{
  "title": "Domains, hypotheses, licensed results, excluded upgrades, and failure tests for BRST–BV quantization",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "brst-bv-anomalies-hypothesis-conclusion-table",
  "caption": "Domains, hypotheses, licensed results, excluded upgrades, and failure tests for BRST–BV quantization",
  "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": "Stationary ideal and reducible gauge generators",
        "html": "Stationary ideal and reducible gauge generators"
      },
      "Required hypotheses": {
        "text": "Regular equations; complete Noether identities and reducibility tower; Koszul–Tate acyclicity in positive antifield number",
        "html": "Regular equations; complete Noether identities and reducibility tower; Koszul–Tate acyclicity in positive antifield number"
      },
      "Licensed result": {
        "text": "A resolution of on-shell functions and a controlled BRST bicomplex",
        "html": "A resolution of on-shell functions and a controlled BRST bicomplex"
      },
      "Excluded converse or upgrade": {
        "text": "Nilpotence alone does not prove exactness or identify the intended quotient",
        "html": "Nilpotence alone does not prove exactness or identify the intended quotient"
      },
      "Adversarial check": {
        "text": "Introduce a dependent equation or missed reducibility relation and exhibit surviving positive-antifield-number homology",
        "html": "Introduce a dependent equation or missed reducibility relation and exhibit surviving positive-antifield-number homology"
      }
    },
    {
      "Object and domain": {
        "text": "Proper classical BV action on the odd cotangent space",
        "html": "Proper classical BV action on the odd cotangent space"
      },
      "Required hypotheses": {
        "text": "Nondegenerate odd symplectic pairing; complete gauge generators; properness; $(S,S)=0$ with consistent degree and sign conventions",
        "html": "Nondegenerate odd symplectic pairing; complete gauge generators; properness; $(S,S)=0$ with consistent degree and sign conventions"
      },
      "Licensed result": {
        "text": "Nilpotent BV differential and a derived critical-locus model",
        "html": "Nilpotent BV differential and a derived critical-locus model"
      },
      "Excluded converse or upgrade": {
        "text": "A classical master action supplies neither a measure nor a quantum theory",
        "html": "A classical master action supplies neither a measure nor a quantum theory"
      },
      "Adversarial check": {
        "text": "Drop an antifield term encoding a field-dependent bracket and calculate $s^2$",
        "html": "Drop an antifield term encoding a field-dependent bracket and calculate $s^2$"
      }
    },
    {
      "Object and domain": {
        "text": "Gauge-fixed BV integral or perturbative functional",
        "html": "Gauge-fixed BV integral or perturbative functional"
      },
      "Required hypotheses": {
        "text": "Regular gauge fermion; Lagrangian slice avoiding singular strata; compatible measure or renormalization; vanishing boundary terms",
        "html": "Regular gauge fermion; Lagrangian slice avoiding singular strata; compatible measure or renormalization; vanishing boundary terms"
      },
      "Licensed result": {
        "text": "Equality of BRST-closed observables under admissible canonical changes",
        "html": "Equality of BRST-closed observables under admissible canonical changes"
      },
      "Excluded converse or upgrade": {
        "text": "Off-shell Green functions need not agree, and no global Gribov-free slice follows",
        "html": "Off-shell Green functions need not agree, and no global Gribov-free slice follows"
      },
      "Adversarial check": {
        "text": "Move the slice through a Faddeev–Popov zero mode or a boundary contribution",
        "html": "Move the slice through a Faddeev–Popov zero mode or a boundary contribution"
      }
    },
    {
      "Object and domain": {
        "text": "Renormalized quantum master identity",
        "html": "Renormalized quantum master identity"
      },
      "Required hypotheses": {
        "text": "Local causal renormalization; consistency of the breaking; trivial obstruction in the relevant local BRST cohomology; admissible finite counterterms",
        "html": "Local causal renormalization; consistency of the breaking; trivial obstruction in the relevant local BRST cohomology; admissible finite counterterms"
      },
      "Licensed result": {
        "text": "Order-by-order restoration of the master identity and formal gauge-invariant observables",
        "html": "Order-by-order restoration of the master identity and formal gauge-invariant observables"
      },
      "Excluded converse or upgrade": {
        "text": "Formal solvability does not imply convergence, a global adiabatic limit, or positivity",
        "html": "Formal solvability does not imply convergence, a global adiabatic limit, or positivity"
      },
      "Adversarial check": {
        "text": "Choose a nontrivial ghost-number-one cocycle and try to remove it with an exact counterterm",
        "html": "Choose a nontrivial ghost-number-one cocycle and try to remove it with an exact counterterm"
      }
    },
    {
      "Object and domain": {
        "text": "Local chiral anomaly in an even-dimensional theory",
        "html": "Local chiral anomaly in an even-dimensional theory"
      },
      "Required hypotheses": {
        "text": "Specified representation, chirality, bundle, trace normalization, local functional class, and descent representative",
        "html": "Specified representation, chirality, bundle, trace normalization, local functional class, and descent representative"
      },
      "Licensed result": {
        "text": "Wess–Zumino-consistent anomaly class and its consistent/covariant-current relation",
        "html": "Wess–Zumino-consistent anomaly class and its consistent/covariant-current relation"
      },
      "Excluded converse or upgrade": {
        "text": "Vanishing de Rham anomaly polynomial does not exclude torsion or large-transformation anomalies",
        "html": "Vanishing de Rham anomaly polynomial does not exclude torsion or large-transformation anomalies"
      },
      "Adversarial check": {
        "text": "Use one $SU(2)$ Weyl doublet: the cubic local test vanishes while the Pfaffian changes sign",
        "html": "Use one $SU(2)$ Weyl doublet: the cubic local test vanishes while the Pfaffian changes sign"
      }
    },
    {
      "Object and domain": {
        "text": "Determinant or Pfaffian line over background-field space",
        "html": "Determinant or Pfaffian line over background-field space"
      },
      "Required hypotheses": {
        "text": "Specified spin structure and representation; family of Fredholm operators; control through zero modes; trivial holonomy on every relevant loop or bordism class",
        "html": "Specified spin structure and representation; family of Fredholm operators; control through zero modes; trivial holonomy on every relevant loop or bordism class"
      },
      "Licensed result": {
        "text": "A globally consistent fermion phase for the tested parameter space",
        "html": "A globally consistent fermion phase for the tested parameter space"
      },
      "Excluded converse or upgrade": {
        "text": "Triviality on one loop or component does not prove global triviality or interacting existence",
        "html": "Triviality on one loop or component does not prove global triviality or interacting existence"
      },
      "Adversarial check": {
        "text": "Build the mapping torus of a large transformation and evaluate its eta phase or mod-two index",
        "html": "Build the mapping torus of a large transformation and evaluate its eta phase or mod-two index"
      }
    },
    {
      "Object and domain": {
        "text": "Candidate anomaly-free chiral gauge theory",
        "html": "Candidate anomaly-free chiral gauge theory"
      },
      "Required hypotheses": {
        "text": "All local and global cancellations; local regulator; all-order identities; cutoff and volume limits; locality; positive representation; nontrivial dynamics",
        "html": "All local and global cancellations; local regulator; all-order identities; cutoff and volume limits; locality; positive representation; nontrivial dynamics"
      },
      "Licensed result": {
        "text": "A physical continuum theory only when every analytic and representation hypothesis is separately proved",
        "html": "A physical continuum theory only when every analytic and representation hypothesis is separately proved"
      },
      "Excluded converse or upgrade": {
        "text": "Charge sums plus a formal BRST charge are not sufficient for a positive Hilbert space",
        "html": "Charge sums plus a formal BRST charge are not sufficient for a positive Hilbert space"
      },
      "Adversarial check": {
        "text": "Demand the missing uniform estimates and positivity theorem after the anomaly arithmetic succeeds",
        "html": "Demand the missing uniform estimates and positivity theorem after the anomaly arithmetic succeeds"
      }
    }
  ]
}