{
  "title": "Thermal Aqft Kms Nonequilibrium Dependency Map",
  "question": "Which additional hypotheses turn a selected time evolution and KMS boundary identity into modular, passivity, phase, relaxation, steady-state, local-temperature, or horizon conclusions?",
  "takeaway": "The automorphism group and full KMS strip identity establish equilibrium relative to one flow; modular realization, complete passivity, phase analysis, mixing, NESS, local thermality, and horizon thermality are separate directional upgrades with additional representation, spectral, scattering, observable, or geometric inputs.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "$C^*$-dynamical system\\\\$(\\mathcal A,\\alpha_t)$, strong continuity\\\\chosen physical time flow"
    },
    {
      "id": "n1",
      "label": "KMS strip at $\\beta$\\\\analytic elements and positivity\\\\$F(t+i\\beta)=\\omega(\\alpha_t(B)A)$"
    },
    {
      "id": "n2",
      "label": "three equilibrium tests\\\\faithful GNS and modular flow\\\\complete passivity; relativistic tube"
    },
    {
      "id": "n3",
      "label": "phase and existence control\\\\factorial extremals; central mixtures\\\\thermal nuclearity and volume limits"
    },
    {
      "id": "n4",
      "label": "additional dynamical input\\\\Liouvillean spectrum or Møller limit\\\\local fields or geometric modular action"
    },
    {
      "id": "n5",
      "label": "licensed scoped conclusions\\\\mixing; NESS and $\\sigma\\geq0$\\\\LTE moments; wedge or horizon KMS"
    },
    {
      "id": "branch",
      "label": "passive for every tensor power\\\\Pusz--Woronowicz hypotheses\\\\KMS or ground state"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "KMS identity",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "add structure",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "decompose",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "new hypotheses",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "conclude",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "branch",
      "label": "work branch",
      "logical_status": "licensed implication"
    }
  ]
}