{
  "title": "Axioms Reconstruction Failure Map",
  "question": "Where does a Wightman reconstruction or structural-theorem argument first fail when a hypothesis is removed?",
  "takeaway": "A positive two-point kernel, selected correlator identities, or a formal interaction picture never replaces positivity of the full hierarchy, cyclicity, spectral support, locality, and theorem-specific analytic hypotheses.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "licensed implication",
      "label": "complete hierarchy $W_n$\\\\all Wightman conditions"
    },
    {
      "id": "a1",
      "row": "licensed implication",
      "label": "cyclic field theory\\\\reconstructed on a common domain"
    },
    {
      "id": "a2",
      "row": "licensed implication",
      "label": "theorem-specific analytic data\\\\tube, Jost, locality"
    },
    {
      "id": "a3",
      "row": "licensed implication",
      "label": "scoped structural conclusion\\\\or Haag obstruction"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "only $W_2$ is positive\\\\arbitrary incompatible $W_4$"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "noncyclic spectator sector\\\\vacuum data miss uniqueness"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "spectrum or locality removed\\\\tube/Jost proof stops"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "false converse or naive\\\\interaction-picture inference"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "quotient and complete",
      "logical_status": "licensed implication"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "establish analytic domain",
      "logical_status": "licensed implication"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "apply named theorem",
      "logical_status": "licensed implication"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "drop full positivity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "drop cyclicity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "drop analytic input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "overstate conclusion",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}