{
  "title": "Domains, hypotheses, licensed results, excluded upgrades, and adversarial checks for defects and categorical symmetry",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "defects-generalized-symmetry-categorical-qft-hypothesis-conclusion-table",
  "caption": "Domains, hypotheses, licensed results, excluded upgrades, and adversarial checks for defects and categorical symmetry",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Required hypotheses",
      "html": "Required hypotheses"
    },
    {
      "text": "Licensed result",
      "html": "Licensed result"
    },
    {
      "text": "Excluded converse or upgrade",
      "html": "Excluded converse or upgrade"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Defects on a stratified two-dimensional spacetime",
        "html": "Defects on a stratified two-dimensional spacetime"
      },
      "Required hypotheses": {
        "text": "Compatible phase labels on every link; typed line and point defects; gluing and isotopy data",
        "html": "Compatible phase labels on every link; typed line and point defects; gluing and isotopy data"
      },
      "Licensed result": {
        "text": "A bicategory of phases, defects, and junctions with defined horizontal and vertical composition",
        "html": "A bicategory of phases, defects, and junctions with defined horizontal and vertical composition"
      },
      "Excluded converse or upgrade": {
        "text": "A drawing of incident strata does not supply a junction or a field-theory amplitude",
        "html": "A drawing of incident strata does not supply a junction or a field-theory amplitude"
      },
      "Adversarial check": {
        "text": "Mismatch the target phase of one wall with the source phase of the next",
        "html": "Mismatch the target phase of one wall with the source phase of the next"
      }
    },
    {
      "Object and domain": {
        "text": "Fusion and module categories",
        "html": "Fusion and module categories"
      },
      "Required hypotheses": {
        "text": "Finite semisimplicity, rigidity, coherent associators and units; compatible module associator",
        "html": "Finite semisimplicity, rigidity, coherent associators and units; compatible module associator"
      },
      "Licensed result": {
        "text": "Finite defect fusion, duals, boundary actions, bimodule composition, and a stated Morita relation",
        "html": "Finite defect fusion, duals, boundary actions, bimodule composition, and a stated Morita relation"
      },
      "Excluded converse or upgrade": {
        "text": "Fusion coefficients alone determine neither the associator nor a QFT realization",
        "html": "Fusion coefficients alone determine neither the associator nor a QFT realization"
      },
      "Adversarial check": {
        "text": "Keep the fusion table but violate a pentagon or snake identity",
        "html": "Keep the fusion table but violate a pentagon or snake identity"
      }
    },
    {
      "Object and domain": {
        "text": "Invertible higher-form defect action",
        "html": "Invertible higher-form defect action"
      },
      "Required hypotheses": {
        "text": "Topological codimension-$(p+1)$ support, group-like fusion, inverse, charge lattice, and global background data",
        "html": "Topological codimension-$(p+1)$ support, group-like fusion, inverse, charge lattice, and global background data"
      },
      "Licensed result": {
        "text": "A $p$-form symmetry action measured by oriented linking characters",
        "html": "A $p$-form symmetry action measured by oriented linking characters"
      },
      "Excluded converse or upgrade": {
        "text": "A deformable noninvertible defect is not a group element",
        "html": "A deformable noninvertible defect is not a group element"
      },
      "Adversarial check": {
        "text": "Fuse the candidate with every label and fail to obtain the transparent defect",
        "html": "Fuse the candidate with every label and fail to obtain the transparent defect"
      }
    },
    {
      "Object and domain": {
        "text": "Noninvertible categorical action",
        "html": "Noninvertible categorical action"
      },
      "Required hypotheses": {
        "text": "Junction spaces, all $F$-symbols, pentagon and triangle equations, duals, and a coherent module category",
        "html": "Junction spaces, all $F$-symbols, pentagon and triangle equations, duals, and a coherent module category"
      },
      "Licensed result": {
        "text": "Allowed sector transitions and composition rules for topological defect endpoints",
        "html": "Allowed sector transitions and composition rules for topological defect endpoints"
      },
      "Excluded converse or upgrade": {
        "text": "Eigenvalues of one fusion matrix are not complete generalized charges",
        "html": "Eigenvalues of one fusion matrix are not complete generalized charges"
      },
      "Adversarial check": {
        "text": "Compare inequivalent module associators with identical action matrices",
        "html": "Compare inequivalent module associators with identical action matrices"
      }
    },
    {
      "Object and domain": {
        "text": "Gauging, equivariantization, or condensation",
        "html": "Gauging, equivariantization, or condensation"
      },
      "Required hypotheses": {
        "text": "Anomaly-free crossed action; or a connected separable commutative algebra and local modules; retained global data",
        "html": "Anomaly-free crossed action; or a connected separable commutative algebra and local modules; retained global data"
      },
      "Licensed result": {
        "text": "The corresponding equivariant, gauged, orbifold, or condensed category in the named finite setting",
        "html": "The corresponding equivariant, gauged, orbifold, or condensed category in the named finite setting"
      },
      "Excluded converse or upgrade": {
        "text": "These constructions are not interchangeable and do not automatically preserve modularity",
        "html": "These constructions are not interchangeable and do not automatically preserve modularity"
      },
      "Adversarial check": {
        "text": "Condense a nonseparable or nonlocal algebra, or omit the crossed-sector obstruction",
        "html": "Condense a nonseparable or nonlocal algebra, or omit the crossed-sector obstruction"
      }
    },
    {
      "Object and domain": {
        "text": "Symmetry TFT boundary or functorial interface",
        "html": "Symmetry TFT boundary or functorial interface"
      },
      "Required hypotheses": {
        "text": "Specified bulk theory, polarization, tangential structure, anomaly matching, balanced actions, and adjunction data",
        "html": "Specified bulk theory, polarization, tangential structure, anomaly matching, balanced actions, and adjunction data"
      },
      "Licensed result": {
        "text": "Topological symmetry and inflow data, gauging interfaces, and coherent wall composition",
        "html": "Topological symmetry and inflow data, gauging interfaces, and coherent wall composition"
      },
      "Excluded converse or upgrade": {
        "text": "The bulk topological theory does not determine boundary Hamiltonians or scaling dimensions",
        "html": "The bulk topological theory does not determine boundary Hamiltonians or scaling dimensions"
      },
      "Adversarial check": {
        "text": "Hold the symmetry TFT fixed while changing boundary dynamics",
        "html": "Hold the symmetry TFT fixed while changing boundary dynamics"
      }
    },
    {
      "Object and domain": {
        "text": "Categorical classification or reconstruction",
        "html": "Categorical classification or reconstruction"
      },
      "Required hypotheses": {
        "text": "Named source class, target category, equivalence, realization functor, completeness assumptions, and semisimplicity or trace replacement",
        "html": "Named source class, target category, equivalence, realization functor, completeness assumptions, and semisimplicity or trace replacement"
      },
      "Licensed result": {
        "text": "Exactly the stated fusion-ring, Morita, center, extended-TQFT, sector, or realization conclusion",
        "html": "Exactly the stated fusion-ring, Morita, center, extended-TQFT, sector, or realization conclusion"
      },
      "Excluded converse or upgrade": {
        "text": "Matching centers or Grothendieck rings does not imply physical QFT equivalence",
        "html": "Matching centers or Grothendieck rings does not imply physical QFT equivalence"
      },
      "Adversarial check": {
        "text": "Change associator, symmetry enrichment, local observables, or extension data while preserving the chosen invariant",
        "html": "Change associator, symmetry enrichment, local observables, or extension data while preserving the chosen invariant"
      }
    }
  ]
}