{
  "title": "Full exchange versus geodesic conformal block",
  "kind": "integration-domain-and-ope-content-comparison",
  "schematic": true,
  "not_to_scale": true,
  "takeaway": "Full-AdS vertex integration gives a contact-sourced Casimir problem and two double-trace towers, while geodesic restriction isolates the physical exchanged block.",
  "scope": "Euclidean AdS_(d+1), tree level, scalar fields, nonderivative cubic vertices, standard quantization, all relevant Casimir eigenvalues distinct for the displayed OPE decomposition, one 12-to-34 channel",
  "conventions": {
    "green_equation": "(-Box + m_chi^2) G_chi = delta_g",
    "casimir_eigenvalue": "C_chi = m_chi^2 L^2 = Delta_chi(Delta_chi-d)",
    "source_sign_dictionary": "J_site = i L_source, so C_site = -C_source and the source convention -L^2 Box becomes +L^2 Box on this page",
    "exchange_equation": "(C_12-C_chi) E_s = -g_12chi g_34chi L^2 D_1234",
    "physical_block": "For Delta_chi not equal to d/2, OPE behavior u^(Delta_chi/2) excludes the shadow behavior u^((d-Delta_chi)/2); at Delta_chi=d/2, the standard limiting and regularity prescription excludes the independent logarithmic branch"
  },
  "panels": [
    {
      "title": "Ordinary exchange: full AdS",
      "vertex_domain": "X and Y range independently over all of AdS",
      "external_legs": "K_1(X), K_2(X), K_3(Y), K_4(Y)",
      "internal_line": "G_chi(X,Y)",
      "casimir_status": "inhomogeneous; the Green-function delta source collapses X to Y and produces the contact diagram",
      "solution_decomposition": {
        "homogeneous": "single-trace O_chi",
        "contact_sourced": "scalar double-trace [O_1 O_2]_(m,0) and [O_3 O_4]_(n,0)"
      },
      "ope_content": [
        "O_chi",
        "[O_1 O_2]_(m,0)",
        "[O_3 O_4]_(n,0)"
      ]
    },
    {
      "title": "Geodesic object: one block",
      "vertex_domain": "X is restricted to gamma_12 and Y to gamma_34",
      "external_legs": "K_1(X), K_2(X), K_3(Y), K_4(Y)",
      "internal_line": "G_chi(X,Y)",
      "casimir_status": "homogeneous; the OPE boundary condition selects the physical branch",
      "ope_content": [
        "one physical (Delta_chi,0) block"
      ]
    }
  ],
  "degeneracy_caveat": "If O_1=O_3 and O_2=O_4, the two sums describe one operator family and must be combined. If only their dimensions coincide, distinct species remain degenerate and require a mixing analysis. A single-trace and double-trace resonance likewise requires a regulated limit and may alter large-N power counting.",
  "crossing_caveat": "Neither panel is by itself a complete crossing-symmetric correlator."
}