{
  "title": "Thermal Phases and AdS Black Holes: validity and failure map",
  "chapter": "Thermal Phases and AdS Black Holes",
  "kind": "validity-failure",
  "schematic": true,
  "not_to_scale": true,
  "stages": [
    "ensemble and boundary data",
    "Euclidean gravitational saddles",
    "action, entropy, and charges",
    "stability and finite-N checks",
    "thermal phase claim"
  ],
  "claims": [
    {
      "object": "Hawking-Page transition",
      "required_declaration": "boundary topology and canonical ensemble",
      "diagnostic": "renormalized free-energy crossing",
      "licensed_conclusion": "dominant semiclassical saddle",
      "unsupported_promotion": "a sharp finite-N transition"
    },
    {
      "object": "charged or rotating saddle",
      "required_declaration": "chemical potentials and regularity",
      "diagnostic": "first law and Hessian stability",
      "licensed_conclusion": "thermodynamics in a stated ensemble",
      "unsupported_promotion": "global stability in every ensemble"
    },
    {
      "object": "higher-derivative entropy",
      "required_declaration": "covariant action and boundary terms",
      "diagnostic": "Wald and Euclidean checks",
      "licensed_conclusion": "corrected saddle entropy",
      "unsupported_promotion": "microscopic state counting"
    }
  ],
  "takeaway": "A holographic phase is an ensemble-specific saddle comparison; classical dominance does not determine exact finite-N spectra or recurrences."
}