{
  "title": "Domains, hypotheses, directional conclusions, excluded upgrades, and failure tests for extended TQFT classifications",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "extended-tqft-bordism-hypothesis-conclusion-table",
  "caption": "Domains, hypotheses, directional conclusions, excluded upgrades, and failure tests for extended TQFT classifications",
  "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": "Unextended oriented bordism functor",
        "html": "Unextended oriented bordism functor"
      },
      "Required hypotheses": {
        "text": "Fixed dimension and oriented bordism category; finite-dimensional target; monoidality, diffeomorphism invariance, nondegenerate duality, and sewing",
        "html": "Fixed dimension and oriented bordism category; finite-dimensional target; monoidality, diffeomorphism invariance, nondegenerate duality, and sewing"
      },
      "Licensed result": {
        "text": "State spaces and bordism maps compatible with disjoint union and gluing",
        "html": "State spaces and bordism maps compatible with disjoint union and gluing"
      },
      "Excluded converse or upgrade": {
        "text": "Closed partition functions alone do not define the functor, and no point value follows",
        "html": "Closed partition functions alone do not define the functor, and no point value follows"
      },
      "Adversarial check": {
        "text": "Assign different maps to two Frobenius-related decompositions of one surface",
        "html": "Assign different maps to two Frobenius-related decompositions of one surface"
      }
    },
    {
      "Object and domain": {
        "text": "Partially or fully extended functor",
        "html": "Partially or fully extended functor"
      },
      "Required hypotheses": {
        "text": "Bordisms with corners through the declared depth; symmetric monoidal higher target; duals and adjoints at every visible level",
        "html": "Bordisms with corners through the declared depth; symmetric monoidal higher target; duals and adjoints at every visible level"
      },
      "Licensed result": {
        "text": "Consistent values on points, strata, junctions, and their compositions through that depth",
        "html": "Consistent values on points, strata, junctions, and their compositions through that depth"
      },
      "Excluded converse or upgrade": {
        "text": "Object duality alone does not provide adjoints for lower morphisms",
        "html": "Object duality alone does not provide adjoints for lower morphisms"
      },
      "Adversarial check": {
        "text": "Use the dual-number algebra or a nonadjointable bimodule and demand a point-level extension",
        "html": "Use the dual-number algebra or a nonadjointable bimodule and demand a point-level extension"
      }
    },
    {
      "Object and domain": {
        "text": "Framed fully extended classification",
        "html": "Framed fully extended classification"
      },
      "Required hypotheses": {
        "text": "Dimension $n$; framed fully extended bordism category; specified symmetric monoidal $(\\infty,n)$-category; fully dualizable object",
        "html": "Dimension $n$; framed fully extended bordism category; specified symmetric monoidal $(\\infty,n)$-category; fully dualizable object"
      },
      "Licensed result": {
        "text": "Equivalence of the functor groupoid with the maximal groupoid of fully dualizable objects",
        "html": "Equivalence of the functor groupoid with the maximal groupoid of fully dualizable objects"
      },
      "Excluded converse or upgrade": {
        "text": "No oriented, positive, unextended, or physical classification follows automatically",
        "html": "No oriented, positive, unextended, or physical classification follows automatically"
      },
      "Adversarial check": {
        "text": "Change the target or omit one adjoint level and compare the resulting functor groupoids",
        "html": "Change the target or omit one adjoint level and compare the resulting functor groupoids"
      }
    },
    {
      "Object and domain": {
        "text": "Oriented, spin, pin, or $G$-structured theory",
        "html": "Oriented, spin, pin, or $G$-structured theory"
      },
      "Required hypotheses": {
        "text": "Declared homomorphism $G\\to O(n)$ and coherent homotopy fixed-point data for the induced action",
        "html": "Declared homomorphism $G\\to O(n)$ and coherent homotopy fixed-point data for the induced action"
      },
      "Licensed result": {
        "text": "A structured refinement of the framed theory",
        "html": "A structured refinement of the framed theory"
      },
      "Excluded converse or upgrade": {
        "text": "An invariant isomorphism without higher coherence is not a homotopy fixed point",
        "html": "An invariant isomorphism without higher coherence is not a homotopy fixed point"
      },
      "Adversarial check": {
        "text": "Forget the spin structure in Arf theory and compare the two torus signs",
        "html": "Forget the spin structure in Arf theory and compare the two torus signs"
      }
    },
    {
      "Object and domain": {
        "text": "Invertible extended theory",
        "html": "Invertible extended theory"
      },
      "Required hypotheses": {
        "text": "Factorization through the Picard target; exact symmetry type; discrete or continuous target; declared deformation and positivity conditions",
        "html": "Factorization through the Picard target; exact symmetry type; discrete or continuous target; declared deformation and positivity conditions"
      },
      "Licensed result": {
        "text": "A spectrum-map or generalized-cohomology classification in the stated regime",
        "html": "A spectrum-map or generalized-cohomology classification in the stated regime"
      },
      "Excluded converse or upgrade": {
        "text": "Noninvertible topological order and microscopic realizability are not included",
        "html": "Noninvertible topological order and microscopic realizability are not included"
      },
      "Adversarial check": {
        "text": "Use a multidimensional state space and attempt to tensor it to the one-dimensional unit",
        "html": "Use a multidimensional state space and attempt to tensor it to the one-dimensional unit"
      }
    },
    {
      "Object and domain": {
        "text": "Relative anomalous theory",
        "html": "Relative anomalous theory"
      },
      "Required hypotheses": {
        "text": "Invertible bulk anomaly functor; compatible boundary truncation; line-valued partition function; coherent trivialization for cancellation",
        "html": "Invertible bulk anomaly functor; compatible boundary truncation; line-valued partition function; coherent trivialization for cancellation"
      },
      "Licensed result": {
        "text": "Functorial anomaly line, inflow transformation, and a precise cancellation condition",
        "html": "Functorial anomaly line, inflow transformation, and a precise cancellation condition"
      },
      "Excluded converse or upgrade": {
        "text": "A local counterterm in one chart does not remove global holonomy",
        "html": "A local counterterm in one chart does not remove global holonomy"
      },
      "Adversarial check": {
        "text": "Transport around a large-gauge loop and compute determinant-line or eta holonomy",
        "html": "Transport around a large-gauge loop and compute determinant-line or eta holonomy"
      }
    },
    {
      "Object and domain": {
        "text": "Hermitian or reflection-positive TQFT with boundaries",
        "html": "Hermitian or reflection-positive TQFT with boundaries"
      },
      "Required hypotheses": {
        "text": "Compatible dagger and reflection structure; positive reflected pairings; module or morphism actions; all sewing and Cardy-type relations required by the model",
        "html": "Compatible dagger and reflection structure; positive reflected pairings; module or morphism actions; all sewing and Cardy-type relations required by the model"
      },
      "Licensed result": {
        "text": "Positive state-space forms and functorial boundary, wall, and junction data",
        "html": "Positive state-space forms and functorial boundary, wall, and junction data"
      },
      "Excluded converse or upgrade": {
        "text": "Dualizability does not imply positivity, and a bare vector-space label is not a boundary condition",
        "html": "Dualizability does not imply positivity, and a bare vector-space label is not a boundary condition"
      },
      "Adversarial check": {
        "text": "Choose a negative Frobenius trace weight or omit the module action",
        "html": "Choose a negative Frobenius trace weight or omit the module action"
      }
    }
  ]
}