{
  "title": "Derived Deformation Homotopical Qft Dependency Map",
  "question": "Which one-way constructions lead from a gauge-theory equation complex to formal moduli, shifted BV geometry, local perturbative quantization, and homotopy-coherent comparison maps, and which additional branch is needed for boundaries and defects?",
  "takeaway": "A derived critical or gauge quotient first retains stabilizers and obstructions; a nilpotent or complete L-infinity structure then defines the formal Maurer–Cartan infinity-groupoid; shifted nondegeneracy licenses BV brackets; regulator removal and local cohomology control quantization; contractions and structured equivalences control effective maps, while boundary gluing and Morita fusion require a separate Lagrangian and dualizability branch.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "derived field object\\\\equations, gauge maps, stabilizers, obstructions\\\\tangent complex in degrees -1, 0, 1, ..."
    },
    {
      "id": "n1",
      "label": "complete formal moduli problem\\\\filtered L-infinity brackets and Maurer–Cartan infinity-groupoid\\\\quasi-isomorphism only in the completed domain"
    },
    {
      "id": "n2",
      "label": "shifted BV or Poisson geometry\\\\closed form and quasi-isomorphism T to L[n]\\\\support, orientation, and boundary-flux control"
    },
    {
      "id": "n3",
      "label": "local perturbative quantization\\\\formal star product or renormalized QME\\\\degree-one local obstruction and degree-zero lifts"
    },
    {
      "id": "n4",
      "label": "homotopy-coherent comparison\\\\chain contraction for tree effective maps\\\\structured biequivalence for categorical replacement"
    },
    {
      "id": "branch",
      "label": "derived boundary intersection and Morita fusion\\\\Lagrangian regularity plus relative tensor products\\\\duals and adjoints for full extension"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "resolve",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "pair",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "quantize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "compare",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "branch",
      "label": "additional gluing hypotheses",
      "logical_status": "licensed implication"
    }
  ]
}