{
  "title": "Mathematical Gauge Theory Failure Map",
  "question": "Which dropped hypothesis first invalidates a quotient, local slice, global gauge, fermion orientation, or continuum-measure claim?",
  "takeaway": "Coarse quotients, reducible backgrounds, nontrivial gauge-group topology, flat lines with nontrivial holonomy, and finite-cutoff weights fail at different checkpoints and require different corrections.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "licensed checkpoints",
      "label": "action groupoid and sector data\\\\stabilizers and allowed arrows retained"
    },
    {
      "id": "a1",
      "row": "licensed checkpoints",
      "label": "local slice and elliptic complex\\\\kernel removed or kept as isotropy"
    },
    {
      "id": "a2",
      "row": "licensed checkpoints",
      "label": "global gauge claim\\\\principal gauge bundle admits a section"
    },
    {
      "id": "a3",
      "row": "licensed checkpoints",
      "label": "fermion measure orientation\\\\Pfaffian line equivariantly trivial"
    },
    {
      "id": "a4",
      "row": "licensed checkpoints",
      "label": "continuum gauge measure\\\\tight limit and reconstruction proved"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "coarse orbit set only\\\\constant or reducible stabilizer erased"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "stabilizer zero mode or remote copy\\\\inverse fails or uniqueness exceeds neighborhood"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "noncontractible based gauge group\\\\Gribov--Singer blocks global section"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "local anomaly vanishes but loop holonomy is $-1$\\\\global Pfaffian sign obstructed"
    },
    {
      "id": "b4",
      "row": "first broken step",
      "label": "finite positive Wilson weight only\\\\no tightness, observable limit, or OS reconstruction"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "localize",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "globalize",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "orient",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a3",
      "to": "a4",
      "label": "remove regulator",
      "logical_status": "ordered diagnostic sequence, not a universal implication chain"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "forget isotropy",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "ignore kernel or range",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "promote local slice",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "ignore holonomy",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a4",
      "to": "b4",
      "label": "overstate cutoff",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}