{
  "relaxed_complex": "RΓ(Z_S,T) is dual to RΓ_c(Z_S,T) (H^i_c ≅ H^{3-i}(Z_S,E[11^∞])^∨): not self-dual; its self-dual partner in the picture is the ordinary complex",
  "ordinary_complex": "RΓ~_ord(Q,T) ≅ RHom(RΓ~_ord(Q,T^∨(1)), Z_11)[−3], degree reflection i ↔ 3−i: H~^1 ↔ H~^2 (Lemma 2.14(ii): H~^2 ≅ Sel_{11^∞}^∨), H~^0 ↔ H~^3",
  "actual_shift": 3,
  "is_two": false,
  "determinant_of_the_ordinary_complex": "det(H~^1) ⊗ det(H~^2)^{-1} = ∧²Sel(T) ⊗ det(Sel_{11^∞}(E/Q)^∨)^{-1} = ∧²Sel(T) ⊗ ∧²Sel(T) = D_MW,11^{⊗2} (Sel_{11^∞} = E(Q)⊗Q_11/Z_11 under the inherited input, so Sel^∨ = Sel(T)^*)",
  "where_the_square_comes_from": "the duality H~^2 ≅ (H~^1)^∨ between degrees 1 and 2 — a free-part square, the one the height pairing trivialises (Round 034 §17's second mechanism); not a torsion square",
  "sign_bookkeeping_in_the_det^{-1}_convention": {
    "degree_1 (torsion E(Q)[11^∞])": "+length",
    "degree_2 (Sha^∨ and the local pieces)": "−length",
    "degree_3 (E(Q)[11^∞]^∨)": "+length",
    "pattern": "𝒜 ∼ #tors² / #Sha — the reciprocal of BSD's #Sha/#tors²; det (not det^{-1}) carries the BSD orientation; the same parity either way"
  },
  "round_034_profile": {
    "their_profile": "H^0 = T, H^1 = M, H^2 = T^∨, centre 1, χ^𝒜 = ℓ(M) − 2ℓ(T)",
    "actual_profile": "H~^1 ⊇ T, H~^2 ⊇ M, H~^3 = T^∨, centre 3/2, χ^𝒜 = 2ℓ(T) − ℓ(M)",
    "relation": "the actual complex shifted by [1] has Round 034's profile; det(C[1]) = det(C)^{-1}, so the exponent flips sign with the shift — the whole-complex shift of Round 034 §82–83, applied globally, not per factor",
    "toy_check": {
      "T": 1,
      "M": 3,
      "chi_actual_degrees_1_2_3": -1,
      "chi_round_034_degrees_0_1_2": 1,
      "one_is_minus_the_other": true
    }
  },
  "forced": false,
  "reading": "the actual shift is 3, not 2; nothing is forced — the BSD exponent pattern is recovered by the global shift [1], with the orientation recorded",
  "agrees": true
}