{
  "Hypothesis_2_11": {
    "(i) H^0(Q,A) = H^0(Q,T^∨(1)) = 0": {
      "holds": true,
      "why": "E(Q)[11^inf] = 0 (torsion trivial; T^∨(1) ≅ A by the Weil pairing)"
    },
    "(ii) H^0(Q_11, T_11^-) = 0": {
      "holds": true,
      "why": "T_11^- = 0 for the relaxed structure"
    },
    "(iii) H^0(Q_11, A_11^-) = H^0(Q_11, (T_11^+)^∨(1)) = 0": {
      "holds": true,
      "why": "A^- = 0; (T^+)^∨(1) = T^∨(1) ≅ E[11^inf] and E(Q_11)[11^inf] = 0 (11 ∤ 16, e = 1)"
    },
    "(iv) H^1_ur(Q_389, T) = H^1_f(Q_389, T)": {
      "holds": true,
      "why": "both are 0: H^1_ur = Z_11(1)/(389-1) = 0 (11 ∤ 388) and H^1(Q_389,T) = 0 (rank 0, E(Q_389)[11^inf] = 0)"
    },
    "T^{I_389} free": {
      "holds": true,
      "why": "split multiplicative: T^{I_389} = Z_11(1), rank 1, saturated"
    },
    "Remark 2.12 (for the ordinary structure, for comparison)": {
      "holds": true,
      "why": "11 ∤ #E(Q)_tors · Tam · #E~(F_11) = 1 · 1 · 16"
    }
  },
  "selmer_complex_vs_RGamma": {
    "RΓ_ur(Q_389,T)": "[T^I --(Fr-1)--> T^I] with T^I = Z_11(1), Fr = 389: H^0 = 0 (389 ≢ 1 mod 11), H^1 = Z_11/388 = 0",
    "RΓ(Q_389,T)": "H^0 = 0, H^1 = 0 (computed), H^2 = E(Q_389)[11^inf]^∨ = 0",
    "RΓ_{/ur}(Q_389,T)": "acyclic",
    "S_p-term": "RΓ(Q_11, T_11^-) = RΓ(Q_11, 0) = 0",
    "hence": "RΓ̃_{F_can}(Q,T) ≃ RΓ(G_{Q,S},T) = RΓ(Z_S,T): CS's complex and MCS's complex are the same object for the relaxed structure",
    "holds": true
  },
  "primes_in_P_1": {
    "397": {
      "#E(F_l)": 374,
      "v_11": 1,
      "dim E(Q_l)[11]": 1,
      "dim H^1(Q_l,E[11])": 2,
      "dim H^1_f": 1,
      "dim H^1_{/F}": 1,
      "in_P_1": true,
      "not_in_S_ram": true
    },
    "991": {
      "#E(F_l)": 1045,
      "v_11": 1,
      "dim E(Q_l)[11]": 1,
      "dim H^1(Q_l,E[11])": 2,
      "dim H^1_f": 1,
      "dim H^1_{/F}": 1,
      "in_P_1": true,
      "not_in_S_ram": true
    }
  },
  "composition": [
    {
      "step": "𝔷_{Q_inf}",
      "ring": "Lambda",
      "complex": "RΓ(Z_S,𝐓)",
      "structure": "relaxed at 11 (RΓ(Z_S,·) is the relaxed Selmer complex)",
      "lattice": "det^{-1}_Lambda RΓ(Z_S,𝐓)",
      "hypotheses": "Conjecture 2.2.2 = the main conjecture (section 2)",
      "status": "basis (CITED_THEOREM)"
    },
    {
      "step": "𝔷_{Q_inf}(1)",
      "ring": "Z_11",
      "complex": "RΓ(Z_S,T) = RΓ̃_{F_can}(Q,T)",
      "structure": "relaxed at 11",
      "lattice": "det^{-1}_{Z_11} RΓ(Z_S,T) = Z_11 (P∧Q) (x) Sel_str(T)",
      "hypotheses": "(2.14) base change; the acyclicity above",
      "status": "basis, nonzero"
    },
    {
      "step": "epsilon_1 = varpi_1(𝔷_{Q_inf}(1) mod 11)",
      "ring": "F_11 = Z_11/11 (n = 1)",
      "complex": "C_PT,n = [H^1_{F^n}(Q,E[11]) → ⊕_{l|n} H^1_{/F}(Q_l,E[11])] ≃ C_Nek/11 (Theorem 2.20)",
      "structure": "relaxed / SS_1",
      "lattice": "SS_1(E[11], F_can, P_1) ≅ det^{-1}(C_PT) = ∧^3 H^1_{F^n}(Q,E[11]) (x) H^1_s(Q_397,E[11])^* (x) H^1_s(Q_991,E[11])^*, free of rank one (Prop 3.3)",
      "degree": "exterior power 3 = chi + nu(n) on the 3-dimensional H^1_{F^n}; local factors of degree 1 each",
      "dual": "H^1_s(Q_l,E[11])^* ≅ H^1_f(Q_l,E[11]) = E(F_l) (x) F_11 = W_l by local Tate duality",
      "hypotheses": "Hyp 2.11 (i)-(iv) verified above; Hyp 2.17 verified at n = 1 with the vertex 397*991 (section 4); Theorem 3.4 itself CITED",
      "status": "basis, nonzero"
    }
  ],
  "structures_consistent_along_the_chain": true,
  "Delta_A": {
    "element": "epsilon_1 (and, in characteristic zero, 𝔷_{Q_inf}(1))",
    "nonzero": true,
    "why": "a basis of a free rank-one module"
  },
  "fitting_ideals": {
    "Cor 3.5": "Fitt^i_{F_11}(Sel_str(E[11])^*) = I_i(epsilon_1) with I_i = Σ_{nu(n) = i} im(epsilon_n)",
    "dim Sel_str(E[11])": 1,
    "I_0": "0",
    "I_1": "F_11 (the strict Selmer group of E[11] is one-dimensional)",
    "meaning": "ideal-level content about the STRICT Selmer group, the dual of the relaxed structure - not about the rank-2 Selmer group"
  },
  "ordinary_channel": {
    "structure": "ordinary (Greenberg), core rank 0",
    "Remark 3.14": "Mazur's main conjecture ⟺ a unique Lambda-basis 𝔷_inf^MSD ↦ 𝓛^MSD; epsilon^MSD ∈ SS_0; Fitt^i(Sel_{11^inf}(E/Q)^∨) = Theta_i^MSD",
    "at_389_a1": "Sel_{11^inf}(E/Q)^∨ ≅ Z_11^2 under the inherited input, so Theta_0 = Theta_1 = 0 and Theta_2 = Z_11: the level-2 Mazur–Tate ideal is the unit ideal",
    "tree_witness": "RUN-068's delta-tilde_{397*991} = 5 ≠ 0 mod 11 is a level-2 witness consistent with Theta_2 = Z_11 (Kurihara's delta_n generate Theta_nu(n))",
    "composed_with_the_relaxed_chain": false,
    "bridge": "Kim Thm 3.11 / the Coleman map (Kato section 17): exp* ∘ loc_p^s carries the relaxed rank-1 objects to the ordinary rank-0 numbers, up to the unit u",
    "note": "a parallel channel, read not composed"
  },
  "rank_one_merge": {
    "SS_1 identified with H^1(Z_S,𝐓), Lambda or the line of 𝓛^MSD because all have rank one": false,
    "rule": "rank one is a property, not an isomorphism; the only identifications used are (2.9), (2.14), Prop 3.3 and Theorem 3.4, each a named canonical map"
  },
  "A_mod11_compat": {
    "Stark line at n": "∧^3 H^1_{F^n}(Q,E[11]) (x) W_397 (x) W_991",
    "B_N line (RUN-103)": "(∧^2 Sel_11)^{-1} (x) W_397 (x) W_991",
    "shared": "the local factor W_397 (x) W_991",
    "different": "the global factor: degree 3 on the relaxed-at-n Selmer group versus degree -2 on the Selmer group",
    "what det(loc_n) ≠ 0 does": "certifies the core vertex (H^1_{(F^n)^*} = 0), which is what makes SS_1 at n a single line - nothing more",
    "equality_of_elements": "UNMEASURED",
    "necessary_for_A": false
  },
  "agrees": true
}