{
  "global_ranks": {
    "H1": 2,
    "H2": 1
  },
  "ordinary_ranks": {
    "H1": 2,
    "H2": 2
  },
  "local_at_11": {
    "H0(Q11,T)": 0,
    "H2(Q11,T)": 0,
    "H1(Q11,T)": 2,
    "H1_f(Q11,T)": 1,
    "H1_s(Q11,T)": 1,
    "H0(Q11,T-)": 0,
    "H2(Q11,T-)": 0,
    "H1(Q11,T-)": 1,
    "why_H0_H2_vanish": "E(Q_11)[11^∞] = 0 (11 ∤ 16, e = 1) kills H^0(Q_11,T) and H^2(Q_11,T) = H^0(Q_11,E[11^∞])^∨; Ẽ(F_11)[11^∞] = 0 (11 ∤ 16) kills H^0(Q_11,T^-) and H^2(Q_11,T^+) = H^0(Q_11,T^-⊗Q/Z)^∨; Ê[11^∞](Q_11) = 0 kills H^2(Q_11,T^-)"
  },
  "local_euler_characteristics_consistent": true,
  "triangle_long_exact_sequence": {
    "terms": [
      "H^1_ord",
      "H^1(Z_S,T)",
      "H^1(Q_11,T^-)",
      "H^2_ord",
      "H^2(Z_S,T)"
    ],
    "ranks": [
      2,
      2,
      1,
      2,
      1
    ],
    "alternating_sum": 0,
    "first_map_is_zero": true,
    "reading": "0 → H^1(Q_11,T^-) → H^2_ord → H^2(Z_S,T) → 0 is exact with ranks 1, 2, 1: the ordinary H^2 (≅ Sel^∨, rank 2) is an extension of the relaxed H^2 (≅ Sel_str^∨, rank 1) by the local singular line at 11"
  },
  "mod_11_shadow": {
    "dims": [
      2,
      2,
      1,
      2,
      1
    ],
    "alternating_sum": 0,
    "rho": 1
  },
  "kummer_audit": {
    "actual_map": "the Kummer map E(Q)⊗Z_11 → H^1(Z_S,T), the inverse limit of the connecting maps of 0 → E[11^n] → E → E → 0; as a map of complexes Λ_{E,11}[−1] → τ_{≤1}RΓ(Z_S,T) → RΓ(Z_S,T)",
    "theorem_source": "Kummer theory; RUN-104: H^1(Z_S,T) = Sel(T) (the quotient H^1/Sel injects into the torsion-free H^1_s(Q_11,T) with rank 0)",
    "integral": true,
    "injective": "yes (E(Q)[11^∞] = 0 and the free part injects)",
    "cokernel": "T_11(Sha(E/Q)) — zero under the inherited input; NUMERIC ONLY",
    "cohomological_degree": 1,
    "rational_quasi_isomorphism_in_degree_1": true,
    "rational_quasi_isomorphism_of_complexes": false,
    "why_not": "H^2(Z_S,T) ⊗ Q ≠ 0 (rank 1): Round 032's concentration hypothesis (6.1) fails; Cone(ι) ≃ H^2(Z_S,T)[−2] is a free rank-one line, not a finite module",
    "consequence": "det H^1 = D_MW,11 is a factor of the line, not the line; the line is D_MW,11 ⊗ det(H^2)^{-1} — H^2 is kept, never killed to fit Round 032"
  },
  "D_MW": "D_MW,11 = det_{Z_11}(E(Q)/tors ⊗ Z_11) = Z_11 · (P ∧ Q)",
  "arithmetic_line": "D_MW,11 ⊗ H_str, H_str := det(H^2(Z_S,T))^{-1}",
  "strict_formula": {
    "exact_sequence": "0 → Sel_str(T) → Sel(T) → H^1_f(Q_11,T) = E(Q_11)⊗̂Z_11 → 0 (onto since the index is 11^0)",
    "index_exponent_k": 0,
    "candidate_generator_mod_11": "u_Q·P − u_P·Q = 6P + 2Q",
    "loc_11_of_candidate_vanishes": true,
    "lies_on_the_strict_line": true,
    "strict_line": "F_11 (P + 4 Q)"
  },
  "second_placement": "H_str ≅ ∧²Sel(T) ⊗ H^1_f(Q_11,T)^{-1} = D_MW,11 ⊗ H^1_f(Q_11,T)^{-1}, hence 𝒜(C_arith) ≅ D_MW,11^{⊗2} ⊗ H^1_f(Q_11,T)^{-1}",
  "agrees": true
}