{
  "title": "Curved Spacetime Aqft Stress Qei Failure Map",
  "question": "Which concrete dropped hypotheses cause degeneracy, undefined local stress tensors, false pointwise or null-energy bounds, invalid focusing upgrades, uncontrolled semiclassical roots, missing global interacting states, or nonunique boundary dynamics?",
  "takeaway": "Unquotiented equation-of-motion directions are symplectic null vectors; a non-Hadamard two-point function blocks coincidence limits; delta sequences and wrongly normalized null boosts defeat stronger QEI claims; finite null segments need not satisfy ANEC; absent contraction stops focusing; Planckian runaway roots, constant infinite-volume couplings, and unspecified Robin data exceed the semiclassical, pAQFT, and boundary theorems.",
  "figure_type": "failure",
  "schematic": true,
  "scale": "not to scale",
  "nodes": [
    {
      "id": "a0",
      "row": "claim checkpoints",
      "label": "globally hyperbolic $P$ and quotient phase space\\\\nondegenerate form and time-slice algebra"
    },
    {
      "id": "a1",
      "row": "claim checkpoints",
      "label": "positive Hadamard state and covariant subtraction\\\\finite conserved stress tensor plus curvature freedom"
    },
    {
      "id": "a2",
      "row": "claim checkpoints",
      "label": "specified field, curve, dimension, and sampler\\\\theorem-specific sampled lower bound"
    },
    {
      "id": "a3",
      "row": "claim checkpoints",
      "label": "named limit or extension hypotheses\\\\conditional ANEC, focusing, stability, pAQFT, or boundary result"
    },
    {
      "id": "b0",
      "row": "first broken step",
      "label": "use $C_0^\\infty$ without quotient, or leave global hyperbolicity without boundary data\\\\$Pu$ remains in the radical; unique $E^\\pm$ is unavailable"
    },
    {
      "id": "b1",
      "row": "first broken step",
      "label": "use a non-Hadamard state or change subtraction scale without matched counterterms\\\\coincidence limit fails or the matter–gravity split becomes inconsistent"
    },
    {
      "id": "b2",
      "row": "first broken step",
      "label": "send $g_\\tau$ to a delta function, boost with fixed proper-time normalization, or isolate a negative null pulse\\\\pointwise, four-dimensional compact-null, or finite-segment ANEC claim fails"
    },
    {
      "id": "b3",
      "row": "first broken step",
      "label": "set initial contraction to zero; retain out-of-domain runaway roots; set $g=1$ globally; or omit Robin data\\\\focusing, stability, global vacuum, or unique boundary theory does not follow"
    }
  ],
  "edges": [
    {
      "from": "a0",
      "to": "a1",
      "label": "local observable",
      "logical_status": "ordered diagnostic sequence, not an implication across theorem classes"
    },
    {
      "from": "a1",
      "to": "a2",
      "label": "sampled theorem",
      "logical_status": "ordered diagnostic sequence, not an implication across theorem classes"
    },
    {
      "from": "a2",
      "to": "a3",
      "label": "named extension",
      "logical_status": "ordered diagnostic sequence, not an implication across theorem classes"
    },
    {
      "from": "a0",
      "to": "b0",
      "label": "omit equation/domain input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a1",
      "to": "b1",
      "label": "omit UV/renormalization input",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a2",
      "to": "b2",
      "label": "change sampling domain or limit",
      "logical_status": "failure under removed or overstated hypothesis"
    },
    {
      "from": "a3",
      "to": "b3",
      "label": "omit extension-specific hypothesis",
      "logical_status": "failure under removed or overstated hypothesis"
    }
  ]
}