{
  "title": "Domains, required hypotheses, directional results, and failure tests for factorization observables",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "factorization-algebras-hypothesis-conclusion-table",
  "caption": "Domains, required hypotheses, directional results, and failure tests for factorization observables",
  "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": "Assignment on opens",
        "html": "Assignment on opens"
      },
      "Required hypotheses": {
        "text": "Symmetric monoidal target; coherent maps for finite disjoint families; unit and equivariance",
        "html": "Symmetric monoidal target; coherent maps for finite disjoint families; unit and equivariance"
      },
      "Licensed result": {
        "text": "Prefactorization algebra",
        "html": "Prefactorization algebra"
      },
      "Excluded converse or upgrade": {
        "text": "Disjoint multiplication alone does not give local-to-global reconstruction",
        "html": "Disjoint multiplication alone does not give local-to-global reconstruction"
      },
      "Adversarial check": {
        "text": "Add a class visible only on one large open and missed by every cover member",
        "html": "Add a class visible only on one large open and missed by every cover member"
      }
    },
    {
      "Object and domain": {
        "text": "Prefactorization algebra on a manifold",
        "html": "Prefactorization algebra on a manifold"
      },
      "Required hypotheses": {
        "text": "Declared Weiss or factorizing covers; homotopically correct Čech object; suitable completion",
        "html": "Declared Weiss or factorizing covers; homotopically correct Čech object; suitable completion"
      },
      "Licensed result": {
        "text": "Factorization algebra with derived local-to-global descent",
        "html": "Factorization algebra with derived local-to-global descent"
      },
      "Excluded converse or upgrade": {
        "text": "Ordinary sheaf descent or underived colimits need not recover multilocal observables",
        "html": "Ordinary sheaf descent or underived colimits need not recover multilocal observables"
      },
      "Adversarial check": {
        "text": "Use a cover in which separated insertion points never occur together",
        "html": "Use a cover in which separated insertion points never occur together"
      }
    },
    {
      "Object and domain": {
        "text": "Classical field complex",
        "html": "Classical field complex"
      },
      "Required hypotheses": {
        "text": "Elliptic Euclidean complex or Green-hyperbolic Lorentzian operator; support conditions; equation-of-motion quotient; wavefront domain where needed",
        "html": "Elliptic Euclidean complex or Green-hyperbolic Lorentzian operator; support conditions; equation-of-motion quotient; wavefront domain where needed"
      },
      "Licensed result": {
        "text": "Classical observable complex with the appropriate shifted BV or Peierls bracket",
        "html": "Classical observable complex with the appropriate shifted BV or Peierls bracket"
      },
      "Excluded converse or upgrade": {
        "text": "Ellipticity does not imply Lorentzian causality, and local functionals alone are not generally bracket-closed",
        "html": "Ellipticity does not imply Lorentzian causality, and local functionals alone are not generally bracket-closed"
      },
      "Adversarial check": {
        "text": "Expose a zero mode, noncompact-support class, or forbidden wavefront product",
        "html": "Expose a zero mode, noncompact-support class, or forbidden wavefront product"
      }
    },
    {
      "Object and domain": {
        "text": "Perturbative quantum observables",
        "html": "Perturbative quantum observables"
      },
      "Required hypotheses": {
        "text": "Gauge-fixed BV resolution; local renormalization; effective quantum master equation; vanishing or cancelled obstruction class",
        "html": "Gauge-fixed BV resolution; local renormalization; effective quantum master equation; vanishing or cancelled obstruction class"
      },
      "Licensed result": {
        "text": "Quantum factorization algebra as a formal, scale-compatible deformation",
        "html": "Quantum factorization algebra as a formal, scale-compatible deformation"
      },
      "Excluded converse or upgrade": {
        "text": "No nonperturbative convergence, positive state, or regulator-independent completion follows",
        "html": "No nonperturbative convergence, positive state, or regulator-independent completion follows"
      },
      "Adversarial check": {
        "text": "Compute a nonzero local anomaly class or introduce a nonlocal counterterm",
        "html": "Compute a nonzero local anomaly class or introduce a nonlocal counterterm"
      }
    },
    {
      "Object and domain": {
        "text": "Locally constant factorization algebra",
        "html": "Locally constant factorization algebra"
      },
      "Required hypotheses": {
        "text": "Disk inclusions are weak equivalences; specified framing, orientation, or other tangential structure; presentable monoidal target",
        "html": "Disk inclusions are weak equivalences; specified framing, orientation, or other tangential structure; presentable monoidal target"
      },
      "Licensed result": {
        "text": "$E_n$ algebra and factorization homology by disk extension and excision",
        "html": "$E_n$ algebra and factorization homology by disk extension and excision"
      },
      "Excluded converse or upgrade": {
        "text": "A generic metric-dependent QFT is not locally constant; an abstract $E_n$ algebra is not automatically a relativistic theory",
        "html": "A generic metric-dependent QFT is not locally constant; an abstract $E_n$ algebra is not automatically a relativistic theory"
      },
      "Adversarial check": {
        "text": "Vary a mass-dependent correlator with disk size or transport framed data around an unlicensed rotation",
        "html": "Vary a mass-dependent correlator with disk size or transport framed data around an unlicensed rotation"
      }
    },
    {
      "Object and domain": {
        "text": "Additive Cauchy-constant AQFT",
        "html": "Additive Cauchy-constant AQFT"
      },
      "Required hypotheses": {
        "text": "Globally hyperbolic Lorentzian regions; Einstein causality; time-slice; time-orderable tuples; compatible monoidal target",
        "html": "Globally hyperbolic Lorentzian regions; Einstein causality; time-slice; time-orderable tuples; compatible monoidal target"
      },
      "Licensed result": {
        "text": "Associated additive Cauchy-constant time-orderable prefactorization algebra",
        "html": "Associated additive Cauchy-constant time-orderable prefactorization algebra"
      },
      "Excluded converse or upgrade": {
        "text": "No Euclidean continuation, Weiss descent, positivity transfer, or BV model is supplied for free",
        "html": "No Euclidean continuation, Weiss descent, positivity transfer, or BV model is supplied for free"
      },
      "Adversarial check": {
        "text": "Choose a causally cyclic tuple with no linear time order",
        "html": "Choose a causally cyclic tuple with no linear time order"
      }
    },
    {
      "Object and domain": {
        "text": "Time-orderable prefactorization algebra",
        "html": "Time-orderable prefactorization algebra"
      },
      "Required hypotheses": {
        "text": "Additivity and Cauchy constancy in the proved strict setting; coherent multiplication reconstructed from Cauchy morphisms",
        "html": "Additivity and Cauchy constancy in the proved strict setting; coherent multiplication reconstructed from Cauchy morphisms"
      },
      "Licensed result": {
        "text": "Strict AQFT comparison and inverse functors in the stated one-category",
        "html": "Strict AQFT comparison and inverse functors in the stated one-category"
      },
      "Excluded converse or upgrade": {
        "text": "The strict theorem does not establish the general infinity-categorical converse",
        "html": "The strict theorem does not establish the general infinity-categorical converse"
      },
      "Adversarial check": {
        "text": "Retain descent but remove time-slice or required higher coherence and test associativity",
        "html": "Retain descent but remove time-slice or required higher coherence and test associativity"
      }
    }
  ]
}