{
  "schema_version": "qft.higher-point-microlocal-spectrum-graph-cone.v1",
  "artifact_id": "qft.artifact.mathematical-qft.microlocal-qft.higher-point-graph-cone-orientation",
  "owner_page_id": "qft.topic.math-qft.microlocal-spectrum-condition",
  "figure": {
    "title": "Higher-Point Microlocal Spectrum Graph Cones",
    "composition": "three vertically stacked panels",
    "schematic": true,
    "scale": "not to scale",
    "dominant_takeaway": "The original graph condition constrains transported covector orientation rather than path causality, and a single frequency flip can preserve base singular support while leaving the allowed cone."
  },
  "conventions": {
    "vertex_order": "field-argument order, not chronological or causal order",
    "graph_class": "finite multigraph; parallel edges may follow different immersed paths",
    "edge_representative": "one geometric edge i--j is represented as i->j for i<j",
    "path_class": "piecewise smooth; not required to be causal",
    "edge_label": "parallel future-causal covector field; a zero label is allowed",
    "vertex_balance": "sum of outgoing-to-later endpoint labels minus incoming-from-earlier endpoint labels",
    "zero_slots": "individual vertex covectors may vanish; only the total zero cotangent tuple is excluded",
    "doubled_edge_equivalence": "equivalently include e and e^{-1}, with the reverse label equal to minus the forward label"
  },
  "variants": [
    {
      "id": "smooth_path",
      "cone": "Gamma_n^sm",
      "path_requirement": "piecewise smooth",
      "strength_rank": 1
    },
    {
      "id": "causal_path",
      "cone": "Gamma_n^c",
      "path_requirement": "causal",
      "strength_rank": 2
    },
    {
      "id": "null_path",
      "cone": "Gamma_n^ll",
      "path_requirement": "null",
      "strength_rank": 3
    }
  ],
  "variant_inclusion": "Gamma_n^ll subset-or-equal Gamma_n^c subset-or-equal Gamma_n^sm",
  "analytic_variant": "On a real-analytic spacetime, analytic muSC replaces the smooth wavefront set by the analytic wavefront set; it is not a path-class synonym.",
  "panels": [
    {
      "id": "edge_data",
      "relation": "gamma_ij is a piecewise-smooth path from x_i to x_j, while xi_ij is a parallel future-causal covector field along it",
      "nonimplication": "the smooth-path definition neither requires gamma_ij to be causal nor identifies xi_ij with the path tangent"
    },
    {
      "id": "allowed_pairing",
      "edges": [
        {
          "source": 1,
          "target": 3,
          "label": "a",
          "frequency": "future null"
        },
        {
          "source": 2,
          "target": 4,
          "label": "b",
          "frequency": "future null"
        }
      ],
      "vertex_tuple": [
        "a_1",
        "b_2",
        "-a_3",
        "-b_4"
      ],
      "assumptions": [
        "a is nonzero"
      ],
      "first_nonzero_slot": 1,
      "first_nonzero_orientation": "future",
      "membership": "in Gamma_4"
    },
    {
      "id": "reversed_frequency",
      "base_paths": "identical to the allowed pairing",
      "changed_edge": {
        "source": 1,
        "target": 3,
        "graph_orientation": "1 -> 3 retained",
        "label_change": "a -> -a",
        "resulting_label": "past causal and forbidden"
      },
      "unchanged_edge": {
        "source": 2,
        "target": 4,
        "label": "b"
      },
      "vertex_tuple": [
        "-a_1",
        "b_2",
        "a_3",
        "-b_4"
      ],
      "assumptions": [
        "a is nonzero"
      ],
      "first_nonzero_slot": 1,
      "first_nonzero_orientation": "past",
      "membership": "not in Gamma_4",
      "singular_support_projection": "unchanged"
    }
  ],
  "flat_space_chain_control": {
    "metric_signature": "+-",
    "prefix_labels": [
      [
        2,
        1
      ],
      [
        3,
        0
      ],
      [
        2,
        -1
      ]
    ],
    "momenta": [
      [
        2,
        1
      ],
      [
        1,
        -1
      ],
      [
        -1,
        -1
      ],
      [
        -2,
        1
      ]
    ],
    "reconstructed_prefixes": [
      [
        2,
        1
      ],
      [
        3,
        0
      ],
      [
        2,
        -1
      ]
    ],
    "total_momentum": [
      0,
      0
    ],
    "all_prefixes_future_causal": true,
    "reconstruction_exact": true,
    "total_conservation_exact": true
  },
  "checks": {
    "allowed_tuple_signed_incidence": true,
    "reversal_changes_first_slot_to_past": true,
    "reversal_keeps_graph_orientation_1_to_3": true,
    "reversal_assumes_nonzero_a": true,
    "reversal_preserves_base_paths": true,
    "smooth_path_not_conflated_with_causal_path": true,
    "partial_zero_slots_permitted": true,
    "flat_chain_reconstructs_prefixes": true,
    "flat_chain_conserves_total_momentum": true
  },
  "sources": [
    {
      "citation": "Brunetti, Fredenhagen, and Koehler, Communications in Mathematical Physics 180 (1996), Definition 4.1, Lemma 4.2, Proposition 4.3, and Theorem 4.6",
      "doi": "https://doi.org/10.1007/BF02099626",
      "open_pdf": "https://arxiv.org/pdf/gr-qc/9510056"
    },
    {
      "citation": "Sanders, Communications in Mathematical Physics 295 (2010), Definitions 2.3-2.4, Proposition 2.5, Lemma 2.6, Theorem 4.2, and Corollary 4.3",
      "doi": "https://doi.org/10.1007/s00220-009-0900-7",
      "open_pdf": "https://arxiv.org/pdf/0903.1021"
    }
  ]
}