{
  "title": "Representative existence and classification frontiers at the 2026-08-10 evidence cutoff",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "existence-classification-problems-hypothesis-conclusion-table",
  "caption": "Representative existence and classification frontiers at the 2026-08-10 evidence cutoff",
  "columns": [
    {
      "text": "Claim target",
      "html": "Claim target"
    },
    {
      "text": "Data that must remain fixed",
      "html": "Data that must remain fixed"
    },
    {
      "text": "Primary theorem or construction class",
      "html": "Primary theorem or construction class"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded upgrade and first open obligation",
      "html": "Excluded upgrade and first open obligation"
    }
  ],
  "rows": [
    {
      "Claim target": {
        "text": "Low-dimensional constructive models",
        "html": "Low-dimensional constructive models"
      },
      "Data that must remain fixed": {
        "text": "Exact interaction, two or three dimensions, ultraviolet and volume regulators, coupling regime, correlation hierarchy, and reconstruction hypotheses",
        "html": "Exact interaction, two or three dimensions, ultraviolet and volume regulators, coupling regime, correlation hierarchy, and reconstruction hypotheses"
      },
      "Primary theorem or construction class": {
        "text": "Model-specific constructive estimates for P ( φ ) 2 , weakly coupled φ 4 3 , or massive Gross–Neveu 2",
        "html": "Model-specific constructive estimates for <var>P</var>(<var>φ</var>)<sub>2</sub>, weakly coupled <var>φ</var><sup>4</sup><sub>3</sub>, or massive Gross–Neveu<sub>2</sub>"
      },
      "Licensed conclusion": {
        "text": "Continuum correlations and further properties only for the stated model and regime",
        "html": "Continuum correlations and further properties only for the stated model and regime"
      },
      "Excluded upgrade and first open obligation": {
        "text": "No transfer to four dimensions and no automatic scattering completeness; prove each added spectral statement separately",
        "html": "No transfer to four dimensions and no automatic scattering completeness; prove each added spectral statement separately"
      }
    },
    {
      "Claim target": {
        "text": "Four-dimensional ferromagnetic scalar scaling",
        "html": "Four-dimensional ferromagnetic scalar scaling"
      },
      "Data that must remain fixed": {
        "text": "Nearest-neighbor Ising-type or lattice-cutoff λ φ 4 class, critical or near-critical scaling, normalized correlations, and corrected hypotheses",
        "html": "Nearest-neighbor Ising-type or lattice-cutoff λ<var>φ</var><sup>4</sup> class, critical or near-critical scaling, normalized correlations, and corrected hypotheses"
      },
      "Primary theorem or construction class": {
        "text": "Aizenman–Duminil-Copin marginal-triviality theorem, 2021 with 2024 corrigendum",
        "html": "Aizenman–Duminil-Copin marginal-triviality theorem, 2021 with 2024 corrigendum"
      },
      "Licensed conclusion": {
        "text": "Gaussianity of subsequential scaling limits in the theorem’s classes",
        "html": "Gaussianity of subsequential scaling limits in the theorem’s classes"
      },
      "Excluded upgrade and first open obligation": {
        "text": "Not a theorem for every scalar action or continuum trajectory; extend the argument or construct a different non-Gaussian target",
        "html": "Not a theorem for every scalar action or continuum trajectory; extend the argument or construct a different non-Gaussian target"
      }
    },
    {
      "Claim target": {
        "text": "Four-dimensional pure Yang–Mills",
        "html": "Four-dimensional pure Yang–Mills"
      },
      "Data that must remain fixed": {
        "text": "Compact simple gauge group, pure gauge fields on Euclidean four-space, axiomatic continuum object, positivity, infinite volume, and vacuum spectrum",
        "html": "Compact simple gauge group, pure gauge fields on Euclidean four-space, axiomatic continuum object, positivity, infinite volume, and vacuum spectrum"
      },
      "Primary theorem or construction class": {
        "text": "Jaffe–Witten problem statement and dated institutional status",
        "html": "Jaffe–Witten problem statement and dated institutional status"
      },
      "Licensed conclusion": {
        "text": "A precise open construction-plus-gap problem; nearby lattice and Yang–Mills–Higgs theorems remain partial evidence",
        "html": "A precise open construction-plus-gap problem; nearby lattice and Yang–Mills–Higgs theorems remain partial evidence"
      },
      "Excluded upgrade and first open obligation": {
        "text": "No constructed pure continuum theory with the required positive gap; prove regulator removal and the separate spectral lower bound",
        "html": "No constructed pure continuum theory with the required positive gap; prove regulator removal and the separate spectral lower bound"
      }
    },
    {
      "Claim target": {
        "text": "Anomaly-free chiral gauge theory",
        "html": "Anomaly-free chiral gauge theory"
      },
      "Data that must remain fixed": {
        "text": "Gauge group and representations, local regulator, exact gauge symmetry, anomaly cancellation, target chirality, positivity, and continuum limit",
        "html": "Gauge group and representations, local regulator, exact gauge symmetry, anomaly cancellation, target chirality, positivity, and continuum limit"
      },
      "Primary theorem or construction class": {
        "text": "Exact lower-dimensional Hamiltonians and partial four-dimensional symmetry-disentangler constructions",
        "html": "Exact lower-dimensional Hamiltonians and partial four-dimensional symmetry-disentangler constructions"
      },
      "Licensed conclusion": {
        "text": "A rigorous route for stated lower-dimensional models and specified four-dimensional symmetry data",
        "html": "A rigorous route for stated lower-dimensional models and specified four-dimensional symmetry data"
      },
      "Excluded upgrade and first open obligation": {
        "text": "Anomaly cancellation alone does not construct the Standard Model; supply the missing local Hamiltonian, interface proof, gauging, and continuum recovery",
        "html": "Anomaly cancellation alone does not construct the Standard Model; supply the missing local Hamiltonian, interface proof, gauging, and continuum recovery"
      }
    },
    {
      "Claim target": {
        "text": "Interacting theory on curved spacetime",
        "html": "Interacting theory on curved spacetime"
      },
      "Data that must remain fixed": {
        "text": "Globally hyperbolic background, compact interaction support, field content, gauge complex, renormalization prescription, and coefficient ring",
        "html": "Globally hyperbolic background, compact interaction support, field content, gauge complex, renormalization prescription, and coefficient ring"
      },
      "Primary theorem or construction class": {
        "text": "Locally covariant perturbative algebras or abstract local C-star relations",
        "html": "Locally covariant perturbative algebras or abstract local C-star relations"
      },
      "Licensed conclusion": {
        "text": "Formal local observables, or an abstract local algebra, at the strength of the selected construction",
        "html": "Formal local observables, or an abstract local algebra, at the strength of the selected construction"
      },
      "Excluded upgrade and first open obligation": {
        "text": "No generic convergent finite-coupling vacuum or physical representation; construct positive states and control global or adiabatic limits",
        "html": "No generic convergent finite-coupling vacuum or physical representation; construct positive states and control global or adiabatic limits"
      }
    },
    {
      "Claim target": {
        "text": "Lattice continuum and universality",
        "html": "Lattice continuum and universality"
      },
      "Data that must remain fixed": {
        "text": "Bare-parameter trajectory, order of ultraviolet and volume limits, observable family, topology, boundary conditions, and symmetry class",
        "html": "Bare-parameter trajectory, order of ultraviolet and volume limits, observable family, topology, boundary conditions, and symmetry class"
      },
      "Primary theorem or construction class": {
        "text": "Uniform moment, correlation, renormalization-group, tightness, or comparison estimates",
        "html": "Uniform moment, correlation, renormalization-group, tightness, or comparison estimates"
      },
      "Licensed conclusion": {
        "text": "Subsequential or full convergence and universality only in the proved topology and observable sector",
        "html": "Subsequential or full convergence and universality only in the proved topology and observable sector"
      },
      "Excluded upgrade and first open obligation": {
        "text": "Finite-size collapse or matching critical exponents is not full QFT convergence; establish uniqueness, positivity, complete correlations, and nontriviality",
        "html": "Finite-size collapse or matching critical exponents is not full QFT convergence; establish uniqueness, positivity, complete correlations, and nontriviality"
      }
    },
    {
      "Claim target": {
        "text": "AQFT–prefactorization comparison",
        "html": "AQFT–prefactorization comparison"
      },
      "Data that must remain fixed": {
        "text": "Lorentzian site, additivity, time-slice conditions, target category, weak-equivalence class, and localization",
        "html": "Lorentzian site, additivity, time-slice conditions, target category, weak-equivalence class, and localization"
      },
      "Primary theorem or construction class": {
        "text": "Benini–Carmona–Grant-Stuart–Schenkel one-categorical equivalence",
        "html": "Benini–Carmona–Grant-Stuart–Schenkel one-categorical equivalence"
      },
      "Licensed conclusion": {
        "text": "Equivalence under the theorem’s additive and target-category hypotheses",
        "html": "Equivalence under the theorem’s additive and target-category hypotheses"
      },
      "Excluded upgrade and first open obligation": {
        "text": "No unrestricted cochain-valued infinity-categorical equivalence; solve the stated infinity-localization problem",
        "html": "No unrestricted cochain-valued infinity-categorical equivalence; solve the stated infinity-localization problem"
      }
    },
    {
      "Claim target": {
        "text": "Conformal-net, VOA, and extended-TQFT classification",
        "html": "Conformal-net, VOA, and extended-TQFT classification"
      },
      "Data that must remain fixed": {
        "text": "Central charge or finiteness class, unitarity and analytic bounds, bordism dimension, tangential structure, extension depth, and target category",
        "html": "Central charge or finiteness class, unitarity and analytic bounds, bordism dimension, tangential structure, extension depth, and target category"
      },
      "Primary theorem or construction class": {
        "text": "Complete subcritical net classifications, conditional VOA–net bridges, and cobordism-hypothesis classifications in fixed targets",
        "html": "Complete subcritical net classifications, conditional VOA–net bridges, and cobordism-hypothesis classifications in fixed targets"
      },
      "Licensed conclusion": {
        "text": "Classification or realization for the explicitly named class",
        "html": "Classification or realization for the explicitly named class"
      },
      "Excluded upgrade and first open obligation": {
        "text": "No classification of all QFTs and no automatic converse; characterize the essential image and prove realization and completeness",
        "html": "No classification of all QFTs and no automatic converse; characterize the essential image and prove realization and completeness"
      }
    }
  ]
}