{
  "title": "Rigorous Renormalization Failure Map",
  "question": "Which dropped hypothesis first blocks factorization, remainder contraction, an all-scale critical orbit, observable convergence, or a scheme-independent continuum conclusion?",
  "takeaway": "Positivity without finite range does not factorize distant polymers, a formal local flow without regulators does not bound the remainder, an untuned relevant coordinate ejects the orbit, and matching beta functions without uniform observable estimates does not establish a common limiting theory.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "positive finite-range $C_j$\\\\independent polymer fluctuations"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "localization plus field and large-set regulators\\\\contractive nonperturbative coordinate"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "tuned relevant data plus summable nonlinear errors\\\\all-scale orbit in the small domain"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "controlled insertions plus terminal estimates and topology\\\\model-specific correlation or scaling limit"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "positive but long-range $C_j$\\\\distant polymers remain correlated; exact factorization is lost"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "omit a field or large-set regulator\\\\large fields or polymers make $\\|K_{j+1}\\|$ uncontrolled"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "shift the bare mass off the critical surface\\\\the relevant component grows and exits the domain"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "match finite-order beta functions only\\\\no uniform remainder, insertion, or continuum comparison follows"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "contract",
      "logical_status": "ordered diagnostic sequence, not an implication across theorem classes"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "stabilize",
      "logical_status": "ordered diagnostic sequence, not an implication across theorem classes"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "limit",
      "logical_status": "ordered diagnostic sequence, not an implication across theorem classes"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "drop finite range",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "drop regulator control",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "mistune relevant data",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "replace convergence by perturbation theory",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}