{
  "title": "Modular Response, Relative Entropy, and Emergent-Gravity Claims: structure map",
  "chapter": "Modular Response, Relative Entropy, and Emergent-Gravity Claims",
  "kind": "structure",
  "schematic": true,
  "not_to_scale": true,
  "stages": [
    "boundary relative entropy",
    "modular flow and first law",
    "bulk canonical energy",
    "positivity and shape checks",
    "conditional gravity inference"
  ],
  "claims": [
    {
      "object": "entanglement first law",
      "required_declaration": "reference state and modular operator",
      "diagnostic": "linear-response identity",
      "licensed_conclusion": "first-order entropy relation",
      "unsupported_promotion": "nonlinear field equations"
    },
    {
      "object": "relative-entropy equality",
      "required_declaration": "boundary and bulk algebras plus area term",
      "diagnostic": "second-order positivity check",
      "licensed_conclusion": "canonical-energy relation in a code sector",
      "unsupported_promotion": "exact equality for all states"
    },
    {
      "object": "field-equation inference",
      "required_declaration": "all balls or shapes and local dynamics assumptions",
      "diagnostic": "constraint and boundary-term check",
      "licensed_conclusion": "conditional linearized equations",
      "unsupported_promotion": "a unique quantum-gravity action"
    }
  ],
  "takeaway": "Information identities constrain gravity only after the dictionary, code sector, semiclassical expansion, and boundary terms are fixed."
}