{
  "schema_version": "qft.hadamard-propagation-cauchy-band.v1",
  "artifact_id": "qft.artifact.curved-spacetime.states-hadamard-microlocal-control.hadamard-propagation-cauchy-band",
  "owner_page_id": "qft.topic.curved-states.propagation-hadamard-property",
  "title": "A complete Cauchy band controls every null bicharacteristic",
  "figure_type": "original qualitative textbook diagram; schematic and not to scale",
  "composition": "two vertically stacked contrast panels",
  "reader_question": "Why does Hadamard control near a complete Cauchy surface propagate globally, while control in an arbitrary small diamond does not?",
  "takeaway": "A neighborhood containing a complete Cauchy surface meets every inextendible null bicharacteristic, whereas a small diamond can miss one; propagation transports the established wavefront relation but neither supplies positivity nor controls missed orbits.",
  "theorem": {
    "setting": {
      "spacetime": "smooth, time-oriented, globally hyperbolic spacetime without boundary",
      "cauchy_surface": "smooth spacelike Cauchy surface Sigma in M",
      "operator": "real, formally self-adjoint, normally hyperbolic scalar operator P with smooth coefficients",
      "kernel": "bidistribution omega_2 with P_x omega_2 = P_x_prime omega_2 = 0 and the canonical antisymmetric part",
      "seed_region": "open neighborhood N containing the complete Cauchy surface Sigma"
    },
    "principal_symbol": "p(x,k)=g_inverse(x)(k,k)",
    "hadamard_relation": "WF(omega_2)={(x,k;x_prime,-k_prime):(x,k)~(x_prime,k_prime), k future-directed null}",
    "relation_definition": "(x,k)~(x_prime,k_prime) means that one null geodesic gamma contains x and x_prime, k is a nonzero null covector with k_sharp tangent to gamma at x, and k_prime=PT_gamma(k)",
    "transport": "The Hamilton flows H_{p,x} and H_{p,x_prime} propagate the two covector entries; k_prime is parallel transport of k along their common null geodesic.",
    "time_slice_reconstruction": "A support-moving map S sends compactly supported tests on M into N and gives omega_2=S_transpose composed with omega_2 restricted to N times N composed with S; the kernel-composition theorem transports the paired relation and excludes partial-zero product-cotangent components.",
    "conclusion": "If the Hadamard wavefront relation holds on N times N, it holds on M times M.",
    "does_not_prove": [
      "positivity of omega_2",
      "existence of a state from arbitrary data",
      "a preferred vacuum or state-selection rule"
    ]
  },
  "panels": [
    {
      "id": "complete_cauchy_band",
      "label": "(a)",
      "seed_region": "N contains all of Sigma",
      "line_style": "solid arrows for controlled null bicharacteristics",
      "marked_objects": [
        "complete Cauchy surface Sigma",
        "open band N containing Sigma",
        "three future-oriented inextendible null bicharacteristics",
        "seed covector x_0,k_0 and transported covector x,k"
      ],
      "exact_geometric_fact": "Every inextendible causal curve, and hence every inextendible null bicharacteristic projection, intersects a Cauchy surface exactly once.",
      "logical_chain": [
        "N contains Sigma",
        "every relevant null orbit crosses N",
        "the local Hadamard relation fixes the allowed covectors on N times N",
        "time-slice kernel reconstruction and wavefront composition propagate the paired relation and exclude partial-zero components"
      ],
      "licensed_conclusion": "global Hadamard wavefront relation on M times M"
    },
    {
      "id": "small_diamond_negative_control",
      "label": "(b)",
      "seed_region": "D intersects only a proper subset of Sigma",
      "line_style": "solid arrow for an orbit sampled by D; dashed arrow and open marker for a missed orbit",
      "marked_objects": [
        "small diamond D",
        "complete Cauchy surface Sigma",
        "one null orbit crossing D",
        "one null orbit missing D"
      ],
      "failed_hypothesis": "the seed region does not contain a complete Cauchy surface",
      "surviving_statement": "Hadamard control is established only on the stated local domain and on bicharacteristic segments reached from it.",
      "forbidden_inference": "Hadamard on D times D implies Hadamard on M times M"
    }
  ],
  "negative_control": {
    "construction": "Choose a null bicharacteristic whose intersection with Sigma lies outside D intersect Sigma.",
    "observation": "The orbit never enters the seed diamond, so data on D times D cannot exclude an additional wavefront component along it.",
    "result": "the global implication is not licensed"
  },
  "conventions": {
    "metric_signature": "(+---)",
    "future_direction": "the first covector in the Hadamard pair is future-directed",
    "second_slot_sign": "the second entry is written -k_prime, where k_prime is parallel transport of k",
    "diagram_time_direction": "future is upward",
    "scale_status": "schematic and not to scale"
  },
  "accessibility": {
    "reading_order": [
      "title and theorem hypotheses",
      "panel (a), complete Cauchy band and global conclusion",
      "panel (b), small-diamond negative control and no global conclusion",
      "claim boundary"
    ],
    "redundant_encodings": [
      "controlled rays use solid lines, filled markers, and direct labels",
      "the missed ray uses a dashed line, an open marker, and a direct label",
      "the complete band and small diamond are labeled directly",
      "no scientific distinction depends on color"
    ],
    "text_alternative": "Two vertically stacked spacetime diagrams contrast the propagation theorem with its negative control. In the first, a neighborhood N containing the complete Cauchy surface Sigma intersects every inextendible null bicharacteristic. Hamilton flow displays the nonzero-slot geometry, while time-slice reconstruction and kernel composition exclude partial-zero components and carry the full Hadamard relation from N times N to M times M. In the second, a small diamond meets only part of Sigma and a dashed null orbit misses it, so Hadamard form on the diamond cannot license a global conclusion. Propagation transports an established singularity relation and does not create positivity."
  },
  "checks": {
    "complete_band_contains_entire_cauchy_surface": true,
    "every_displayed_controlled_orbit_crosses_band": true,
    "transport_in_both_distribution_arguments_is_explicit": true,
    "partial_zero_components_are_controlled_by_time_slice_reconstruction": true,
    "small_diamond_is_not_a_cauchy_neighborhood": true,
    "missed_orbit_is_explicit": true,
    "forbidden_global_inference_is_explicit": true,
    "positivity_is_not_attributed_to_propagation": true
  },
  "public_files": {
    "svg": "/figures/curved-spacetime/hadamard-propagation-cauchy-band.svg",
    "semantic_json": "/figures/curved-spacetime/hadamard-propagation-cauchy-band.json"
  }
}