{
  "title": "Thermal Aqft Kms Nonequilibrium Failure Map",
  "question": "Which counterexamples block the common upgrades from stationarity to KMS, passivity to complete passivity, thermal-state existence to mixing, a steady average to a unique NESS, or local horizon geometry to Hawking radiation?",
  "takeaway": "A stationary non-Gibbs state can fail KMS, a passive state can activate on tensor powers, phase multiplicity and conserved Liouvillean modes block uniqueness and mixing, finite reservoirs recur, one local thermal moment does not determine a temperature distribution, and local Rindler geometry does not select a global black-hole state.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "full KMS strip identity\\\\equilibrium for the selected flow"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "complete passivity plus theorem hypotheses\\\\KMS or ground state"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "nuclearity plus separate spectral control\\\\existence and stated mixing result"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "scattering or local/horizon theorem inputs\\\\NESS, LTE, or geometric KMS only"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "stationary non-Gibbs populations\\\\invariance survives; KMS boundary fails"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "one-copy passive non-Gibbs state\\\\tensor powers activate extractable work"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "disjoint phases or conserved Liouvillean mode\\\\KMS exists; uniqueness or decay fails"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "finite recurrence, moment ambiguity, or local Rindler analogy\\\\unique NESS, universal temperature, or Hawking flux fails"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "work test",
      "logical_status": "ordered diagnostic sequence, not an implication across claim classes"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "dynamics",
      "logical_status": "ordered diagnostic sequence, not an implication across claim classes"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "application",
      "logical_status": "ordered diagnostic sequence, not an implication across claim classes"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "omit strip",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "omit tensor stability",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "omit spectral input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "omit global construction",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}