{
  "Hypothesis_4_1": {
    "(i) p ∤ Tam(E/Q) · ∏_{v|p} #Ẽ_v(F_v)": {
      "holds": true,
      "evidence": "Tam = c_389 = 1 (RUN-067), #Ẽ(F_11) = 16; 11 ∤ 16",
      "label": "COMPUTED_HERE"
    },
    "(ii) the image of ρ: G_Q → Aut(E[p]) contains a Cartan subgroup": {
      "holds": true,
      "evidence": "ρ̄_11 is surjective (RUN-069, six maximal-subgroup witnesses), and GL_2(F_11) contains its own Cartan subgroups",
      "label": "DERIVED_FROM_VERIFIED_INPUTS"
    }
  },
  "literal_type": {
    "domain": "S_p^{(k)} × S_p^{(k)}, where S_p(E/Q) = lim_n Sel_{p^n}(E/Q) and S_p = S^{(1)} ⊃ S^{(2)} ⊃ … ⊃ S^{(p)} is Bertolini–Darmon's filtration",
    "codomain": "Q^k = I^k/I^{k+1} — a free rank-one Z_11-module only after a topological generator of Γ is chosen; that choice is a gauge, of the same species as RUN-103's augmentation generators",
    "selmer_filtration": "(4.2.1) Sel^{(k)} = Sel_{p^n}(E/Q) ∩ 𝓘^{k-1} Sel_{p^n}(E/Q_n), via Lemma 4.4's isomorphism Sel_{p^n}(E/Q) ≅ Sel_{p^n}(E/L)^G",
    "order_k": "1 ≤ k ≤ p − 1 = 10; k = 1 is the usual p-adic height pairing",
    "order_used_here": 1,
    "order_in_range": true,
    "symmetry": "symmetric (a height pairing); Nekovář's is defined as β^{(k)}(x) ∪_k y with the k-th derived Bockstein",
    "sign": "Theorem 4.2: ⟨,⟩^BD_k = -1 · ⟨,⟩^Nek_k — the identification carries a sign",
    "integral_or_rational": "valued in Q^k, a quotient of integral ideals of Λ; neither map is a real number",
    "hypotheses": "Hypothesis 4.1 for the BD side; Remark 4.3: Nekovář's pairing needs no such hypothesis and allows any k ≥ 1"
  },
  "identification": "PASS (CITED_THEOREM: MCS Theorem 4.2, hypotheses instantiated here)",
  "identification_is_not_nondegeneracy": true,
  "declaration_matches_the_paper": {
    "range_of_k": true,
    "codomain_is_I^k/I^{k+1}": true,
    "sign": true,
    "rule": "the codomain is a quotient of augmentation ideals; rewriting it as Z_11 to meet another line's shape is the refused move"
  },
  "nondegeneracy": "UNMEASURED",
  "det_h_p": {
    "map": "det(h_p^{(k)}) : D_MW,11^{⊗2} → (Q^k)^{⊗2}",
    "target_presumed_Z_11": false,
    "nonzero": "UNMEASURED — at k = 1 this is the nonvanishing of the p-adic regulator (Schneider's conjecture), open for a rank-2 curve; Theorem 4.2 identifies two pairings and says nothing about rank or radical",
    "what_would_be_needed": "a nondegeneracy theorem for the p-adic height on Sel(T) ⊗ Q_11, or a computation of det h_p ≠ 0 to a proven precision"
  },
  "C_p": "PARTIAL",
  "agrees": true
}