{
  "title": "Vertex Operator Algebras Net Bridges Failure Map",
  "question": "Which missing formal identity, semisimplicity hypothesis, energy estimate, closure-level locality theorem, point-field generation condition, or extension comparison first breaks the claimed VOA-net bridge?",
  "takeaway": "Finite mode tests do not prove a VOA, C2-cofiniteness alone permits logarithmic modules, core commutation does not prove strong locality, a net need not have enough recoverable point fields, and matching characters or sector data do not prove extension equivalence.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "directional checks",
      "label": "VOA state--field map\\\\all modes satisfy formal identities"
    },
    {
      "id": "a1",
      "row": "directional checks",
      "label": "tensor or finiteness theorem\\\\module class and convergence fixed"
    },
    {
      "id": "a2",
      "row": "directional checks",
      "label": "unitary smeared fields\\\\energy bounds and common core"
    },
    {
      "id": "a3",
      "row": "directional checks",
      "label": "conformal net construction\\\\strong locality and covariance"
    },
    {
      "id": "a4",
      "row": "directional checks",
      "label": "extension or converse bridge\\\\matched functor and hypotheses"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "match finitely many modes only\\\\no uniform locality exponent or Jacobi identity"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "$C_2$-cofinite but nonrational input\\\\indecomposables and logarithms survive"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "vanishing commutator on one dense core\\\\closures need not commute strongly"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "net without controlled point-field generation\\\\no recovered VOA or essential surjectivity"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "matching characters, branching, or sectors only\\\\no multiplication, natural equivalence, or converse"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "modules",
      "logical_status": "ordered first-failure tests, not reversible equivalences"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "Hilbert",
      "logical_status": "ordered first-failure tests, not reversible equivalences"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "locality",
      "logical_status": "ordered first-failure tests, not reversible equivalences"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "bridge",
      "logical_status": "ordered first-failure tests, not reversible equivalences"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "truncate identities",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "drop semisimplicity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "skip closure theorem",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "assume point fields",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "reverse comparison",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}