{
  "title": "Constructive Qft Failure Map",
  "question": "Which concrete dropped hypotheses block normalization, regulator removal, reconstruction, or scattering while preserving a narrower mathematically correct statement?",
  "takeaway": "An unbounded-below interaction fails before any limit, nonuniform estimates fail to identify a joint limit, absent OS or spectral hypotheses block reconstruction or scattering, and changing dimension, coupling regime, or observable blocks transfer of a model theorem.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "stable density: $0<Z_{\\Lambda,\\varepsilon}<\\infty$\\\\finite-cutoff probability law"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "uniform bounds plus uniqueness\\\\declared joint-limit topology"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "full OS package and then isolated particle shell\\\\relativistic model and Haag--Ruelle states"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "fixed dimension, scaling regime, and observable class\\\\model-specific constructive conclusion"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "$P(u)=-u^4$ makes $Z=\\infty$ along the constant mode\\\\no probability law; formal coefficients do not survive"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "UV and volume limits are swapped without a joint bound, or coexistence gives two subsequences\\\\tightness or one ordered limit survives; uniqueness does not"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "reflection positivity fails, or the lowest spectrum is continuum with no isolated shell\\\\a Euclidean law or gapped theory may survive; OS or scattering does not"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "$\\phi^4_3$ counterterms are used in $d=4$, or a weak-coupling gauge-field limit is asked to control Wilson loops\\\\finite regulators or narrower observables survive; enlarged existence claim does not"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "limit",
      "logical_status": "ordered tests; no converse is implied"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "OS",
      "logical_status": "ordered tests; no converse is implied"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "scope",
      "logical_status": "ordered tests; no converse is implied"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "unstable action",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "nonuniform or nonunique limit",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "missing positivity or shell",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "domain mismatch",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}