{
  "title": "Rigorous Renormalization Dependency Map",
  "question": "How do a positive finite-range covariance decomposition, normed polymer coordinates, an exact scale map, tuned relevant parameters, and observable insertions combine to yield a rigorous critical asymptotic or scaling limit?",
  "takeaway": "Finite-range factorization makes the polymer representation local, regulators make the extracted remainder contractive, and a nonlinear stability theorem selects an all-scale critical orbit; only a separately controlled observable flow and terminal-scale limit license a model-specific correlation or scaling theorem.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "regulated object\\\\$Z_0=\\int e^{-V_0(\\phi)}d\\mu_C(\\phi)$\\\\finite torus, cutoff, named observables"
    },
    {
      "id": "n1",
      "label": "positive covariance split\\\\$C=\\sum_j C_j$\\\\finite range and derivative bounds"
    },
    {
      "id": "n2",
      "label": "normed coordinates\\\\$Z_j=e^{-u_j|\\Lambda|}(I_j\\circ K_j)(\\Lambda)$\\\\local $V_j$, polymer $K_j$, regulators"
    },
    {
      "id": "n3",
      "label": "exact scale map\\\\$x_{j+1}=R_j(x_j)$\\\\localization plus $\\|K_{j+1}\\|_{j+1}\\le q\\|K_j\\|_j+\\mathrm{Err}_j$"
    },
    {
      "id": "n4",
      "label": "tuned critical trajectory\\\\relevant counterterms fixed by boundary data\\\\marginal flow plus stable manifold"
    },
    {
      "id": "n5",
      "label": "observable and terminal flow\\\\insertions through coalescence scale\\\\critical asymptotic or stated scaling limit"
    },
    {
      "id": "branch",
      "label": "choose bare mass and vacuum counterterm\\\\cancel growing relevant components\\\\generic forward orbit is not critical"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "decompose",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "factorize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "integrate",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "tune",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "observables",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "branch",
      "label": "boundary-value input",
      "logical_status": "licensed implication"
    }
  ]
}