{
  "generators_on_curve": true,
  "formal_depth": {
    "P": {
      "multiplier": 16,
      "v_11_of_x": -2,
      "in_E_1": true,
      "depth_j": 1,
      "unit_t_over_11j_mod_11": 9
    },
    "Q": {
      "multiplier": 16,
      "v_11_of_x": -2,
      "in_E_1": true,
      "depth_j": 1,
      "unit_t_over_11j_mod_11": 6
    }
  },
  "additivity_check": true,
  "local": {
    "E(Q_11)[11^inf] = 0": {
      "holds": true,
      "why": "#E~(F_11) = 16, 11 ∤ 16; E_1(Q_11) torsion-free since e = 1 < 10"
    },
    "E(Q_389)[11^inf] = 0": {
      "holds": true,
      "why": "c_389 = 1 (RUN-067), E_0/E_1 = F_389^x of order 388 = 4·97, E_1 torsion-free"
    },
    "H^1(Q_389, T) = 0": {
      "holds": true,
      "why": "rank 0 (l ≠ p) and H^1(Q_389,T)_tor = E(Q_389)[11^inf] = 0; also H^1_ur = Z_11(1)/(389 - 1) = 0 since 11 ∤ 388"
    }
  },
  "poitou_tate": {
    "(2.13) at alpha = 1": "0 → H^1_rel(Q,T) → H^1(Z_S,T) → ⊕_{l|N} H^1(Q_l,T)/H^1_f → Sel_str(E[11^inf])^∨ → H^2(Z_S,T) → ⊕_{l|Np} H^2(Q_l,T) → H^0(Q,T^*)^∨ → 0",
    "H^1(Q_389,T)/H^1_f": "0 (CS: H^1_f = H^1_tor = H^1 for l | N), so H^1(Z_S,T) = H^1_rel(Q,T)",
    "H^2(Q_l,T) = H^0(Q_l,T^*)^∨ = E(Q_l)[11^inf]^∨": "0 at 11 and at 389 (computed above)",
    "H^0(Q,T^*)": "0 by (tor_Q)",
    "consequence": "H^2(Z_S,T) ≅ Sel_str(E[11^inf])^∨ canonically, where Sel_str = ker(Sel_{11^inf}(E/Q) → E(Q_11) (x) Q_11/Z_11)"
  },
  "euler_characteristic": "rank H^1(Z_S,T) - rank H^2(Z_S,T) = 1 (Tate: -rank T^- = -1 with H^0 = 0)",
  "loc_11": {
    "image_of_E(Q)(x)Z_11_in_E(Q_11)(x)^Z_11": "11^0 Z_11 (depth of 16P, 16Q in the formal group: 1, 1)",
    "surjective_onto_E(Q_11)(x)Q_11/Z_11": true,
    "u_P": 9,
    "u_Q": 6,
    "rank_of_loc_11_mod_11_on_E(Q)/11": 1,
    "strict_selmer_line_mod_11": "F_11 (P + 4 Q)",
    "strict_slope": 4,
    "index_exponent_k": 0
  },
  "conditional_on_inherited_Sha": {
    "input": "INHERITED - Sha(E/Q)[11] = 0 is carried by the P5 documents (RUN-011, RUN-022) and has never been verified by this line",
    "corank_Sel_{11^inf}(E/Q)": "2 + corank Sha[11^inf] = 2 under the inherited input",
    "corank_Sel_str": "corank Sel - 1 = 1 (loc_11 surjective onto a corank-1 divisible group)",
    "rank_H2": 1,
    "rank_H1": 2,
    "H2_torsion": "Z/11^0 = 0",
    "H1(Z_S,T) = Sel(T) = E(Q)(x)Z_11": "H^1/Sel injects into the torsion-free H^1_s(Q_11,T) with rank 0, so it is 0; Sel(T) = E(Q)(x)Z_11 once T_11(Sha) = 0",
    "cross_check_isotropy": "the image of H^1(Z_S,V) in the 2-dimensional H^1(Q_11,V) is its own annihilator (Poitou–Tate), so 1-dimensional; it contains loc_11(Sel(V)) = H^1_f(Q_11,V), which is 1-dimensional and nonzero (16P in E_1 \\ E_2); hence H^1(Z_S,V) = Sel(V), of dimension 2 - the same rank by a second route",
    "unconditional": "rank H^1 = 1 + rank H^2 and H^2 ≅ Sel_str^∨ hold without the inherited input; only the numbers 2, 1 need it"
  },
  "ranks_at_trivial_character": {
    "H1": 2,
    "H2": 1
  },
  "determinant_line": {
    "det^{-1}_{Z_11} RΓ(Z_S,T)": "det(H^1) (x) det^{-1}(H^2) (CS's convention, section 2.3)",
    "det(H^1)": "∧^2 (Z_11 P ⊕ Z_11 Q) = Z_11 · (P ∧ Q)  [degree +2 in H^1]",
    "det^{-1}(H^2)": "Hom(H^2, Z_11) = Hom(Sel_str^∨, Z_11) = T_11(Sel_str(E[11^inf])) = Sel_str(T) = Z_11 · R_str, R_str = a_Q P - a_P Q with a_P, a_Q the 11-adic units u_P, u_Q lift  [degree -1 in H^2]",
    "R_str mod 11": "F_11 (P + 4 Q)",
    "placement": "𝔷_{Q_inf}(1) is a Z_11-basis of Z_11 (P ∧ Q) (x) Sel_str(T): the rank-2 determinant tensored with the strict-Selmer line",
    "degrees": [
      2,
      -1
    ],
    "torsion_factor": "none (H^1 torsion-free by (tor_Q); H^2 torsion-free since k = 0)"
  },
  "descent": {
    "(str-fin) at alpha = 1": "FAILS: Sel_str has corank 1",
    "CS theta / Prop 2.3.1": "not available at the trivial character (kappa_1(1) would also have to be nonzero)",
    "so": "𝔷_{Q_inf}(1) has NO image in H^1(Z_S,T); its home is the line above",
    "descends_to_H1": false
  },
  "specialisation": {
    "map": "(2.14): det^{-1}_Lambda RΓ(Z_S,𝐓) (x)_{Lambda, 1} Z_11 ≅ det^{-1}_{Z_11} RΓ(Z_S,T)  [FK06 Prop 1.6.5(3)]",
    "status": "CITED (standard base change of determinants of perfect complexes)",
    "basis_to_basis": true,
    "so_nonzero": "a basis of a free rank-one Z_11-module is nonzero: Delta_A^{char 0} ≠ 0 at once"
  },
  "characteristic_zero_element_comes_from": "the main conjecture (section 2), through (2.14)",
  "scalar": "UNMEASURED - the unit epsilon with 𝔷_{Q_inf}(1) = epsilon · (P∧Q)(x)R_str is not computed; Kim's u lives in (Z_p/I_n)^x, a mod-11 quantity",
  "agrees": true
}