{
  "rows": [
    {
      "complex": "A",
      "name": "RΓ(Z_S,T) (Castella–Sano)",
      "ring": "Z_11",
      "local_at_11": "none (all of H^1(Q_11,T); the relaxed condition)",
      "local_at_389": "none (Z_S = Z[1/11·389]: unramified outside S is the only condition, and H^1(Q_389,T) = 0 anyway)",
      "degrees": "perfect, cohomology in [1, 2] (H^0 = T^{G_Q} = 0 by trivial torsion; H^3 = 0 for p odd)",
      "duality": "Poitou–Tate/Artin–Verdier: dual to the compactly supported complex RΓ_c(Z_S,T), H^i_c ≅ H^{3-i}(Z_S, E[11^∞])^∨ — NOT self-dual",
      "shift": "3 (i ↔ 3−i) against RΓ_c",
      "theorem": "Fukaya–Kato (perfectness); CS §2.2–2.3"
    },
    {
      "complex": "B",
      "name": "RΓ~_{F_can}(Q,T) (Macias Castillo–Sano, Nekovář; T^+ = T, T^- = 0)",
      "ring": "Z_11",
      "local_at_11": "RΓ(Q_11, T^-) = RΓ(Q_11, 0) = 0",
      "local_at_389": "RΓ_{/ur}(Q_389,T), acyclic (RUN-104: H^1_ur = Z_11(1)/388 = 0, H^1(Q_389,T) = 0, H^2 = 0)",
      "degrees": "Cone(...)[−1]: same cohomology as A",
      "duality": "dual to the strict-at-11 complex (F* = strict)",
      "shift": "3",
      "theorem": "MCS Definition 2.2; RUN-104's acyclicity: B ≃ A"
    },
    {
      "complex": "C",
      "name": "C_PT,n = [H^1_{F^n}(Q,E[11^n]) → ⊕_{l|n} H^1_{/F}(Q_l,E[11^n])] (Poitou–Tate model)",
      "ring": "Z/11^n",
      "local_at_11": "relaxed",
      "local_at_389": "H^1_f = 0",
      "degrees": "[1, 2]",
      "duality": "H^2(C_PT) = H^1_{F*}(Q,E[11^n]^*(1))^* (MCS Lemma 2.19(ii))",
      "shift": "n/a (finite model)",
      "theorem": "MCS Theorem 2.20: C_PT ≃ C_Nek/11^n under Hyp 2.11 + a core vertex (RUN-104 verified both at n = 1)"
    },
    {
      "complex": "D",
      "name": "RΓ~_ord(Q,T) (the ordinary Greenberg datum T^- = T_11(Ẽ), Example 2.1) — the self-dual hub",
      "ring": "Z_11",
      "local_at_11": "im(H^1(Q_11,T^+) → H^1(Q_11,T)) = H^1_f(Q_11,T) (ordinary; good ordinary since a_11 = -4, 11 ∤ 16)",
      "local_at_389": "RΓ_{/ur}(Q_389,T), acyclic",
      "degrees": "[1, 3] a priori; here H~^3 ≅ H~^0(Q,E[11^∞])^∨ ⊆ H^0(Q,E[11^∞])^∨ = 0, so [1, 2]",
      "duality": "self-dual: H~^2 ≅ H^1_{ord*}(Q,E[11^∞])^∨ = Sel_{11^∞}(E/Q)^∨ (Lemma 2.14(ii); Hyp 2.11 by Remark 2.12: 11 ∤ 1·1·16)",
      "shift": "3 (i ↔ 3−i): degrees 1 and 2 are dual, 0 and 3 are dual",
      "theorem": "MCS Definition 2.2 and Lemma 2.14; Nekovář's duality (Selmer complexes §6.3; the shift is the standard one and is not re-read this round)"
    }
  ],
  "same_object": "A ≃ B (RUN-104); C ≃ B/11^n (Theorem 2.20); D is a different complex, related to A by the triangle",
  "triangle": "RΓ~_ord(Q,T) → RΓ(Z_S,T) → RΓ(Q_11,T^-) ⊕ RΓ_{/ur}(Q_389,T) →  (MCS Definition 2.2, by the definition of the cone)",
  "C_arith": "RΓ(Z_S,T)",
  "verdict": "PASS: RΓ(Z_S,T) is the actual arithmetic perfect complex for B",
  "agrees": true
}