{
  "exact_object": "H^2(Z_S,T) ≅ H^1_{F_can*}(Q, T^∨(1))^∨ = Sel_str(E[11^∞])^∨ — the Pontryagin dual of the strict-at-11 Selmer group of the divisible module",
  "theorem": "MCS Lemma 2.14(ii) for the relaxed datum (Hyp 2.11 verified in RUN-104); independently CS (2.13) with every local H^2 vanishing (RUN-104)",
  "free_rank": 1,
  "free_rank_conditional_on": "INHERITED - Sha(E/Q)[11] = 0 carried by the programme's documents, never verified by this line; NUMERIC ONLY (unconditionally: corank Sel_str = corank Sel_{11^∞}(E/Q) − 1)",
  "torsion": "Z/11^0 = 0: the torsion of Sel_str^∨ is the cokernel of loc_11 on E(Q)⊗Z_11 → E(Q_11)⊗̂Z_11, of order 11^k with k = 0 (16P, 16Q at depth 1), plus Sha[11^∞]^∨-type torsion (inherited zero)",
  "dual_object": "Sel_str(E[11^∞]) = ker(Sel_{11^∞}(E/Q) → E(Q_11) ⊗ Q_11/Z_11); its Tate module Sel_str(T) = ker(Sel(T) → E(Q_11)⊗̂Z_11) = Z_11 · R_str",
  "local_condition_provenance": "strict at 11 (the dual of relaxed); at 389 the finite condition H^1_f = 0 = H^1 (no condition to impose); unramified elsewhere",
  "splitting": "no splitting claimed: H^2 is torsion-free here (k = 0, inherited Sha zero), so H^2 = H^2_free and det(H^2)^{-1} needs no choice",
  "H_str": "H_str := det(H^2)^{-1} = Hom(H^2, Z_11) ≅ T_11(Sel_str(E[11^∞])) = Sel_str(T) = Z_11 · R_str, R_str ≡ P + 4Q (mod 11)",
  "C_input": "H_str, a rank-one lattice inside Sel(T) — the strict/free line; trivialising it is C's derived-height question, not B's",
  "agrees": true
}