{
  "title": "Derived Deformation Homotopical Qft Failure Map",
  "question": "Which concrete countertests expose truncation, completion, boundary, locality, dualizability, and comparison failures before a formal model is mistaken for an equivalent quantum theory?",
  "takeaway": "An orbit set loses stabilizers and obstructions, an uncompleted L-infinity series need not define a finite displacement, unrestricted Maxwell boundary data create symplectic flux, diagonal contractions or nonlocal primitives fail locality, and a non-chain projection or unstructured strictification preserves too little for physical equivalence.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checks",
      "label": "derived equation and gauge complex\\\\$H^0,H^1,H^2$ data retained"
    },
    {
      "id": "a1",
      "row": "claim checks",
      "label": "nilpotent or complete filtered formal moduli\\\\weak equivalence of Maurer–Cartan infinity-groupoids"
    },
    {
      "id": "a2",
      "row": "claim checks",
      "label": "closed shifted pairing and Lagrangian boundary maps\\\\BV bracket and zero-mode-preserving derived gluing"
    },
    {
      "id": "a3",
      "row": "claim checks",
      "label": "regular contraction or renormalized local extension\\\\formal associative product and local QME lift"
    },
    {
      "id": "a4",
      "row": "claim checks",
      "label": "valid contraction, relative tensor products, and structured equivalence\\\\tree effective map or coherent categorical comparison"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "ordinary orbit at a reducible flat connection\\\\$H^0$ stabilizer and $H^2$ obstruction data disappear"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "formal quasi-isomorphism evaluated outside its filtration\\\\the Taylor series need not converge or reach other components"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "Maxwell field on a boundary with unrestricted traces\\\\nonzero symplectic flux invalidates closedness and Lagrangianity"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "local $\\phi^4$ contraction on the diagonal or Green-operator primitive\\\\the first is undefined and the second is nonlocal"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "projection that is not a chain map, or strictification after forgetting topology and domains\\\\selected amplitudes or fusion survive, physical equivalence does not"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "complete",
      "logical_status": "ordered diagnostic sequence; a failure removes only the named derived, formal, local, or structured conclusion"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "pair",
      "logical_status": "ordered diagnostic sequence; a failure removes only the named derived, formal, local, or structured conclusion"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "renormalize",
      "logical_status": "ordered diagnostic sequence; a failure removes only the named derived, formal, local, or structured conclusion"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "compare",
      "logical_status": "ordered diagnostic sequence; a failure removes only the named derived, formal, local, or structured conclusion"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "derived directions truncated",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "completion omitted",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "boundary flux survives",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "locality fails",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "structure forgotten",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}