{
  "schema": "OMEGA_TYPED_BRIDGES/1",
  "date": "2026-11-10",
  "bridge_records": 5,
  "lean_namespace": "OmegaBridges except existing Omega placement theorems",
  "bridges": [
    {
      "id": "DIM-PLANAR-PLACEMENT",
      "source": "marked planar OMEGA curve with a>0",
      "target": "isometric placement in an ambient real normed space",
      "map": "z -> alpha+E(z), E real-linear and isometric",
      "preserves": [
        "selected G=a*phi",
        "selected C=a*pi",
        "G/(G+C)",
        "homothety commutation"
      ],
      "evidence_kind": "EXISTING_LEAN_SOURCE_READ",
      "proof_source": "sources/Placement.lean",
      "theorems": [
        "Omega.provenance_all_placements",
        "Omega.placed_homothety"
      ],
      "rebuilt_in_this_supplement": false
    },
    {
      "id": "DIM-600CELL-ICOSAHEDRON",
      "source": "12 neighbors of p=(-1,0,0,0) in pinned 600-cell",
      "target": "standard 3D icosahedral sampling nodes of radius sqrt(2+phi)",
      "map": "L(p)=2*phi*(p1,p3,p2)",
      "inverse": "u -> (-phi/2,u0/(2*phi),u2/(2*phi),u1/(2*phi))",
      "evidence_kind": "LEAN_KERNEL_CHECKED_FINITE",
      "proof_source": "OmegaBridges.lean",
      "theorems": [
        "local_neighbor_count",
        "local_neighbor_slice",
        "local_map_exact",
        "local_is_standard_icosahedron",
        "local_radius",
        "every_degree_five_moment"
      ],
      "scope": "Real normalization and identification with sphere integrals are written in PROOFS.md"
    },
    {
      "id": "DIM-ICOSAHEDRAL-LINEAR-RECOVERY",
      "source": "linear field with coefficient vector in Z[phi]^3",
      "target": "12 exact sample values in Z[phi]",
      "map": "w -> (u dot w)_u",
      "inverse": "(3-phi)/20 * sum_u u*sample_u",
      "evidence_kind": "LEAN_KERNEL_CHECKED_UNIVERSAL",
      "proof_source": "OmegaBridges.lean",
      "theorems": [
        "icosahedral_frame_identity",
        "linear_field_reconstruction",
        "linear_samples_injective"
      ],
      "scope": "Integer-pair representation; exact division on valid images"
    },
    {
      "id": "ALG-GOLDEN-FIBONACCI-BLOCK",
      "source": "golden-coordinate-derived 600-cell signal coefficients",
      "target": "multiplication by alpha=13+6phi and polynomial 13I+6F of Fibonacci matrix F",
      "map": "(a,b)->(13a+6b,6a+19b)",
      "evidence_kind": "LEAN_KERNEL_CHECKED",
      "proof_source": "OmegaBridges.lean",
      "theorems": [
        "all_480_geometric_mode_entries",
        "golden_step_is_multiplication",
        "golden_step_is_fibonacci_polynomial"
      ]
    },
    {
      "id": "ALG-NORM-SPECTRAL-FACTOR",
      "source": "golden update block B",
      "target": "norm scaling and spectral polynomial X^2-32X+211",
      "map": "N(a,b)=a^2+ab-b^2; N(B^n p)=211^n N(p)",
      "evidence_kind": "LEAN_KERNEL_CHECKED_UNIVERSAL",
      "proof_source": "OmegaBridges.lean",
      "theorems": [
        "norm_multiplicative",
        "conjugate_multiplier",
        "golden_iter_norm",
        "golden_step_spectral_recurrence"
      ],
      "scope": "Polynomial on the geometric invariant block; other matrix factors are separate"
    }
  ],
  "composed_route": {
    "path": [
      "DIM-600CELL-ICOSAHEDRON",
      "DIM-ICOSAHEDRAL-LINEAR-RECOVERY",
      "specified finite coefficient codec",
      "existing witnessed-history recovery"
    ],
    "condition": "Retain source class, codec, framing, and valid history inverse",
    "status": "WRITTEN_COMPOSITION_WITH_EXPLICIT_INTERFACES"
  }
}