{
  "schema_version": "qft.propagation-singularities-hamilton-flow-caustic.v1",
  "artifact_id": "qft.artifact.mathematical-qft.microlocal-qft.hamilton-flow-caustic-continuation",
  "owner_page_id": "qft.topic.math-qft.propagation-singularities-hyperbolic-fields",
  "title": "Source-Free Hamilton Transport Through a Caustic",
  "figure_type": "original qualitative textbook diagram; boundaryless schematic and not to scale",
  "composition": "two vertically stacked panels",
  "reader_question": "Why can a geometric-optics amplitude fail at a caustic although the wavefront continues, and exactly where can a source change that conclusion?",
  "takeaway": "On a real-principal-type interval disjoint from WF(f), Hamilton flow transports established WF(u) membership smoothly in the nonzero cotangent bundle; a caustic is a rank failure of the projection of a propagated Lagrangian family to spacetime, not an endpoint of the lifted bicharacteristics.",
  "theorem": {
    "operator": "properly supported scalar classical pseudodifferential operator P",
    "principal_symbol": "real homogeneous p with dp nonzero on Char(P)",
    "characteristic_set": "Char(P)={p=0} in T-star M minus the zero section",
    "equation": "Pu=f",
    "source_free_condition": "gamma(I) intersect WF(f) is empty",
    "conclusion": "WF(u) membership is all-or-none on the connected Hamilton orbit segment gamma(I)",
    "source_stop": "at gamma(s_0) in WF(f), the source-free theorem gives no comparison across s_0"
  },
  "panels": [
    {
      "id": "characteristic_phase_space",
      "label": "(a)",
      "marked_objects": [
        "nonzero characteristic set Char(P)",
        "oriented Hamilton orbit gamma with gamma-dot=H_p",
        "two marked wavefront covectors on a source-free interval",
        "a separate source wavefront covector gamma(s_0) in WF(f)",
        "dashed undetermined continuation after the source marker"
      ],
      "licensed_relation": "solid transport on a connected interval disjoint from WF(f)",
      "unlicensed_relation": "no propagation comparison across a covector in WF(f)"
    },
    {
      "id": "spacetime_projection_caustic",
      "label": "(b)",
      "marked_objects": [
        "smooth characteristic Lagrangian Lambda",
        "projection pi from Lambda to spacetime",
        "four projected null rays",
        "a caustic where rank(d pi restricted to Lambda) drops",
        "continued individual Hamilton trajectories"
      ],
      "coordinate_chart_failure": "a single graph or eikonal chart ceases to be valid at the caustic",
      "amplitude_failure": "a branchwise Van Vleck or stationary-phase amplitude may diverge at the caustic",
      "invariant_survival": "the lifted Lagrangian and individual Hamilton trajectories continue smoothly"
    }
  ],
  "exact_null_caustic_model": {
    "spacetime": "1+2 Minkowski with metric dt^2-dx^2-dz^2",
    "homogeneous_phase": "Phi=x*mu+z*lambda-t*sqrt(mu^2+lambda^2)-mu^3/(3*lambda^2), lambda>0",
    "homogeneity": "degree one in (mu,lambda) on the conic chart lambda>0",
    "critical_equations": [
      "0=partial_mu Phi=x-t*mu/sqrt(mu^2+lambda^2)-mu^2/lambda^2",
      "0=partial_lambda Phi=z-t*lambda/sqrt(mu^2+lambda^2)+2*mu^3/(3*lambda^3)"
    ],
    "fiber_ratio": "s=mu/lambda",
    "base_map": "q(t,s)=(t,t*s/sqrt(1+s^2)+s^2,t/sqrt(1+s^2)-2*s^3/3)",
    "covector": "k(s,lambda)=lambda*(-sqrt(1+s^2),s,1)",
    "null_control": "k_t^2-k_x^2-k_z^2=0",
    "hamilton_proportionality": "k_sharp=-lambda*sqrt(1+s^2)*partial_t q",
    "rank_loss": "at t=s=0, partial_t q=(1,0,1) while partial_s q=(0,0,0)",
    "nonzero_lift": "at s=0, k=lambda*(-1,0,1) is nonzero and partial_s k=(0,lambda,0)",
    "spatial_front_at_t_zero": "9*z^2=4*x^3 with x>=0",
    "sample_controls": [
      {
        "input": {
          "t": -1.25,
          "s": -0.75,
          "lambda": 2
        },
        "base": [
          -1.25,
          1.3125,
          -0.71875
        ],
        "covector": [
          -2.5,
          -1.5,
          2
        ],
        "q_tangent": [
          1,
          -0.6,
          0.8
        ],
        "q_family_tangent": [
          0,
          -2.14,
          -1.605
        ],
        "critical_residuals": [
          0,
          0
        ],
        "null_norm": 0,
        "canonical_pairings": [
          0,
          0
        ],
        "raised_covector_minus_hamilton_tangent": [
          0,
          0,
          0
        ]
      },
      {
        "input": {
          "t": 0.8,
          "s": 0.4,
          "lambda": 1.5
        },
        "base": [
          0.8,
          0.457112541083283,
          0.700114686041541
        ],
        "covector": [
          -1.61554944214035,
          0.6,
          1.5
        ],
        "q_tangent": [
          1,
          0.371390676354104,
          0.928476690885259
        ],
        "q_family_tangent": [
          0,
          1.44032875233466,
          -0.576131500933865
        ],
        "critical_residuals": [
          0,
          0
        ],
        "null_norm": 0,
        "canonical_pairings": [
          0,
          0
        ],
        "raised_covector_minus_hamilton_tangent": [
          0,
          0,
          0
        ]
      },
      {
        "input": {
          "t": 1.1,
          "s": 1.2,
          "lambda": 0.75
        },
        "base": [
          1.1,
          2.28504340755711,
          -0.447797160369072
        ],
        "covector": [
          -1.171537451386,
          0.9,
          0.75
        ],
        "q_tangent": [
          1,
          0.768221279597376,
          0.64018439966448
        ],
        "q_family_tangent": [
          0,
          2.68860772116022,
          -3.22632926539226
        ],
        "critical_residuals": [
          0,
          0
        ],
        "null_norm": 0,
        "canonical_pairings": [
          0,
          0
        ],
        "raised_covector_minus_hamilton_tangent": [
          0,
          0,
          0
        ]
      }
    ],
    "origin_control": {
      "input": {
        "t": 0,
        "s": 0,
        "lambda": 2
      },
      "q_s": [
        0,
        0,
        0
      ],
      "q_ss": [
        0,
        2,
        0
      ],
      "projection_rank_generic": 2,
      "projection_rank_at_caustic": 1,
      "lifted_tangents": [
        [
          1,
          0,
          1,
          0,
          0,
          0
        ],
        [
          0,
          0,
          0,
          0,
          2,
          0
        ],
        [
          0,
          0,
          0,
          -1,
          0,
          1
        ]
      ],
      "lifted_tangent_rank": 3
    },
    "cusp_controls": [
      {
        "s": -1.25,
        "x": 1.5625,
        "z": 1.30208333333333,
        "residual_9z2_minus_4x3": -1.77635683940025e-15
      },
      {
        "s": -0.4,
        "x": 0.16,
        "z": 0.0426666666666667,
        "residual_9z2_minus_4x3": 0
      },
      {
        "s": 0.65,
        "x": 0.4225,
        "z": -0.183083333333333,
        "residual_9z2_minus_4x3": 0
      },
      {
        "s": 1.1,
        "x": 1.21,
        "z": -0.887333333333334,
        "residual_9z2_minus_4x3": 1.77635683940025e-15
      }
    ]
  },
  "assumptions": [
    "nonzero cotangent covectors",
    "scalar real homogeneous principal symbol",
    "dp is nonzero on the characteristic set",
    "boundaryless source-free interval for the solid propagation claim",
    "global hyperbolicity is not required for the local theorem"
  ],
  "conventions": {
    "metric_signature": "(+---)",
    "klein_gordon_representative": "p=g_inverse(k,k)",
    "klein_gordon_projection": "a null geodesic up to harmless Hamiltonian rescaling, with k parallel transported",
    "time_orientation": "the propagation theorem preserves either covector orientation and does not select one",
    "scale_status": "schematic and not to scale"
  },
  "does_not_determine": [
    "amplitude or distributional order",
    "Maslov phase",
    "energy or dispersive decay",
    "geodesic completeness",
    "boundary reflection or diffraction",
    "existence or positivity of a quantum state"
  ],
  "independent_controls": [
    {
      "id": "minkowski_control",
      "model": "p=k_0 squared minus spatial k squared",
      "result": "k is constant and x is an affine null line"
    },
    {
      "id": "ultrastatic_sphere_control",
      "model": "R times a round S^2 of radius R, with spherical arclength s and a transverse Jacobi field proportional to sin(s/R)",
      "result": "the Jacobi field vanishes at the antipode while geodesic and Hamilton flow remain smooth through s=pi R"
    }
  ],
  "accessibility": {
    "reading_order": [
      "title and scope",
      "panel (a), source-free transport and source stop condition",
      "panel (b), folded spacetime projection and continued lifted flow",
      "claim boundary"
    ],
    "redundant_encodings": [
      "licensed propagation uses a solid arrow, filled markers, and direct text",
      "the source stop uses an open marker, a cross, a dashed arrow, and direct text",
      "the caustic uses ray convergence, an open marker, a dashed guide, and direct text",
      "no scientific distinction depends on color"
    ],
    "text_alternative": "Two stacked schematics. Above, a solid Hamilton orbit on the nonzero characteristic set carries a marked wavefront covector across a source-free interval; a dashed interruption at a source wavefront covector marks where the theorem gives no continuation rule. Below, a smooth characteristic Lagrangian projects to null rays that focus at a caustic and continue, while the lifted covectors remain on smooth Hamilton trajectories."
  },
  "checks": {
    "zero_section_excluded": true,
    "source_free_interval_explicit": true,
    "source_marker_is_in_WF_f": true,
    "no_solid_implication_crosses_source_marker": true,
    "caustic_is_projection_rank_loss": true,
    "exact_model_phase_is_degree_one": true,
    "exact_model_covector_is_nonzero_and_null_at_caustic": true,
    "individual_lifted_trajectories_continue": true,
    "boundary_reflection_absent": true,
    "time_orientation_not_selected": true,
    "amplitude_and_decay_claims_excluded": true
  },
  "public_files": {
    "svg": "/figures/mathematical-qft/propagation-singularities-hamilton-flow-caustic.svg",
    "semantic_json": "/figures/mathematical-qft/propagation-singularities-hamilton-flow-caustic.json"
  }
}