{
  "title": "Derived and homotopical QFT objects, hypotheses, licensed conclusions, and decisive failure tests.",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "derived-deformation-homotopical-qft-hypothesis-conclusion-table",
  "caption": "Derived and homotopical QFT objects, hypotheses, licensed conclusions, and decisive failure 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 adversarial check",
      "html": "Excluded converse or adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Derived critical locus or gauge quotient",
        "html": "Derived critical locus or gauge quotient"
      },
      "Essential hypotheses": {
        "text": "Specified equation section and gauge complex; compatible grading; chosen smooth, algebraic, or elliptic category",
        "html": "Specified equation section and gauge complex; compatible grading; chosen smooth, algebraic, or elliptic category"
      },
      "Licensed conclusion": {
        "text": "A tangent complex whose cohomology records stabilizers, deformations, equations, and obstructions",
        "html": "A tangent complex whose cohomology records stabilizers, deformations, equations, and obstructions"
      },
      "Excluded converse or adversarial check": {
        "text": "An ordinary orbit set at a reducible flat connection deletes both stabilizer and obstruction groups",
        "html": "An ordinary orbit set at a reducible flat connection deletes both stabilizer and obstruction groups"
      }
    },
    {
      "Object and domain": {
        "text": "L∞ formal moduli problem",
        "html": "L∞ formal moduli problem"
      },
      "Essential hypotheses": {
        "text": "Nilpotent base or complete descending filtration; filtration-compatible brackets and morphisms; characteristic-zero comparison where invoked",
        "html": "Nilpotent base or complete descending filtration; filtration-compatible brackets and morphisms; characteristic-zero comparison where invoked"
      },
      "Licensed conclusion": {
        "text": "A Maurer–Cartan infinity-groupoid invariant under the stated filtered quasi-isomorphism",
        "html": "A Maurer–Cartan infinity-groupoid invariant under the stated filtered quasi-isomorphism"
      },
      "Excluded converse or adversarial check": {
        "text": "A formal series evaluated at a finite unfiltered displacement need not converge or give a global moduli equivalence",
        "html": "A formal series evaluated at a finite unfiltered displacement need not converge or give a global moduli equivalence"
      }
    },
    {
      "Object and domain": {
        "text": "Shifted BV or Poisson geometry",
        "html": "Shifted BV or Poisson geometry"
      },
      "Essential hypotheses": {
        "text": "Perfect tangent and cotangent complexes; closed form; quasi-isomorphic nondegeneracy map; orientation and support pairing",
        "html": "Perfect tangent and cotangent complexes; closed form; quasi-isomorphic nondegeneracy map; orientation and support pairing"
      },
      "Licensed conclusion": {
        "text": "A shifted bracket on derived functions and, for Lagrangian maps, shifted structure on a homotopy intersection",
        "html": "A shifted bracket on derived functions and, for Lagrangian maps, shifted structure on a homotopy intersection"
      },
      "Excluded converse or adversarial check": {
        "text": "Unrestricted Maxwell boundary data produce nonzero symplectic flux, so the bulk pairing is not closed",
        "html": "Unrestricted Maxwell boundary data produce nonzero symplectic flux, so the bulk pairing is not closed"
      }
    },
    {
      "Object and domain": {
        "text": "Formal star product on observables",
        "html": "Formal star product on observables"
      },
      "Essential hypotheses": {
        "text": "Poisson structure and formal parameter; for the direct free-field formula, regular functional derivatives and a defined propagator contraction",
        "html": "Poisson structure and formal parameter; for the direct free-field formula, regular functional derivatives and a defined propagator contraction"
      },
      "Licensed conclusion": {
        "text": "An associative formal deformation with first commutator $i\\hbar$ times the Poisson bracket",
        "html": "An associative formal deformation with first commutator $i\\hbar$ times the Poisson bracket"
      },
      "Excluded converse or adversarial check": {
        "text": "Local nonlinear functionals put singular kernels on diagonals; associativity on regular functionals does not supply the missing extension",
        "html": "Local nonlinear functionals put singular kernels on diagonals; associativity on regular functionals does not supply the missing extension"
      }
    },
    {
      "Object and domain": {
        "text": "Perturbative BV quantization",
        "html": "Perturbative BV quantization"
      },
      "Essential hypotheses": {
        "text": "Regulator and removal limit; local deformation complex; lower-order master equation; admissible symmetry and scaling conditions",
        "html": "Regulator and removal limit; local deformation complex; lower-order master equation; admissible symmetry and scaling conditions"
      },
      "Licensed conclusion": {
        "text": "A degree-one obstruction class; when it vanishes, lifts forming a degree-zero cohomological torsor",
        "html": "A degree-one obstruction class; when it vanishes, lifts forming a degree-zero cohomological torsor"
      },
      "Excluded converse or adversarial check": {
        "text": "A Green-operator primitive can be nonlocal; vanishing at one order does not remove higher obstructions or framing dependence",
        "html": "A Green-operator primitive can be nonlocal; vanishing at one order does not remove higher obstructions or framing dependence"
      }
    },
    {
      "Object and domain": {
        "text": "Boundary gluing and Morita composition",
        "html": "Boundary gluing and Morita composition"
      },
      "Essential hypotheses": {
        "text": "Homotopy fiber products; Lagrangian regularity for shifted claims; existing relative tensor products; duals and adjoints for full extension",
        "html": "Homotopy fiber products; Lagrangian regularity for shifted claims; existing relative tensor products; duals and adjoints for full extension"
      },
      "Licensed conclusion": {
        "text": "Zero-mode-preserving gluing and coherent fusion of algebras, bimodules, and intertwiners",
        "html": "Zero-mode-preserving gluing and coherent fusion of algebras, bimodules, and intertwiners"
      },
      "Excluded converse or adversarial check": {
        "text": "An ordinary transverse intersection loses harmonic cokernels; a nondualizable infinite algebra does not define a fully extended theory",
        "html": "An ordinary transverse intersection loses harmonic cokernels; a nondualizable infinite algebra does not define a fully extended theory"
      }
    },
    {
      "Object and domain": {
        "text": "Effective transfer or categorical strictification",
        "html": "Effective transfer or categorical strictification"
      },
      "Essential hypotheses": {
        "text": "Chain contraction for transfer; convergence or formal filtration as stated; local equivalence and essential surjectivity for biequivalence; named extra structures",
        "html": "Chain contraction for transfer; convergence or formal filtration as stated; local equivalence and essential surjectivity for biequivalence; named extra structures"
      },
      "Licensed conclusion": {
        "text": "Transferred tree-level brackets or a strict two-category preserving the specified homotopy type",
        "html": "Transferred tree-level brackets or a strict two-category preserving the specified homotopy type"
      },
      "Excluded converse or adversarial check": {
        "text": "A non-chain projection fails at the unary identity; forgetting topology, domains, adjoints, or observables leaves only an algebraic surrogate",
        "html": "A non-chain projection fails at the unary identity; forgetting topology, domains, adjoints, or observables leaves only an algebraic surrogate"
      }
    }
  ]
}