A dimension-one anyon can have nontrivial monodromy, so dimension does not fix statistics"
      }
    },
    {
      "Object and domain": {
        "text": "Compact-group field reconstruction",
        "html": "Compact-group field reconstruction"
      },
      "Essential hypotheses": {
        "text": "Full symmetric rigid C*-tensor category; simple unit; sums, subobjects, conjugates; concrete DHR action",
        "html": "Full symmetric rigid C*-tensor category; simple unit; sums, subobjects, conjugates; concrete DHR action"
      },
      "Licensed conclusion": {
        "text": "A compact $G$, a complete normal field system, and fixed-point observables",
        "html": "A compact $G$, a complete normal field system, and fixed-point observables"
      },
      "Excluded converse or adversarial check": {
        "text": "A modular braided category violates symmetry; a compact action alone does not prove sector completeness",
        "html": "A modular braided category violates symmetry; a compact action alone does not prove sector completeness"
      }
    },
    {
      "Object and domain": {
        "text": "Cone-localized massive or topological charge",
        "html": "Cone-localized massive or topological charge"
      },
      "Essential hypotheses": {
        "text": "Exterior equivalence for spacelike cones; transportability; cone duality; controlled infinite-string limit",
        "html": "Exterior equivalence for spacelike cones; transportability; cone duality; controlled infinite-string limit"
      },
      "Licensed conclusion": {
        "text": "BF tensor category and topology-sensitive braiding",
        "html": "BF tensor category and topology-sensitive braiding"
      },
      "Excluded converse or adversarial check": {
        "text": "A loop linking the semi-infinite tail prevents shrinking the charge into a double cone",
        "html": "A loop linking the semi-infinite tail prevents shrinking the charge into a double cone"
      }
    },
    {
      "Object and domain": {
        "text": "Infrared or interpolating representation",
        "html": "Infrared or interpolating representation"
      },
      "Essential hypotheses": {
        "text": "Specified restricted algebra and charge-class norm criterion, or specified left/right vacuum limits and positive energy",
        "html": "Specified restricted algebra and charge-class norm criterion, or specified left/right vacuum limits and positive energy"
      },
      "Licensed conclusion": {
        "text": "A relative infrared charge class, or a kink sector labeled by two vacua",
        "html": "A relative infrared charge class, or a kink sector labeled by two vacua"
      },
      "Excluded converse or adversarial check": {
        "text": "Equal total charge does not fix a soft cloud; one vacuum cannot match both ends of a kink",
        "html": "Equal total charge does not fix a soft cloud; one vacuum cannot match both ends of a kink"
      }
    },
    {
      "Object and domain": {
        "text": "Completeness and scale comparison",
        "html": "Completeness and scale comparison"
      },
      "Essential hypotheses": {
        "text": "Quantified admissible class; full concrete sector action; all scaling-limit points or a uniqueness theorem; uniform charge control",
        "html": "Quantified admissible class; full concrete sector action; all scaling-limit points or a uniqueness theorem; uniform charge control"
      },
      "Licensed conclusion": {
        "text": "Completeness or preservation only for the named class and scale relation",
        "html": "Completeness or preservation only for the named class and scale relation"
      },
      "Excluded converse or adversarial check": {
        "text": "Fusion rules omit associators, braiding, local action, excluded sectors, and charges that emerge or disappear in a limit",
        "html": "Fusion rules omit associators, braiding, local action, excluded sectors, and charges that emerge or disappear in a limit"
      }
    }
  ]
}