{
  "schema": "OMEGA_EXPLICIT_JOIN/1",
  "date": "2026-11-07",
  "status": "written proof with exact finite certificates; new synthesis, not claimed deposited",
  "objects": {
    "scalar_affine": "Omega_gamma(t)=1+gamma*(t-1)",
    "real_network": "F_gamma(v)=1+gamma*P*(v-1)",
    "geometric_embedding": "E(t,w)=t*1+C*w; C has unit 600-cell vertices as rows",
    "integer_network": "T(d)=floor(Q*M*d/D)",
    "full_record": "(T(d), Q*M*d-D*T(d))",
    "mod8_record": "(T(d), d mod 8)",
    "integer_geometry_embedding": "E_Z(t,x,z)=t*1+C0*x+C1*z, with 2C=C0+phi*C1; t integer, x,z in Z^4"
  },
  "relations": [
    {
      "name": "marked_scale",
      "identity": "chi(lambda*a)=Omega_lambda(chi(a))",
      "domain": "a>0, fixed labeled template/calibration"
    },
    {
      "name": "five_mode_join",
      "identity": "F_gamma(E(t,w))=E(Omega_gamma(t),gamma*(13+6*phi)/25*w)",
      "domain": "t in R, w in R^4"
    },
    {
      "name": "integer_golden_coordinate_join",
      "identity": "M E_Z(t,x,z)=E_Z(25t,13x+6z,6x+19z)",
      "domain": "(t,x,z) in Z x Z^4 x Z^4",
      "inverse": "t=sum(d)/120; x=(2C0^T+C1^T)d/120; z=(C0^T+3C1^T)d/120",
      "warning": "M preserves this image; T need not. Witnesses recover the original image point."
    },
    {
      "name": "scalar_integer_restriction",
      "identity": "T(n*1)=floor(Q*n/S)*1; r(n*1)=25*(Q*n-S*floor(Q*n/S))*1",
      "domain": "n in Z"
    },
    {
      "name": "valid_witness_translation",
      "identity": "J(full(d))=mod8(d), H(mod8(d))=full(d); HJ=Id, JH=Id",
      "domain": "valid record images with reduced output retained"
    },
    {
      "name": "ideal_rational_defect",
      "identity": "F_lambda(1+d)=1+T(d)+r(d)/D+(lambda-q)*P*d",
      "domain": "d in Z^120"
    },
    {
      "name": "history_join",
      "identity": "DecodeHistory(EncodeHistory(d))=d",
      "domain": "any finite horizon; ordered witnesses and terminal retained"
    }
  ],
  "not_claimed": [
    "graph forced by scalar constants",
    "all source theories equivalent",
    "bit optimum found",
    "fast general decoder",
    "full Lean formalization"
  ]
}