{
  "theorem": "LR Theorem C1.4",
  "statement": "there EXISTS a class kappa_d(f,g) in H^1(Q, (1/X) V_c(f,g)^*) interpolating the finite-level Beilinson-Flach classes _d BF[f_k, g_l] at integers (k, l) with k != r",
  "hypothesis": "d in Z_{>1} with (d, 6S) = 1",
  "d": 5,
  "d_greater_than_1": true,
  "S": [
    2,
    11,
    19,
    389
  ],
  "six_times_S": 975612,
  "gcd_d_with_6S": 1,
  "coprime": true,
  "per_prime_coprimality": {
    "2": true,
    "3": true,
    "11": true,
    "19": true,
    "389": true
  },
  "matches_pc001": "PC-001 and Attack 09 record exactly this: d = 5 with gcd(5, 6*8*389*11) = 1",
  "existence_verdict": "PASS-EXISTENCE",
  "what_it_does_not_give": "no coordinates, and no finite-level representative in hand — the theorem gives existence and an interpolation property, not a computed class",
  "label": "CITED_THEOREM for the existence; COMPUTED_HERE for the hypothesis on d",
  "agrees": true
}