{
  "title": "Constructive Qft Dependency Map",
  "question": "Which one-way estimates carry a regulated Euclidean field from a Gaussian covariance through an interacting density and cutoff removal to OS reconstruction and, under additional spectral hypotheses, particle scattering?",
  "takeaway": "A stable finite-cutoff density is only the start: uniform correlation or multiscale bounds are needed for regulator limits, the full OS package is needed for a relativistic theory, and an isolated mass shell is needed for Haag–Ruelle scattering; scalar, fermionic, stochastic, and gauge models supply these inputs by different mechanisms.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "Gaussian reference law $\\mu_C$ on $\\mathcal S'$\\\\continuous positive covariance; reflection-positive kernel when required"
    },
    {
      "id": "n1",
      "label": "regulated interacting object $\\mu_{\\Lambda,\\varepsilon}$\\\\Wick polynomial or Grassmann determinant; stability and normalization"
    },
    {
      "id": "n2",
      "label": "uniform constructive bounds\\\\cluster or correlation estimates; multiscale counterterm control; tightness"
    },
    {
      "id": "n3",
      "label": "cutoff-independent Schwinger hierarchy\\\\specified volume/UV limit, topology, observables, and non-Gaussian check"
    },
    {
      "id": "n4",
      "label": "OS reconstruction\\\\symmetry, Euclidean covariance, reflection positivity, growth and clustering"
    },
    {
      "id": "n5",
      "label": "particle and scattering structure\\\\mass gap plus isolated shell and Haag--Ruelle hypotheses"
    },
    {
      "id": "branch",
      "label": "$P(\\phi)_2,\\ \\phi^4_3$: Wick/cluster or RG bounds\\\\Gross--Neveu/Yukawa: determinant and multiscale bounds\\\\dynamic $\\Phi^4_3$: renormalized SPDE convergence\\\\gauge--Higgs: weak-coupling local Gaussian comparison"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "weight",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "bounds",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "limit",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "OS",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "shell",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "branch",
      "label": "model-specific control",
      "logical_status": "licensed implication"
    }
  ]
}