{
  "title": "Gauge-geometric objects, hypotheses, licensed conclusions, and boundary tests",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "mathematical-gauge-theory-hypothesis-conclusion-table",
  "caption": "Gauge-geometric objects, hypotheses, licensed conclusions, and boundary tests",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Essential hypotheses",
      "html": "Essential hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or extension",
      "html": "Excluded converse or extension"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Connection action groupoid",
        "html": "Connection action groupoid"
      },
      "Essential hypotheses": {
        "text": "Fixed principal bundle, Sobolev index above the multiplication threshold, and a declared based or boundary gauge group.",
        "html": "Fixed principal bundle, Sobolev index above the multiplication threshold, and a declared based or boundary gauge group."
      },
      "Licensed conclusion": {
        "text": "Gauge arrows, stabilizers, and families are retained; gauge-invariant functions descend to the coarse orbit space.",
        "html": "Gauge arrows, stabilizers, and families are retained; gauge-invariant functions descend to the coarse orbit space."
      },
      "Excluded converse or extension": {
        "text": "The coarse orbit set does not recover automorphisms, boundary symmetries, or gluing data.",
        "html": "The coarse orbit set does not recover automorphisms, boundary symmetries, or gluing data."
      },
      "Adversarial check": {
        "text": "On $U(1)$ connections over $S^1$, replace the groupoid by holonomy values and observe that constant $U(1)$ isotropy disappears.",
        "html": "On $U(1)$ connections over $S^1$, replace the groupoid by holonomy values and observe that constant $U(1)$ isotropy disappears."
      }
    },
    {
      "Object and domain": {
        "text": "Orbit-type stratum",
        "html": "Orbit-type stratum"
      },
      "Essential hypotheses": {
        "text": "Proper action or gauge-theoretic slice hypotheses, with stabilizer conjugacy type held fixed.",
        "html": "Proper action or gauge-theoretic slice hypotheses, with stabilizer conjugacy type held fixed."
      },
      "Licensed conclusion": {
        "text": "A neighborhood is modeled by a stabilizer quotient of a slice, and orbit dimension includes the stabilizer dimension.",
        "html": "A neighborhood is modeled by a stabilizer quotient of a slice, and orbit dimension includes the stabilizer dimension."
      },
      "Excluded converse or extension": {
        "text": "A free-action manifold formula cannot be continued across reducible points where isotropy jumps.",
        "html": "A free-action manifold formula cannot be continued across reducible points where isotropy jumps."
      },
      "Adversarial check": {
        "text": "Apply the free quotient formula at $U=\\pm\\mathbf1$ under $SU(2)$ conjugation; it predicts a nonzero orbit although the orbit is a point.",
        "html": "Apply the free quotient formula at $U=\\pm\\mathbf1$ under $SU(2)$ conjugation; it predicts a nonzero orbit although the orbit is a point."
      }
    },
    {
      "Object and domain": {
        "text": "Bundle sector and allowed gauge group",
        "html": "Bundle sector and allowed gauge group"
      },
      "Essential hypotheses": {
        "text": "Bundle classification, faithful global form, matter representations, boundary conditions, and allowed components of the gauge group.",
        "html": "Bundle classification, faithful global form, matter representations, boundary conditions, and allowed components of the gauge group."
      },
      "Licensed conclusion": {
        "text": "Configuration space is a disjoint union over admitted bundle classes, quotiented within each class by the declared transformations.",
        "html": "Configuration space is a disjoint union over admitted bundle classes, quotiented within each class by the declared transformations."
      },
      "Excluded converse or extension": {
        "text": "A large transformation need not change bundle class, and a Lie algebra does not determine the sector set.",
        "html": "A large transformation need not change bundle class, and a Lie algebra does not determine the sector set."
      },
      "Adversarial check": {
        "text": "Put $c_1=1$ on $S^2$ and demand one global potential, or quotient by a transformation forbidden at the boundary.",
        "html": "Put $c_1=1$ on $S^2$ and demand one global potential, or quotient by a transformation forbidden at the boundary."
      }
    },
    {
      "Object and domain": {
        "text": "Coulomb slice and Faddeev–Popov operator",
        "html": "Coulomb slice and Faddeev–Popov operator"
      },
      "Essential hypotheses": {
        "text": "Closed base or elliptic boundary domain, controlled stabilizer, $k>d/2+1$, and invertibility of $d_A^*d_A$ on the chosen gauge algebra.",
        "html": "Closed base or elliptic boundary domain, controlled stabilizer, $k>d/2+1$, and invertibility of $d_A^*d_A$ on the chosen gauge algebra."
      },
      "Licensed conclusion": {
        "text": "A unique small transformation carries nearby fields to a local slice, modulo the stabilizer.",
        "html": "A unique small transformation carries nearby fields to a local slice, modulo the stabilizer."
      },
      "Excluded converse or extension": {
        "text": "Local invertibility does not give one representative on every orbit or exclude distant copies.",
        "html": "Local invertibility does not give one representative on every orbit or exclude distant copies."
      },
      "Adversarial check": {
        "text": "Insert $0\\neq\\xi\\in\\ker d_A$ at a reducible connection and watch the inverse-function hypothesis fail.",
        "html": "Insert $0\\neq\\xi\\in\\ker d_A$ at a reducible connection and watch the inverse-function hypothesis fail."
      }
    },
    {
      "Object and domain": {
        "text": "Elliptic deformation complex and global gauge bundle",
        "html": "Elliptic deformation complex and global gauge bundle"
      },
      "Essential hypotheses": {
        "text": "Field equation making a complex, elliptic symbol, Fredholm domains, and separately the topology of the based gauge group.",
        "html": "Field equation making a complex, elliptic symbol, Fredholm domains, and separately the topology of the based gauge group."
      },
      "Licensed conclusion": {
        "text": "Stabilizer, tangent, and obstruction groups are finite-dimensional; nontrivial universal-bundle topology can forbid a global section.",
        "html": "Stabilizer, tangent, and obstruction groups are finite-dimensional; nontrivial universal-bundle topology can forbid a global section."
      },
      "Excluded converse or extension": {
        "text": "Ellipticity and a Fredholm index do not imply unobstructedness, compactness, or global gauge uniqueness.",
        "html": "Ellipticity and a Fredholm index do not imply unobstructedness, compactness, or global gauge uniqueness."
      },
      "Adversarial check": {
        "text": "Promote the perturbative slice on $S^3$ globally despite $\\pi_0\\operatorname{Map}_*(S^3,SU(2))\\cong\\mathbb Z$.",
        "html": "Promote the perturbative slice on $S^3$ globally despite $\\pi_0\\operatorname{Map}_*(S^3,SU(2))\\cong\\mathbb Z$."
      }
    },
    {
      "Object and domain": {
        "text": "Derived moduli and determinant or Pfaffian line",
        "html": "Derived moduli and determinant or Pfaffian line"
      },
      "Essential hypotheses": {
        "text": "Derived mapping-stack hypotheses or a smooth Fredholm family; for a Pfaffian, the required real structure and dimension; gauge-equivariant gluing.",
        "html": "Derived mapping-stack hypotheses or a smooth Fredholm family; for a Pfaffian, the required real structure and dimension; gauge-equivariant gluing."
      },
      "Licensed conclusion": {
        "text": "Automorphisms and obstructions occupy separate tangent degrees, while curvature and holonomy locate local and global fermion anomalies.",
        "html": "Automorphisms and obstructions occupy separate tangent degrees, while curvature and holonomy locate local and global fermion anomalies."
      },
      "Excluded converse or extension": {
        "text": "Derived structure does not remove obstructions, and cancellation of a local anomaly polynomial does not trivialize line holonomy.",
        "html": "Derived structure does not remove obstructions, and cancellation of a local anomaly polynomial does not trivialize line holonomy."
      },
      "Adversarial check": {
        "text": "Keep only closed points at a reducible representation, or transport one $SU(2)$ doublet around the nontrivial large-gauge loop.",
        "html": "Keep only closed points at a reducible representation, or transport one $SU(2)$ doublet around the nontrivial large-gauge loop."
      }
    },
    {
      "Object and domain": {
        "text": "Regulated or continuum gauge measure",
        "html": "Regulated or continuum gauge measure"
      },
      "Essential hypotheses": {
        "text": "At finite cutoff, compact integration domain and positive action; for a continuum result, uniform tightness, observable convergence, Euclidean axioms, and reconstruction.",
        "html": "At finite cutoff, compact integration domain and positive action; for a continuum result, uniform tightness, observable convergence, Euclidean axioms, and reconstruction."
      },
      "Licensed conclusion": {
        "text": "A finite lattice gives a positive gauge-invariant probability law; a continuum QFT follows only when the additional limit theorems are proved.",
        "html": "A finite lattice gives a positive gauge-invariant probability law; a continuum QFT follows only when the additional limit theorems are proved."
      },
      "Excluded converse or extension": {
        "text": "Finite-cutoff positivity, formal gauge fixing, or a determinant symbol does not imply a regulator-independent interacting measure.",
        "html": "Finite-cutoff positivity, formal gauge fixing, or a determinant symbol does not imply a regulator-independent interacting measure."
      },
      "Adversarial check": {
        "text": "Present one Wilson lattice law as the continuum theory and demand the missing topology, tightness, renormalization, and reconstruction estimates.",
        "html": "Present one Wilson lattice law as the continuum theory and demand the missing topology, tightness, renormalization, and reconstruction estimates."
      }
    }
  ]
}