{
  "title": "Bv Bfv Boundaries Corners Gluing Failure Map",
  "question": "Which concrete missing hypotheses break BV–BFV compatibility, admissible boundary reduction, cut-independent and associative gluing, anomaly cancellation, or promotion from a topological model to a four-dimensional non-Abelian continuum theory?",
  "takeaway": "Discarding the variational defect breaks the relative master equation; a non-Lagrangian or non-invariant boundary condition leaves flux; omitted zero modes or corners make gluing cut-dependent; local cancellation can leave global anomaly holonomy; and topological perturbative formulas do not supply four-dimensional non-Abelian renormalization, regulator removal, or positivity.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checks",
      "label": "relative bulk--boundary data\\\\BV--BFV compatibility"
    },
    {
      "id": "a1",
      "row": "claim checks",
      "label": "Lagrangian $Q$-invariant boundary condition\\\\controlled BFV reduction"
    },
    {
      "id": "a2",
      "row": "claim checks",
      "label": "polarization, residual modes, and corner data\\\\state and coherent composition"
    },
    {
      "id": "a3",
      "row": "claim checks",
      "label": "renormalized mQME and trivial anomaly classes\\\\formal quantum gluing"
    },
    {
      "id": "a4",
      "row": "claim checks",
      "label": "named model and analytic domain\\\\exactly the proved conclusion"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "drop the boundary primitive from the bulk variation\\\\closed CME is false and no BFV charge is induced"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "choose a non-Lagrangian subspace or quotient charged motion\\\\symplectic flux remains or physical directions disappear"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "omit harmonic modes or shared-corner terms\\\\partition function depends on the cut or gluing order"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "cancel only a local variation while holonomy survives\\\\relative QME or global gauge invariance fails"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "promote BF or 2D YM formulas to four-dimensional non-Abelian theory\\\\renormalized boundary observables, limits, and positivity remain unproved"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "reduce",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct models"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "prepare state",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct models"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "renormalize",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct models"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "apply theorem",
      "logical_status": "ordered diagnostic sequence, not an implication across distinct models"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "discard defect",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "wrong boundary data",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "omit residual or corner input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "retain anomaly",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "overstate model",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}