{
  "title": "Locally Covariant Dynamical Locality Failure Map",
  "question": "Which concrete dropped hypotheses block propagator covariance, relative Cauchy evolution, kinematic–dynamic equality, same-physics propagation, preferred natural states, and boundary covariance?",
  "takeaway": "A non-causally-convex image breaks the Green-operator square, a nonglobally hyperbolic perturbation removes the Cauchy isomorphisms, scalar zero modes and Maxwell fluxes are exterior-invariant but not locally generated, diagonal theories defeat one-spacetime equivalence, and changing boundary data or confusing a state space with one natural state blocks the remaining upgrades.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "causally convex Loc morphism\\\\intertwined propagator and CCR map"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "admissible compact $h$ plus time-slice\\\\relative Cauchy automorphism"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "model support theorem\\\\$\\mathcal A^{\\rm kin}=\\mathcal A^{\\rm dyn}$"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "dynamical locality plus theorem-specific inputs\\\\SPASs or natural-state no-go"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "image lets a causal curve leave and re-enter\\\\$E_N\\psi_*\\neq\\psi_*E_M$; commutator map fails"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "$g+h$ not globally hyperbolic or no time-slice\\\\$\\mathcal A(j^\\pm)^{-1}$ is unavailable"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "massless constant mode or universal Maxwell flux\\\\dynamical subobject is larger than kinematic"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "diagonal species change, missing Reeh–Schlieder, or changed Robin data\\\\equivalence, no-go, or boundary-functor claim stops"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "Cauchy maps",
      "logical_status": "ordered diagnostic sequence, not an implication across claim classes"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "fixed points",
      "logical_status": "ordered diagnostic sequence, not an implication across claim classes"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "named theorem",
      "logical_status": "ordered diagnostic sequence, not an implication across claim classes"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "drop causal convexity",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "leave source category",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "retain global mode",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "missing theorem input",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}