{
  "title": "Framework objects, licensed conclusion types, failure boundaries, and physical return routes across eleven study arcs",
  "scope": "Qualitative framework, theorem-status, failure-boundary, and physical-return comparison across eleven study arcs; no quantitative data are tabulated.",
  "table_id": "mathematical-qft-theorem-status-framework-handoff-table",
  "caption": "Framework objects, licensed conclusion types, failure boundaries, and physical return routes across eleven study arcs",
  "columns": [
    {
      "text": "Study arc",
      "html": "Study arc"
    },
    {
      "text": "Primitive objects",
      "html": "Primitive objects"
    },
    {
      "text": "Main conclusion type",
      "html": "Main conclusion type"
    },
    {
      "text": "Decisive hypotheses",
      "html": "Decisive hypotheses"
    },
    {
      "text": "Excluded upgrade",
      "html": "Excluded upgrade"
    },
    {
      "text": "Physical return",
      "html": "Physical return"
    }
  ],
  "rows": [
    {
      "Study arc": {
        "text": "Claims, Wightman fields, and Euclidean reconstruction",
        "html": "Claims, Wightman fields, and Euclidean reconstruction"
      },
      "Primitive objects": {
        "text": "Typed theorem records, operator-valued distributions, Wightman functions, Schwinger hierarchies",
        "html": "Typed theorem records, operator-valued distributions, Wightman functions, Schwinger hierarchies"
      },
      "Main conclusion type": {
        "text": "Axiom consequences and directional Hilbert-space reconstruction",
        "html": "Axiom consequences and directional Hilbert-space reconstruction"
      },
      "Decisive hypotheses": {
        "text": "Common domains, spectral support, locality, full positivity hierarchy, Euclidean growth and regularity",
        "html": "Common domains, spectral support, locality, full positivity hierarchy, Euclidean growth and regularity"
      },
      "Excluded upgrade": {
        "text": "Matching low-point functions or Euclidean covariance alone does not establish existence or equivalence",
        "html": "Matching low-point functions or Euclidean covariance alone does not establish existence or equivalence"
      },
      "Physical return": {
        "text": "Relativistic fields, correlators, and foundational structural claims",
        "html": "Relativistic fields, correlators, and foundational structural claims"
      }
    },
    {
      "Study arc": {
        "text": "Particles, scattering, and local nets",
        "html": "Particles, scattering, and local nets"
      },
      "Primitive objects": {
        "text": "Local observables, asymptotic fields, detector limits, quasilocal algebras, states and representations",
        "html": "Local observables, asymptotic fields, detector limits, quasilocal algebras, states and representations"
      },
      "Main conclusion type": {
        "text": "Particle construction, scattering limits, net construction, or representation comparison",
        "html": "Particle construction, scattering limits, net construction, or representation comparison"
      },
      "Decisive hypotheses": {
        "text": "Mass isolation or infrared substitute, locality, stability, clustering, selection criterion, controlled asymptotic limit",
        "html": "Mass isolation or infrared substitute, locality, stability, clustering, selection criterion, controlled asymptotic limit"
      },
      "Excluded upgrade": {
        "text": "LSZ notation does not imply Haag–Ruelle hypotheses or asymptotic completeness; an abstract net need not recover point fields",
        "html": "LSZ notation does not imply Haag–Ruelle hypotheses or asymptotic completeness; an abstract net need not recover point fields"
      },
      "Physical return": {
        "text": "Scattering amplitudes, infrared observables, and physical particle interpretation",
        "html": "Scattering amplitudes, infrared observables, and physical particle interpretation"
      }
    },
    {
      "Study arc": {
        "text": "Modular theory, sectors, and local covariance",
        "html": "Modular theory, sectors, and local covariance"
      },
      "Primitive objects": {
        "text": "Von Neumann algebras, modular objects, localized endomorphisms, tensor categories, spacetime functors",
        "html": "Von Neumann algebras, modular objects, localized endomorphisms, tensor categories, spacetime functors"
      },
      "Main conclusion type": {
        "text": "Geometric modular action, split inclusion, gauge reconstruction, or natural dynamics",
        "html": "Geometric modular action, split inclusion, gauge reconstruction, or natural dynamics"
      },
      "Decisive hypotheses": {
        "text": "Faithful cyclic separating states, nuclearity bounds, localization and conjugates, typed spacetime morphisms, time-slice property",
        "html": "Faithful cyclic separating states, nuclearity bounds, localization and conjugates, typed spacetime morphisms, time-slice property"
      },
      "Excluded upgrade": {
        "text": "The split property is not sharp tensor factorization; DHR reconstruction and dynamical locality do not apply outside their stated domains",
        "html": "The split property is not sharp tensor factorization; DHR reconstruction and dynamical locality do not apply outside their stated domains"
      },
      "Physical return": {
        "text": "Thermal structure, charge sectors, gauge symmetry, and curved-spacetime dynamics",
        "html": "Thermal structure, charge sectors, gauge symmetry, and curved-spacetime dynamics"
      }
    },
    {
      "Study arc": {
        "text": "Thermality, information, and measurement",
        "html": "Thermality, information, and measurement"
      },
      "Primitive objects": {
        "text": "C*-dynamical systems, KMS states, weights, normal maps, instruments, recovery channels, correctable algebras",
        "html": "C*-dynamical systems, KMS states, weights, normal maps, instruments, recovery channels, correctable algebras"
      },
      "Main conclusion type": {
        "text": "Equilibrium characterization, passivity, entropy inequality, local measurement, or recovery theorem",
        "html": "Equilibrium characterization, passivity, entropy inequality, local measurement, or recovery theorem"
      },
      "Decisive hypotheses": {
        "text": "Named automorphism group, state normality, modular or split assumptions, locality and causal ordering, exact recovery criterion",
        "html": "Named automorphism group, state normality, modular or split assumptions, locality and causal ordering, exact recovery criterion"
      },
      "Excluded upgrade": {
        "text": "A KMS identity is not detector thermality, and a sharp type-III local algebra has no canonical density matrix",
        "html": "A KMS identity is not detector thermality, and a sharp type-III local algebra has no canonical density matrix"
      },
      "Physical return": {
        "text": "Thermal, nonequilibrium, measurement, and quantum-information applications",
        "html": "Thermal, nonequilibrium, measurement, and quantum-information applications"
      }
    },
    {
      "Study arc": {
        "text": "Constructive QFT and rigorous renormalization",
        "html": "Constructive QFT and rigorous renormalization"
      },
      "Primitive objects": {
        "text": "Cutoff measures, Schwinger functions, polymer activities, Banach-space RG trajectories, scaling limits",
        "html": "Cutoff measures, Schwinger functions, polymer activities, Banach-space RG trajectories, scaling limits"
      },
      "Main conclusion type": {
        "text": "Measure construction, cutoff removal, controlled flow, universality, or continuum limit",
        "html": "Measure construction, cutoff removal, controlled flow, universality, or continuum limit"
      },
      "Decisive hypotheses": {
        "text": "Stability, uniform estimates, tightness, tuned relevant directions, contraction, limit topology, retained observables",
        "html": "Stability, uniform estimates, tightness, tuned relevant directions, contraction, limit topology, retained observables"
      },
      "Excluded upgrade": {
        "text": "A finite regulated model, numerical flow, or perturbative beta function is not a nonperturbative continuum construction",
        "html": "A finite regulated model, numerical flow, or perturbative beta function is not a nonperturbative continuum construction"
      },
      "Physical return": {
        "text": "Renormalization, lattice, and nonperturbative model claims",
        "html": "Renormalization, lattice, and nonperturbative model claims"
      }
    },
    {
      "Study arc": {
        "text": "Causal, microlocal, and curved-spacetime QFT",
        "html": "Causal, microlocal, and curved-spacetime QFT"
      },
      "Primitive objects": {
        "text": "Microcausal functionals, time-ordered products, wavefront sets, Hadamard states, local fields and stress tensors",
        "html": "Microcausal functionals, time-ordered products, wavefront sets, Hadamard states, local fields and stress tensors"
      },
      "Main conclusion type": {
        "text": "Formal interacting construction, distribution extension, local-covariant renormalization, or energy inequality",
        "html": "Formal interacting construction, distribution extension, local-covariant renormalization, or energy inequality"
      },
      "Decisive hypotheses": {
        "text": "Causal factorization, formal coefficient ring, scaling-degree and wavefront conditions, local covariance, state and sampling class",
        "html": "Causal factorization, formal coefficient ring, scaling-degree and wavefront conditions, local covariance, state and sampling class"
      },
      "Excluded upgrade": {
        "text": "Formal construction does not imply convergence; a quantum energy inequality does not automatically imply ANEC or solve backreaction",
        "html": "Formal construction does not imply convergence; a quantum energy inequality does not automatically imply ANEC or solve backreaction"
      },
      "Physical return": {
        "text": "Renormalized observables, curved-spacetime fields, horizons, and semiclassical gravity",
        "html": "Renormalized observables, curved-spacetime fields, horizons, and semiclassical gravity"
      }
    },
    {
      "Study arc": {
        "text": "Factorization and derived QFT",
        "html": "Factorization and derived QFT"
      },
      "Primitive objects": {
        "text": "Prefactorization algebras, Weiss cosheaves, elliptic complexes, derived critical loci, deformation and obstruction complexes",
        "html": "Prefactorization algebras, Weiss cosheaves, elliptic complexes, derived critical loci, deformation and obstruction complexes"
      },
      "Main conclusion type": {
        "text": "Descent, local-to-global computation, formal quantization, or direction-specific framework comparison",
        "html": "Descent, local-to-global computation, formal quantization, or direction-specific framework comparison"
      },
      "Decisive hypotheses": {
        "text": "Cover class, target category, additivity and time-slice assumptions, gradings, completions, obstruction vanishing",
        "html": "Cover class, target category, additivity and time-slice assumptions, gradings, completions, obstruction vanishing"
      },
      "Excluded upgrade": {
        "text": "One comparison functor is not an unrestricted AQFT equivalence, and a derived formal moduli problem is not a nonperturbative Hilbert-space theory",
        "html": "One comparison functor is not an unrestricted AQFT equivalence, and a derived formal moduli problem is not a nonperturbative Hilbert-space theory"
      },
      "Physical return": {
        "text": "Local observables, effective field theory, and geometric or topological sectors",
        "html": "Local observables, effective field theory, and geometric or topological sectors"
      }
    },
    {
      "Study arc": {
        "text": "Gauge geometry, BRST–BV, and BV–BFV",
        "html": "Gauge geometry, BRST–BV, and BV–BFV"
      },
      "Primitive objects": {
        "text": "Orbit groupoids, bundle sectors, resolution complexes, BV actions, anomaly classes, boundary phase spaces and relative states",
        "html": "Orbit groupoids, bundle sectors, resolution complexes, BV actions, anomaly classes, boundary phase spaces and relative states"
      },
      "Main conclusion type": {
        "text": "Local slice, cohomological resolution, perturbative quantization, anomaly cancellation, or boundary gluing",
        "html": "Local slice, cohomological resolution, perturbative quantization, anomaly cancellation, or boundary gluing"
      },
      "Decisive hypotheses": {
        "text": "Global sector data, properness or derived reduction, master equations, renormalization, Lagrangian boundary data, residual modes and corners",
        "html": "Global sector data, properness or derived reduction, master equations, renormalization, Lagrangian boundary data, residual modes and corners"
      },
      "Excluded upgrade": {
        "text": "Faddeev–Popov notation does not solve the global orbit problem; topological gluing formulas do not construct four-dimensional Yang–Mills",
        "html": "Faddeev–Popov notation does not solve the global orbit problem; topological gluing formulas do not construct four-dimensional Yang–Mills"
      },
      "Physical return": {
        "text": "Gauge theory, anomalies, edge modes, and bounded-region field theory",
        "html": "Gauge theory, anomalies, edge modes, and bounded-region field theory"
      }
    },
    {
      "Study arc": {
        "text": "Conformal nets and vertex operator algebras",
        "html": "Conformal nets and vertex operator algebras"
      },
      "Primitive objects": {
        "text": "Interval nets, positive-energy representations, subfactors, Q-systems, graded vertex algebras and modules",
        "html": "Interval nets, positive-energy representations, subfactors, Q-systems, graded vertex algebras and modules"
      },
      "Main conclusion type": {
        "text": "Sector or extension classification, tensor-category structure, VOA-to-net construction, or restricted converse",
        "html": "Sector or extension classification, tensor-category structure, VOA-to-net construction, or restricted converse"
      },
      "Decisive hypotheses": {
        "text": "Covariance and positivity, finite-index or rationality assumptions, unitarity, energy bounds, strong locality, cofiniteness and completion",
        "html": "Covariance and positivity, finite-index or rationality assumptions, unitarity, energy bounds, strong locality, cofiniteness and completion"
      },
      "Excluded upgrade": {
        "text": "Neither rational examples nor strong locality establish a universal equivalence between conformal nets and VOAs",
        "html": "Neither rational examples nor strong locality establish a universal equivalence between conformal nets and VOAs"
      },
      "Physical return": {
        "text": "Two-dimensional CFT, chiral algebras, modular data, and conformal constructions",
        "html": "Two-dimensional CFT, chiral algebras, modular data, and conformal constructions"
      }
    },
    {
      "Study arc": {
        "text": "Extended TQFT, defects, and categorical symmetry",
        "html": "Extended TQFT, defects, and categorical symmetry"
      },
      "Primitive objects": {
        "text": "Bordism categories, higher targets, fully dualizable objects, defect data, fusion and module categories, symmetry TFTs",
        "html": "Bordism categories, higher targets, fully dualizable objects, defect data, fusion and module categories, symmetry TFTs"
      },
      "Main conclusion type": {
        "text": "Functorial construction, extended classification, gauging, condensation, or anomaly realization",
        "html": "Functorial construction, extended classification, gauging, condensation, or anomaly realization"
      },
      "Decisive hypotheses": {
        "text": "Dimension, tangential structure, extension depth, target category, dualizability, positivity, semisimplicity or its replacement, anomaly cancellation",
        "html": "Dimension, tangential structure, extension depth, target category, dualizability, positivity, semisimplicity or its replacement, anomaly cancellation"
      },
      "Excluded upgrade": {
        "text": "The cobordism hypothesis does not classify arbitrary physical QFTs, and fusion data alone do not prove completeness of defects or charges",
        "html": "The cobordism hypothesis does not classify arbitrary physical QFTs, and fusion data alone do not prove completeness of defects or charges"
      },
      "Physical return": {
        "text": "Topological phases, generalized symmetry, conformal defects, and many-body applications",
        "html": "Topological phases, generalized symmetry, conformal defects, and many-body applications"
      }
    },
    {
      "Study arc": {
        "text": "Existence and classification frontiers",
        "html": "Existence and classification frontiers"
      },
      "Primitive objects": {
        "text": "Model families, regulator systems, continuum limits, comparison functors, proposed classification data",
        "html": "Model families, regulator systems, continuum limits, comparison functors, proposed classification data"
      },
      "Main conclusion type": {
        "text": "Dated existence, mass-gap, universality, equivalence, or classification statement",
        "html": "Dated existence, mass-gap, universality, equivalence, or classification statement"
      },
      "Decisive hypotheses": {
        "text": "Exact dimension and fields, observables, topology of limits, nontriviality, positivity or reconstruction, source version, contrary results",
        "html": "Exact dimension and fields, observables, topology of limits, nontriviality, positivity or reconstruction, source version, contrary results"
      },
      "Excluded upgrade": {
        "text": "Physical evidence, perturbation theory, a partial model theorem, or a classification invariant cannot close the remaining proof obligations",
        "html": "Physical evidence, perturbation theory, a partial model theorem, or a classification invariant cannot close the remaining proof obligations"
      },
      "Physical return": {
        "text": "The physical model together with a dated research record",
        "html": "The physical model together with a dated research record"
      }
    }
  ]
}