{
  "item": "C4 / C1",
  "C4": {
    "membership": "CLOSED",
    "membership_source": "RUN-093, COMPUTED_HERE",
    "nonvanishing": "CLOSED",
    "nonvanishing_source": "CITED_THEOREM",
    "nonvanishing_chain": [
      "LR Theorem A6.1 (= Bellaiche-Chenevier [BC06, Thm 1]): reducibility ideal is exactly m",
      "LR section A6.2: hence all four representations are non-split",
      "LR Corollary A6.3(i): V(f_beta) is case (3) of Theorem A2.1 — non-split and CRYSTALLINE at p",
      "case (3)'s class lies in H^1(Q, Q_11(chi_8)(1)); crystalline puts it in H^1_f; non-split makes it nonzero",
      "RUN-092: dim H^1_f = 1, spanned by kappa(epsilon)"
    ],
    "direction": "CLOSED",
    "scalar": "OPEN",
    "conclusion": "[Theta_aux] = c_ES/Kum * kappa(epsilon), c_ES/Kum in Q_11^x"
  },
  "C1": {
    "partner_independence": "CITED_THEOREM",
    "why": "C_pi = C = c*_f(r) is a functional of the auxiliary family alone: PARTNER-INDEPENDENT, and nonzero. RUN-090 carried this as certificate item C1's unproved core.",
    "hypotheses_instantiated": {
      "f_beta non-critical": true,
      "partner ordinary": true
    },
    "still_missing": "the BF classes' coordinates and any finite-level representative"
  },
  "B1_B2_existence": {
    "verdict": "PASS-EXISTENCE",
    "theorem": "LR Theorem C1.4",
    "hypothesis_on_d": {
      "d": 5,
      "condition": "d in Z_{>1} with (d, 6S) = 1",
      "gcd_with_6S": 1,
      "holds": true
    },
    "label": "CITED_THEOREM for the existence; COMPUTED_HERE for the hypothesis on d",
    "note": "the same class serves both partners: kappa_d(f, g) is built for the PAIR, so BF_E and BF_0 are the same theorem applied with g = the two curves"
  },
  "B3_B4_computability": "UNMEASURED",
  "gate": "src96 · lr-hypotheses",
  "level": "E",
  "mutation": "relabel the nonvanishing as COMPUTED_HERE, claim the scalar, add an over-strong partner hypothesis, or flip the Ext twist",
  "agrees": true
}