{
  "title": "Domains, required hypotheses, directional results, and failure tests for BV–BFV boundaries and gluing",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "bv-bfv-boundaries-corners-gluing-hypothesis-conclusion-table",
  "caption": "Domains, required hypotheses, directional results, and failure tests for BV–BFV boundaries and gluing",
  "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": "Bulk BV theory on $M$ with boundary",
        "html": "Bulk BV theory on $M$ with boundary"
      },
      "Required hypotheses": {
        "text": "Projectable $Q_M$; surjective boundary restriction; exact or prequantum boundary form; controlled variational signs and domains",
        "html": "Projectable $Q_M$; surjective boundary restriction; exact or prequantum boundary form; controlled variational signs and domains"
      },
      "Licensed result": {
        "text": "Boundary BFV phase space, charge, and modified classical master equation",
        "html": "Boundary BFV phase space, charge, and modified classical master equation"
      },
      "Excluded converse or upgrade": {
        "text": "A closed-manifold CME cannot be used after discarding the surface term",
        "html": "A closed-manifold CME cannot be used after discarding the surface term"
      },
      "Adversarial check": {
        "text": "Vary the action and retain every integration-by-parts contribution",
        "html": "Vary the action and retain every integration-by-parts contribution"
      }
    },
    {
      "Object and domain": {
        "text": "Boundary constraint system",
        "html": "Boundary constraint system"
      },
      "Required hypotheses": {
        "text": "Regular coisotropic zero locus; declared gauge subgroup; smooth or derived reduction; integrable generators",
        "html": "Regular coisotropic zero locus; declared gauge subgroup; smooth or derived reduction; integrable generators"
      },
      "Licensed result": {
        "text": "BFV resolution and reduced boundary observables",
        "html": "BFV resolution and reduced boundary observables"
      },
      "Excluded converse or upgrade": {
        "text": "A transformation with nonzero surface charge is not automatically gauge",
        "html": "A transformation with nonzero surface charge is not automatically gauge"
      },
      "Adversarial check": {
        "text": "Quotient a Maxwell flux sector by arbitrary boundary parameters and count the lost directions",
        "html": "Quotient a Maxwell flux sector by arbitrary boundary parameters and count the lost directions"
      }
    },
    {
      "Object and domain": {
        "text": "Boundary condition or polarization",
        "html": "Boundary condition or polarization"
      },
      "Required hypotheses": {
        "text": "Lagrangianity; $Q^\\partial$ tangency; compatible primitive; finite or controlled intersection with evolution relation",
        "html": "Lagrangianity; $Q^\\partial$ tangency; compatible primitive; finite or controlled intersection with evolution relation"
      },
      "Licensed result": {
        "text": "Admissible boundary problem and state-space polarization",
        "html": "Admissible boundary problem and state-space polarization"
      },
      "Excluded converse or upgrade": {
        "text": "Dirichlet-looking constraints need not cancel symplectic flux",
        "html": "Dirichlet-looking constraints need not cancel symplectic flux"
      },
      "Adversarial check": {
        "text": "Evaluate the boundary Green form on allowed variations",
        "html": "Evaluate the boundary Green form on allowed variations"
      }
    },
    {
      "Object and domain": {
        "text": "Manifold with corners",
        "html": "Manifold with corners"
      },
      "Required hypotheses": {
        "text": "Iterated restriction; shifted symplectic degrees; compatible corner charges and endpoint polarizations; refinement coherence",
        "html": "Iterated restriction; shifted symplectic degrees; compatible corner charges and endpoint polarizations; refinement coherence"
      },
      "Licensed result": {
        "text": "Extended BV–BFV data and associative face composition in the proved model",
        "html": "Extended BV–BFV data and associative face composition in the proved model"
      },
      "Excluded converse or upgrade": {
        "text": "Codimension-one compatibility does not erase corner terms",
        "html": "Codimension-one compatibility does not erase corner terms"
      },
      "Adversarial check": {
        "text": "Compare two orders of gluing three faces",
        "html": "Compare two orders of gluing three faces"
      }
    },
    {
      "Object and domain": {
        "text": "Perturbative quantum state",
        "html": "Perturbative quantum state"
      },
      "Required hypotheses": {
        "text": "Gauge fixing; finite residual sector; renormalized BFV operator; local and global anomaly cancellation; mQME",
        "html": "Gauge fixing; finite residual sector; renormalized BFV operator; local and global anomaly cancellation; mQME"
      },
      "Licensed result": {
        "text": "Cohomology class of a formal boundary state",
        "html": "Cohomology class of a formal boundary state"
      },
      "Excluded converse or upgrade": {
        "text": "No convergent measure, positivity, or global vacuum follows from a formal QME",
        "html": "No convergent measure, positivity, or global vacuum follows from a formal QME"
      },
      "Adversarial check": {
        "text": "Retain a nonzero anomaly class or determinant-line holonomy",
        "html": "Retain a nonzero anomaly class or determinant-line holonomy"
      }
    },
    {
      "Object and domain": {
        "text": "Cut $M=M_1\\cup_\\Sigma M_2$",
        "html": "Cut $M=M_1\\cup_\\Sigma M_2$"
      },
      "Required hypotheses": {
        "text": "Clean or derived intersection; complementary polarizations; matching corner data; one copy of each residual mode; BV pushforward",
        "html": "Clean or derived intersection; complementary polarizations; matching corner data; one copy of each residual mode; BV pushforward"
      },
      "Licensed result": {
        "text": "Paired state satisfying the mQME and composition up to the declared equivalence",
        "html": "Paired state satisfying the mQME and composition up to the declared equivalence"
      },
      "Excluded converse or upgrade": {
        "text": "Multiplication followed by an unspecified functional integral is not a gluing theorem",
        "html": "Multiplication followed by an unspecified functional integral is not a gluing theorem"
      },
      "Adversarial check": {
        "text": "Double-count a harmonic mode and compare determinant degrees",
        "html": "Double-count a harmonic mode and compare determinant degrees"
      }
    },
    {
      "Object and domain": {
        "text": "Boundary model or extrapolation",
        "html": "Boundary model or extrapolation"
      },
      "Required hypotheses": {
        "text": "Named BF, Chern–Simons, 2D Yang–Mills, free stratified, or smoothened Abelian theorem with its geometry and order",
        "html": "Named BF, Chern–Simons, 2D Yang–Mills, free stratified, or smoothened Abelian theorem with its geometry and order"
      },
      "Licensed result": {
        "text": "Exactly the model-specific classical, perturbative, or gluing statement proved",
        "html": "Exactly the model-specific classical, perturbative, or gluing statement proved"
      },
      "Excluded converse or upgrade": {
        "text": "No general four-dimensional non-Abelian continuum existence or positivity theorem follows",
        "html": "No general four-dimensional non-Abelian continuum existence or positivity theorem follows"
      },
      "Adversarial check": {
        "text": "Replace a Gaussian BF cylinder by 4D Yang–Mills and list every missing analytic input",
        "html": "Replace a Gaussian BF cylinder by 4D Yang–Mills and list every missing analytic input"
      }
    }
  ]
}