{
  "title": "Euclidean Os Reconstruction Dependency Map",
  "question": "Which Euclidean objects and Osterwalder–Schrader hypotheses lead to a positive Hilbert space, controlled Lorentzian boundary values, vacuum uniqueness, and a qualified spectral gap?",
  "takeaway": "The complete Schwinger hierarchy, uniform OS growth, Euclidean covariance and symmetry, and reflection positivity have separate directed roles: they produce the null quotient and positive semigroup before ordered-tube continuation yields a local positive-energy Wightman theory; clustering adds vacuum and gap conclusions but not a particle pole.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "Euclidean law $\\mu$ or full $S_n$\\\\on $\\mathcal S'(\\mathbb R^d)$, all $n$"
    },
    {
      "id": "n1",
      "label": "$E(d)$ covariance, symmetry, E0$'$/E0$''$\\\\$\\langle\\Theta F,F\\rangle_E\\geq0$ on $\\mathcal A_+$"
    },
    {
      "id": "n2",
      "label": "$\\mathcal D_+/\\mathcal N\\to\\mathcal H$\\\\$T_E(\\tau)=e^{-\\tau H}$, $H\\geq0$"
    },
    {
      "id": "n3",
      "label": "ordered $\\tau_1>\\cdots>\\tau_n$ regions\\\\permuted forward tubes, singular loci excluded"
    },
    {
      "id": "n4",
      "label": "tempered boundaries $W_n$\\\\$\\operatorname{supp}\\widetilde W_n\\subset\\overline V_+$"
    },
    {
      "id": "n5",
      "label": "local positive-energy Wightman QFT\\\\unique cyclic theory; cluster $\\Rightarrow\\ker H=\\mathbb C\\Omega$"
    },
    {
      "id": "branch",
      "label": "graded order reversal\\\\oriented link reflection\\\\gauge-invariant $\\mathcal A_+$"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "moments",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "quotient",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "semigroup",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "boundary",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "reconstruct",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "branch",
      "label": "fermion and gauge variants",
      "logical_status": "licensed implication"
    }
  ]
}