{
  "title": "Superselection Statistics Gauge Reconstruction Dependency Map",
  "question": "How does a vacuum-relative representation selection branch into compact DHR sectors, cone-localized BF sectors, future-light-cone charge classes, and two-vacuum kink sectors, and which additional hypotheses alone lead to compact-gauge reconstruction?",
  "takeaway": "The localization region fixes the charge object: only transportable double-cone endomorphisms with finite statistics, conjugates, and symmetric exchange proceed through Doplicher–Roberts duality to a compact group and field algebra; cones retain braiding, light-cone restriction coarsens infrared sectors, and kinks retain ordered vacuum boundaries.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "vacuum net $O\\mapsto\\mathcal A(O)$ and admissible representations\\\\choose reference $\\pi_0$ and measurement algebra"
    },
    {
      "id": "n1",
      "label": "selection region and transportability\\\\double cone $O$, spacelike cone $C$, future light cone $V$, or opposite wedges"
    },
    {
      "id": "n2",
      "label": "compact DHR branch\\\\$\\rho|_{\\mathcal A(O')}=\\mathrm{id}$; transporters and intertwiners\\\\C*-tensor category by composition"
    },
    {
      "id": "n3",
      "label": "finite-statistics structure\\\\conjugate equations, $d(\\rho)<\\infty$, statistics operator $\\varepsilon$"
    },
    {
      "id": "n4",
      "label": "symmetric rigid full DHR category\\\\fiber functor $E$ and $G=\\operatorname{Aut}_\\otimes(E)$\\\\field algebra $\\mathfrak F$, $\\mathfrak F^G=\\mathfrak A$"
    },
    {
      "id": "branch",
      "label": "spacelike cone $C$\\\\BF morphisms and braid statistics\\\\future light cone $V$\\\\simple charge classes\\\\opposite wedges $W_L,W_R$\\\\kink $(a,b)$ with matched boundary composition"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "choose algebra",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "localize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "conjugates",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "DR theorem",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "branch",
      "label": "non-DHR branches",
      "logical_status": "licensed implication"
    }
  ]
}