{
  "title": "Bv Bfv Boundaries Corners Gluing Dependency Map",
  "question": "Which one-way hypotheses connect a bulk BV variational problem to boundary BFV phase space, admissible boundary and residual data, a renormalized relative quantum state, corner-coherent BV pushforward, and a model-specific gluing result?",
  "takeaway": "The bulk variational defect induces the BFV primitive and charge; a Lagrangian boundary condition or polarization and complete residual sector are additional data; the modified QME requires renormalization and anomaly cancellation; and gluing requires matching corners, clean or derived reduction, and BV pushforward in a model where those constructions have been proved.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "bulk BV variation\\\\cohomological fields and degree $-1$ form\\\\surface defect retained"
    },
    {
      "id": "n1",
      "label": "boundary BFV data\\\\degree-zero symplectic form and charge\\\\projectable restriction map"
    },
    {
      "id": "n2",
      "label": "admissible boundary data\\\\Lagrangian polarization and $Q$ tangency\\\\complete residual cohomology"
    },
    {
      "id": "n3",
      "label": "relative quantum state\\\\renormalized BFV operator and mQME\\\\local and global anomalies removed"
    },
    {
      "id": "n4",
      "label": "corner-coherent gluing\\\\matched strata and clean or derived intersection\\\\pairing followed by BV pushforward"
    },
    {
      "id": "n5",
      "label": "model-specific conclusion\\\\BF, Chern--Simons, 2D YM, free boundary theory\\\\only the proved perturbative or exact scope"
    },
    {
      "id": "branch",
      "label": "AKSZ target and brane $\\Rightarrow$ classical examples\\\\edge extension $\\Rightarrow$ dressed observables and surface symmetry\\\\stratified descent $\\Rightarrow$ boundary and defect factorization maps"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "vary",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "polarize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "quantize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "compose",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "identify model",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "branch",
      "label": "separate boundary structures",
      "logical_status": "licensed implication"
    }
  ]
}