{
  "title": "Extended Tqft Bordism Failure Map",
  "question": "Which concrete counterexamples expose a failed sewing relation, insufficient adjoints, missing tangential fixed-point data, false invertibility, untrivialized anomaly holonomy, absent reflection positivity, or an unsupported physical-realization claim?",
  "takeaway": "Incompatible Frobenius contractions defeat decomposition independence; the dual numbers separate unextended Frobenius data from full Morita dualizability; Arf theory detects forgotten spin structure; multidimensional state spaces defeat tensor invertibility; determinant-line holonomy defeats local trivialization; negative trace weights and bare vector-space boundary labels defeat positive or functorial boundary claims; and none of the repaired mathematical conclusions alone constructs a microscopic QFT.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checks",
      "label": "sewing relations hold\\\\unextended functor"
    },
    {
      "id": "a1",
      "row": "claim checks",
      "label": "all required adjoints exist\\\\fully extended framed theory"
    },
    {
      "id": "a2",
      "row": "claim checks",
      "label": "fixed point and Picard tests pass\\\\structured or invertible theory"
    },
    {
      "id": "a3",
      "row": "claim checks",
      "label": "global trivialization and positive pairings\\\\absolute positive theory"
    },
    {
      "id": "a4",
      "row": "claim checks",
      "label": "declared realization theorem\\\\physical model only in its proved regime"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "violate the Frobenius relation\\\\two decompositions give different maps"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "use $\\mathbb C[\\epsilon]/(\\epsilon^2)$ in the Morita target\\\\object dual exists but evaluation lacks required adjoints"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "forget spin in Arf theory or use $\\dim Z(\\Sigma)>1$\\\\structured descent or tensor invertibility fails"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "trivialize one anomaly chart or choose a negative trace weight\\\\global holonomy or a negative norm remains"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "present only a functor or spectrum class\\\\no Hamiltonian, gap, locality, or continuum construction follows"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "extend",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "refine",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "tests",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "realize",
      "logical_status": "ordered diagnostic sequence, not a proof that later hypotheses follow from earlier ones"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "drop sewing",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "omit higher adjoints",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "omit fixed point or Picard test",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "omit global or positivity test",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "omit realization proof",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}