{
  "title": "Curved Spacetime Aqft Stress Qei Dependency Map",
  "question": "Which one-way hypotheses connect globally hyperbolic free-field propagation to Hadamard states, renormalized stress tensors, sampled energy bounds, controlled ANEC limits, and conditional focusing, and where do semiclassical, perturbative, and boundary constructions branch?",
  "takeaway": "Global hyperbolicity and the equation-of-motion quotient construct the free algebra; a positive Hadamard two-point function licenses local composite fields; local covariance, conservation, and scaling classify stress-tensor freedom; positivity plus an exact sampler gives a QEI; only a separate complete-geodesic limit and geometric hypotheses lead onward to ANEC or focusing, while semiclassical response, pAQFT, and boundary dynamics require independent extra data.",
  "figure_type": "dependency",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "n0",
      "label": "geometry and operator\\\\$(M,g)$ globally hyperbolic, no boundary\\\\$P=\\Box_g+m^2+\\xi R$, unique $E^\\pm$"
    },
    {
      "id": "n1",
      "label": "causal phase space and algebra\\\\$\\mathcal E=C_0^\\infty/PC_0^\\infty$\\\\$\\sigma([f],[h])=\\int fEh$; CCR and time-slice"
    },
    {
      "id": "n2",
      "label": "admissible state\\\\positive bisolution, antisymmetric part $iE$\\\\Hadamard $\\operatorname{WF}(\\omega_2)$"
    },
    {
      "id": "n3",
      "label": "local stress tensor\\\\Hadamard subtraction plus Ward correction\\\\$T_{ab}^{\\rm ren}+\\alpha_1m^4g_{ab}+\\alpha_2m^2G_{ab}+\\alpha_3I_{ab}+\\alpha_4J_{ab}$"
    },
    {
      "id": "n4",
      "label": "sampled positivity theorem\\\\free field, Hadamard class, timelike curve\\\\$\\int g^2\\langle T_{uu}\\rangle\\geq-Q[g]$"
    },
    {
      "id": "n5",
      "label": "additional limit and geometry\\\\complete affine null line plus infrared control\\\\ANEC; focusing also needs contraction and global causality"
    },
    {
      "id": "branch",
      "label": "matched couplings plus retarded response $\\Rightarrow$ conditional semiclassical stability\\\\compact interaction plus causal factorization $\\Rightarrow$ local formal pAQFT\\\\fixed positive self-adjoint boundary extension $\\Rightarrow$ boundary-dependent theory"
    }
  ],
  "edges": [
    {
      "from": "n0",
      "to": "n1",
      "label": "quotient",
      "logical_status": "licensed implication"
    },
    {
      "from": "n1",
      "to": "n2",
      "label": "choose state",
      "logical_status": "licensed implication"
    },
    {
      "from": "n2",
      "to": "n3",
      "label": "renormalize",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "n4",
      "label": "sample",
      "logical_status": "licensed implication"
    },
    {
      "from": "n4",
      "to": "n5",
      "label": "controlled limit",
      "logical_status": "licensed implication"
    },
    {
      "from": "n3",
      "to": "branch",
      "label": "separate extensions",
      "logical_status": "licensed implication"
    }
  ]
}