{
  "title": "Normed RG objects, hypotheses, directional conclusions, and decisive failure tests",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "rigorous-renormalization-hypothesis-conclusion-table",
  "caption": "Normed RG objects, hypotheses, directional conclusions, and decisive failure tests",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Required hypotheses",
      "html": "Required hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or upgrade",
      "html": "Excluded converse or upgrade"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Lattice covariance $C=(-\\Delta+m^2)^{-1}$ on a torus",
        "html": "Lattice covariance $C=(-\\Delta+m^2)^{-1}$ on a torus"
      },
      "Required hypotheses": {
        "text": "A positive finite-range decomposition with uniform derivative bounds and a specified mass-scale regime",
        "html": "A positive finite-range decomposition with uniform derivative bounds and a specified mass-scale regime"
      },
      "Licensed conclusion": {
        "text": "Sequential Gaussian integration with exact factorization for sufficiently separated polymers",
        "html": "Sequential Gaussian integration with exact factorization for sufficiently separated polymers"
      },
      "Excluded converse or upgrade": {
        "text": "A momentum-shell partition by itself does not give finite-range independence or positivity of every piece",
        "html": "A momentum-shell partition by itself does not give finite-range independence or positivity of every piece"
      },
      "Adversarial check": {
        "text": "Put two polymers farther apart than the nominal range and test whether their fluctuation covariance really vanishes",
        "html": "Put two polymers farther apart than the nominal range and test whether their fluctuation covariance really vanishes"
      }
    },
    {
      "Object and domain": {
        "text": "One RG step $R_j:(V_j,K_j)\\mapsto(V_{j+1},K_{j+1})$",
        "html": "One RG step $R_j:(V_j,K_j)\\mapsto(V_{j+1},K_{j+1})$"
      },
      "Required hypotheses": {
        "text": "A scale-dependent domain, localization operator, field and large-set regulators, and differentiable norm estimates",
        "html": "A scale-dependent domain, localization operator, field and large-set regulators, and differentiable norm estimates"
      },
      "Licensed conclusion": {
        "text": "An exact coordinate identity and a contractive bound for the extracted nonperturbative remainder",
        "html": "An exact coordinate identity and a contractive bound for the extracted nonperturbative remainder"
      },
      "Excluded converse or upgrade": {
        "text": "A truncated beta function neither defines the exact map nor bounds the discarded activity",
        "html": "A truncated beta function neither defines the exact map nor bounds the discarded activity"
      },
      "Adversarial check": {
        "text": "Delete the large-field regulator and evaluate the norm on a high-amplitude field configuration",
        "html": "Delete the large-field regulator and evaluate the norm on a high-amplitude field configuration"
      }
    },
    {
      "Object and domain": {
        "text": "All-scale bulk trajectory",
        "html": "All-scale bulk trajectory"
      },
      "Required hypotheses": {
        "text": "Tuned relevant initial coordinates, a controlled marginal recursion, stable-manifold hypotheses, and summable nonlinear errors",
        "html": "Tuned relevant initial coordinates, a controlled marginal recursion, stable-manifold hypotheses, and summable nonlinear errors"
      },
      "Licensed conclusion": {
        "text": "An orbit that remains in the small domain and approaches the designated infrared behavior",
        "html": "An orbit that remains in the small domain and approaches the designated infrared behavior"
      },
      "Excluded converse or upgrade": {
        "text": "Forward iteration from generic bare data does not land on the critical manifold, and linear stability is not nonlinear stability",
        "html": "Forward iteration from generic bare data does not land on the critical manifold, and linear stability is not nonlinear stability"
      },
      "Adversarial check": {
        "text": "Perturb the bare mass in the unstable direction and check that the deviation grows before the terminal scale",
        "html": "Perturb the bare mass in the unstable direction and check that the deviation grows before the terminal scale"
      }
    },
    {
      "Object and domain": {
        "text": "Critical observable with insertions",
        "html": "Critical observable with insertions"
      },
      "Required hypotheses": {
        "text": "A controlled bulk orbit, renormalized observable coordinates, coalescence-scale estimates, and convergence in a stated topology",
        "html": "A controlled bulk orbit, renormalized observable coordinates, coalescence-scale estimates, and convergence in a stated topology"
      },
      "Licensed conclusion": {
        "text": "A model-specific two-point asymptotic, susceptibility law, or Gaussian scaling limit",
        "html": "A model-specific two-point asymptotic, susceptibility law, or Gaussian scaling limit"
      },
      "Excluded converse or upgrade": {
        "text": "Bulk free-energy control does not automatically prove correlation-function convergence or universality",
        "html": "Bulk free-energy control does not automatically prove correlation-function convergence or universality"
      },
      "Adversarial check": {
        "text": "Insert two sources, follow their mixed coupling, and verify that the claimed amplitude survives the terminal-scale remainder",
        "html": "Insert two sources, follow their mixed coupling, and verify that the claimed amplitude survives the terminal-scale remainder"
      }
    },
    {
      "Object and domain": {
        "text": "Weak four-dimensional lattice $|\\varphi|^4$ model near criticality",
        "html": "Weak four-dimensional lattice $|\\varphi|^4$ model near criticality"
      },
      "Required hypotheses": {
        "text": "Small positive coupling, dimension four, tuned critical mass, and the lattice and observable hypotheses of the cited theorem",
        "html": "Small positive coupling, dimension four, tuned critical mass, and the lattice and observable hypotheses of the cited theorem"
      },
      "Licensed conclusion": {
        "text": "Gaussian critical scaling with proved logarithmic corrections, including the $n$-dependent susceptibility exponent",
        "html": "Gaussian critical scaling with proved logarithmic corrections, including the $n$-dependent susceptibility exponent"
      },
      "Excluded converse or upgrade": {
        "text": "These results do not yield a non-Gaussian four-dimensional continuum measure or settle every strong-coupling lattice action",
        "html": "These results do not yield a non-Gaussian four-dimensional continuum measure or settle every strong-coupling lattice action"
      },
      "Adversarial check": {
        "text": "Change the dimension or leave the small-coupling domain and test which summability and marginal-flow estimates fail",
        "html": "Change the dimension or leave the small-coupling domain and test which summability and marginal-flow estimates fail"
      }
    },
    {
      "Object and domain": {
        "text": "Supersymmetric walk or sectorized fermionic model",
        "html": "Supersymmetric walk or sectorized fermionic model"
      },
      "Required hypotheses": {
        "text": "For the walk, an exact boson–fermion representation and supersymmetric localization; for a Fermi surface, regular geometry, sector bounds, and the theorem's temperature range",
        "html": "For the walk, an exact boson–fermion representation and supersymmetric localization; for a Fermi surface, regular geometry, sector bounds, and the theorem's temperature range"
      },
      "Licensed conclusion": {
        "text": "Walk critical asymptotics or a restricted weak-coupling Fermi-liquid construction with controlled kernels",
        "html": "Walk critical asymptotics or a restricted weak-coupling Fermi-liquid construction with controlled kernels"
      },
      "Excluded converse or upgrade": {
        "text": "Scalar polymer estimates do not transfer automatically to Grassmann sectors, nesting, van Hove points, or zero temperature",
        "html": "Scalar polymer estimates do not transfer automatically to Grassmann sectors, nesting, van Hove points, or zero temperature"
      },
      "Adversarial check": {
        "text": "Move the Fermi level to a singular or nested surface, or break the supersymmetric identity, and locate the first lost estimate",
        "html": "Move the Fermi level to a singular or nested surface, or break the supersymmetric identity, and locate the first lost estimate"
      }
    },
    {
      "Object and domain": {
        "text": "Two regulators or RG coordinate systems",
        "html": "Two regulators or RG coordinate systems"
      },
      "Required hypotheses": {
        "text": "A common microscopic model, explicit maps between domains, uniform error bounds, and separately proved convergence of the same observables",
        "html": "A common microscopic model, explicit maps between domains, uniform error bounds, and separately proved convergence of the same observables"
      },
      "Licensed conclusion": {
        "text": "A conditional comparison of selected universal quantities or equivalent limiting correlations",
        "html": "A conditional comparison of selected universal quantities or equivalent limiting correlations"
      },
      "Excluded converse or upgrade": {
        "text": "Matching low-order beta coefficients does not prove conjugacy of exact maps, regulator independence, or existence of a continuum QFT",
        "html": "Matching low-order beta coefficients does not prove conjugacy of exact maps, regulator independence, or existence of a continuum QFT"
      },
      "Adversarial check": {
        "text": "Add an irrelevant term whose accumulated error is not summable and test whether the proposed comparison still closes",
        "html": "Add an irrelevant term whose accumulated error is not summable and test whether the proposed comparison still closes"
      }
    }
  ]
}