{
  "title": "Euclidean Os Reconstruction Failure Map",
  "question": "Which concrete counterexamples stop the OS reconstruction chain when all-order positivity, reflection positivity, uniform growth, or a valid converse is removed?",
  "takeaway": "A continued two-point function is not a theory, ordinary covariance positivity is not reflection positivity, fixed-order temperedness does not supply the OS II growth bound, and exponential clustering can locate a spectral threshold without producing an isolated particle; graded and link reflections and contour domains must also be checked separately.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "licensed implication",
      "label": "complete hierarchy $S_n$\\\\positive for every polynomial"
    },
    {
      "id": "a1",
      "row": "licensed implication",
      "label": "OS form on $\\mathcal A_+$\\\\quotient $\\mathcal D_+/\\mathcal N$"
    },
    {
      "id": "a2",
      "row": "licensed implication",
      "label": "E0$'$/E0$''$ plus ordered tubes\\\\tempered $W_n$ boundaries"
    },
    {
      "id": "a3",
      "row": "licensed implication",
      "label": "quantitative clustering\\\\spectral conclusion in a dense sector"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "only $S_2$ valid, set $S_4=0$\\\\two-point continuation survives only"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "$C(p^2)>0$ but one residue is negative\\\\Gaussian law, no OS Hilbert space"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "$k_n=n^2$ or a contour crosses a pinch\\\\no controlled tempered boundary"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "continuous $\\rho$ starts at $m^2>0$\\\\gap scale, no particle pole"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "quotient",
      "logical_status": "licensed implication"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "continue",
      "logical_status": "licensed implication"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "spectral theorem",
      "logical_status": "licensed implication"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "drop all-$n$ positivity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "drop reflection positivity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "drop uniform growth or leave the tube",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "apply a false converse",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}