{
  "title": "Locally Covariant Dynamical Locality Dependency Map",
  "question": "How do causally convex spacetime embeddings, a covariant algebra functor, the time-slice axiom, relative Cauchy evolution, and exterior-background invariance lead to dynamical locality and its conditional same-physics and natural-state conclusions?",
  "takeaway": "Loc geometry types every algebra map; functoriality, Einstein causality, and time-slice are separate inputs; only time-slice constructs relative Cauchy evolution, whose fixed subobjects can then be compared with kinematic regional algebras, while stress response, SPASs, and no-natural-state results each require additional named hypotheses.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "source $\\mathsf{Loc}$\\\\$M=(M,g,o,t)$ globally hyperbolic\\\\$\\psi^*g_N=g_M$, causal convexity"
    },
    {
      "id": "n1",
      "label": "theory $\\mathcal A:\\mathsf{Loc}\\to\\mathsf{Phys}$\\\\identity and composition\\\\Einstein causality; time-slice separate"
    },
    {
      "id": "n2",
      "label": "natural data\\\\field $\\Phi:\\mathcal D\\Rightarrow\\mathcal A$\\\\theory map $\\zeta:\\mathcal A\\Rightarrow\\mathcal B$"
    },
    {
      "id": "n3",
      "label": "compact admissible $h$\\\\four Cauchy isomorphisms\\\\$\\operatorname{rce}_M[h]=(\\tau_M^-)^{-1}\\tau_M^+$"
    },
    {
      "id": "n4",
      "label": "two regional subobjects\\\\$\\mathcal A^{\\rm kin}(M;O)$ from inclusion\\\\$\\mathcal A^{\\rm dyn}(M;O)$ from exterior-rce fixed points"
    },
    {
      "id": "n5",
      "label": "dynamical locality\\\\$\\mathcal A^{\\rm kin}\\cong\\mathcal A^{\\rm dyn}$\\\\SPASs; natural-state no-go needs extra assumptions"
    },
    {
      "id": "branch",
      "label": "Hadamard GNS plus common domain\\\\$\\delta_h\\operatorname{rce}=-\\frac{i}{2}[T(h),\\cdot]$\\\\weak free-scalar theorem"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "functor",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "naturality",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "time-slice",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "invariants",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "compare",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "branch",
      "label": "stress response",
      "logical_status": "licensed implication"
    }
  ]
}