{
  "rank_H0(R, T^*(1))": 1,
  "why_rank_one": "the Weil pairing makes det T = Z_11(1), on which complex conjugation acts by -1; so its eigenvalues on T are +1 and -1",
  "structures": {
    "relaxed (Kato's canonical structure F_can)": {
      "T_11^+": "T",
      "T_11^-": "0",
      "rank_T_minus": 0,
      "chi_by_Prop_2_4_ii": 1,
      "local_condition_at_11": "H^1_F(Q_11, X) = H^1(Q_11, X): everything",
      "Stark_module": "SS_1"
    },
    "ordinary (Greenberg, Example 2.1)": {
      "T_11^+": "ker(T → T_11(E~))",
      "T_11^-": "T_11(E~)",
      "rank_T_minus": 1,
      "chi_by_Prop_2_4_ii": 0,
      "local_condition_at_11": "im(H^1(Q_11, X^+) → H^1(Q_11, X))",
      "Stark_module": "SS_0 (Remark 3.14; Lemma 3.8's Heegner analogue also has 0)"
    }
  },
  "relaxed_core_rank": 1,
  "vertex_397_991": {
    "n": "397 * 991",
    "A_n": "E[11] (n = 1)",
    "dim_H1(Z_S,E[11])": 2,
    "dim_Sel_str(E[11])": 1,
    "rho": 1,
    "dim_H1_{F^n}(Q,E[11])": 3,
    "local_dims": [
      1,
      1
    ],
    "chi_by_2_2_2": 1,
    "core_vertex_certificate": {
      "free": "F_11-vector spaces",
      "local_free_rank_one": true,
      "H^1_{(F^n)^*}(Q,E[11]) = 0": true,
      "why": "the (F^n)^*-Selmer group is strict at 11 and at 397, 991, hence inside ker(loc_n on Sel_11) = 0 since det M_loc = 2 ≠ 0 (RUN-022: Sel_both = 0)",
      "conditional_on": "INHERITED - Sha(E/Q)[11] = 0 is carried by the P5 documents (RUN-011, RUN-022) and has never been verified by this line"
    }
  },
  "Hypothesis_2_17": {
    "chi(F) by (2.2.2)": 1,
    "chi(script F) by Prop 2.4(ii)": 1,
    "equal": true,
    "status": "PASS at n = 1 with the tree's vertex (conditional on the inherited Sha[11] = 0); the general existence of core vertices for F_can under (sur) is CITED (MR04 / BSS19), not read"
  },
  "is_it_two": false,
  "instruction": "not 2: the identification with a rank-2 Stark object is NOT forced; the degree and placement are stated instead (section 3)",
  "agrees": true
}