{
  "title": "Superselection Statistics Gauge Reconstruction Failure Map",
  "question": "Which concrete obstructions invalidate compact localization, symmetric gauge reconstruction, or a completeness claim while leaving a narrower sector statement intact?",
  "takeaway": "A translated thermal discriminator or Gauss flux defeats double-cone localization, a linked ribbon defeats shrinking a BF charge, nontrivial monodromy defeats the symmetric compact-group theorem, and missing concrete action or nonunique scaling limits defeat completeness even when fusion data remain valid.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "double-cone exterior equivalence plus transportability\\\\compact DHR endomorphism"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "spacelike-cone equivalence plus controlled string limit\\\\BF sector with braid data"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "full symmetric rigid finite DHR category\\\\compact $G$ and normal field system"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "quantified representation class plus concrete action and scale control\\\\class-relative completeness"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "KMS energy density or QED flux remains outside every bounded $O$\\\\local normality or conserved charge survives, DHR selection does not"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "toric-code loop links a semi-infinite ribbon\\\\cone localization and $D(\\mathbb Z_2)$ braiding survive, compact localization does not"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "$M_{\\rho,\\sigma}=\\varepsilon_{\\sigma,\\rho}\\varepsilon_{\\rho,\\sigma}\\neq1$\\\\braided category survives, ordinary $\\operatorname{Rep}(G)$ reconstruction does not"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "fusion rules omit associators, local action, infrasectors, kinks, or another scaling-limit net\\\\partial sector data survive, completeness and net classification do not"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "localize",
      "logical_status": "ordered diagnostic sequence, not an implication across localization classes"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "categorify",
      "logical_status": "ordered diagnostic sequence, not an implication across localization classes"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "reconstruct",
      "logical_status": "ordered diagnostic sequence, not an implication across localization classes"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "noncompact discriminator",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "linking loop",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "nontrivial monodromy",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "missing sectors or limits",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}