{
  "hypotheses": {
    "p > 3": {
      "holds": true,
      "label": "COMPUTED_HERE",
      "evidence": "p = 11"
    },
    "(sur) rho-bar_11 surjective": {
      "holds": true,
      "label": "DERIVED_FROM_VERIFIED_INPUTS",
      "evidence": "RUN-069 (src69/src71): six maximal-subgroup witnesses"
    },
    "good ordinary at 11": {
      "holds": true,
      "label": "COMPUTED_HERE",
      "evidence": "#E(F_11) = 16, a_11 = -4, 11 ∤ a_11"
    },
    "(tor_Q) E(Q)[11] = 0": {
      "holds": true,
      "label": "DERIVED_FROM_VERIFIED_INPUTS",
      "evidence": "RUN-011 (src12): torsion trivial; also implied by (sur)"
    },
    "H^2(Z_S, 𝐓) Lambda-torsion": {
      "holds": true,
      "label": "CITED_THEOREM",
      "evidence": "Kato Thm 12.4(1), as cited in CS section 2.2; not re-read"
    },
    "the MSD main conjecture at (389.a1, 11)": {
      "holds": true,
      "label": "CITED_THEOREM",
      "evidence": "CS section 1.1.4, proof of Theorem B: the MSD main conjecture in the good ordinary case is proved in [48, 7] = [Wan15, BCS25]; equivalent to [24, Conj. 12.10] = Kato, hence to Conjecture 2.2.2 (Proposition 2.2.3); the internal hypotheses of Wan15 and BCS25 are NOT re-read by this line (UNMEASURED beyond CS's list)"
    },
    "Kato Conj 12.10 ⟺ the MSD conjecture": {
      "holds": true,
      "label": "CITED_THEOREM",
      "evidence": "CS: 'known to be equivalent' (Kato section 17); not re-read"
    },
    "Prop 2.2.3: Conj 12.10 ⟺ Conj 2.2.2": {
      "holds": true,
      "label": "CITED_THEOREM",
      "evidence": "proved in CS section 2.2 through Kato93 I.2.1.5"
    }
  },
  "instantiable_hypotheses_all_hold": true,
  "remarks": {
    "(p ∤ c_phi)": {
      "needed_for": "Theorem A / Conjecture B (Kurihara), NOT for the main conjecture 2.2.2",
      "status": "EXTERNAL - not needed for the main conjecture; Mazur78 Cor 4.1 gives 11 ∤ c_phi for the strong Weil curve since 11^2 ∤ 389, optimality of 389.a1 being external (LMFDB)",
      "11^2 divides 389": false
    },
    "SU-type ramified prime": {
      "v_389(Delta)": 1,
      "rho-bar ramified at 389 iff 11 ∤ v_389(Delta)": true,
      "note": "the hypothesis Skinner–Urban would need; CS's Theorem B route through [Wan15, BCS25] lists only p > 3, (sur), good ordinary"
    },
    "discriminant": 389
  },
  "route": "[Wan15, BCS25] ⟹ MSD main conjecture ⟹ (Kato) Conj 12.10 ⟹ (Prop 2.2.3) Conjecture 2.2.2 ⟹ 𝔷_{Q_inf} exists and is a basis",
  "KATO_DETERMINANT_IMC": "PASS/CITED_THEOREM",
  "unmeasured_conditions": [
    "the internal hypotheses of Wan15 and BCS25",
    "Kato Thm 12.4(1) (read through CS only)",
    "optimality of 389.a1 / the Manin constant (not needed here)"
  ],
  "agrees": true
}