{
  "title": "Factorization Algebras Failure Map",
  "question": "Which concrete missing hypotheses separate coherent disjoint products from descent, valid field-theoretic observables from ill-defined products, unobstructed quantization from anomalies, and the strict Lorentzian comparison from a general higher-categorical equivalence?",
  "takeaway": "A product-only assignment can miss global classes; ordinary covers can miss finite configurations; zero modes and wavefront violations can break field-theoretic observables; nonzero local anomaly classes obstruct the quantum master equation; and descent or local constancy cannot replace tangential data, Cauchy constancy, time ordering, positivity, completion, or higher coherence.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "coherent disjoint maps\\\\prefactorization algebra"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "derived Weiss descent in a suitable target\\\\factorization algebra"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "valid field complex and analytic domain\\\\classical observable factorization algebra"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "local QME obstruction removed\\\\formal quantum observables"
    },
    {
      "id": "a4",
      "row": "claim checkpoints",
      "label": "additive Cauchy-constant time-orderable data\\\\strict Lorentzian comparison"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "add a generator visible only on one large open\\\\cover diagram cannot reconstruct it"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "use an ordinary cover missing separated insertions\\\\bilocal support is absent from every member"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "retain zero modes or multiply forbidden wavefront sets\\\\quotient, bracket, or product is ill-defined"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "leave a nonzero anomaly or allow nonlocal counterterms\\\\effective QME or locality fails"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "omit time-slice, tangential data, positivity, or higher coherence\\\\generic factorization-to-AQFT equivalence does not follow"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "descent",
      "logical_status": "ordered diagnostic sequence, not an implication across theory classes"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "field input",
      "logical_status": "ordered diagnostic sequence, not an implication across theory classes"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "quantize",
      "logical_status": "ordered diagnostic sequence, not an implication across theory classes"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "compare",
      "logical_status": "ordered diagnostic sequence, not an implication across theory classes"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "omit global class",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "wrong cover",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "omit analytic input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "nonzero obstruction",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "overstate theorem",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}