{
  "rows": [
    {
      "object": "coefficient ring",
      "paper": "Lambda = Z_p[[Gamma]], Gamma = Gal(Q_inf/Q) cyclotomic; Q(Lambda) its fraction field",
      "ours": "Z_11[[Gamma]]",
      "kind": "ring"
    },
    {
      "object": "Galois module",
      "paper": "T = T_p(E) with its cyclotomic deformation 𝐓; S = {inf} ∪ {l | Np}; Z_S = Z[1/Np]",
      "ours": "T = T_11(389.a1), 𝐓 = T (x) Lambda, S = {inf, 11, 389}, Z_S = Z[1/4279]",
      "kind": "module"
    },
    {
      "object": "complex",
      "paper": "RΓ(Z_S, 𝐓): a perfect complex of Lambda-modules, acyclic outside degrees 1 and 2 [FK06]",
      "ours": "the same complex at (389.a1, 11); Euler characteristic -rank T^- = -1",
      "kind": "perfect complex",
      "status": "CITED (perfectness: FK06 through CS)"
    },
    {
      "object": "torsion hypothesis",
      "paper": "H^2(Z_S, 𝐓) is Lambda-torsion [Kato Thm 12.4(1)]",
      "ours": "holds by Kato's theorem; a hypothesis of Conjecture 2.2.2 and the reason (2.9) exists",
      "kind": "hypothesis",
      "status": "CITED_THEOREM (Kato 12.4(1), read only through CS)"
    },
    {
      "object": "determinant line",
      "paper": "det^{-1}_Lambda RΓ(Z_S, 𝐓): an invertible Lambda-module",
      "ours": "canonical: the determinant of a perfect complex needs no choice",
      "kind": "line",
      "status": "PASS (canonical)"
    },
    {
      "object": "the map (2.9)",
      "paper": "Q(Lambda) (x) det^{-1} RΓ ≅ Q(Lambda) (x) H^1(Z_S, 𝐓), canonical, from H^2 torsion and Euler characteristic -1; Remark 2.2.1: an integral map det^{-1}RΓ → H^1 [BS21 Thm 2.18], injective iff H^2 is torsion",
      "ours": "available; its integral form is injective here (H^2 torsion), so a preimage of an element of H^1 is unique when it exists",
      "kind": "canonical isomorphism",
      "status": "PASS (canonical)"
    },
    {
      "object": "distinguished Kato element",
      "paper": "z_inf^{(S)} = 𝒫_N · z_inf in H^1(Z_S, 𝐓), 𝒫_N = ∏_{l | N} 𝒫_l, 𝒫_l = (1 - α_l l^{-1} γ_l)(1 - β_l l^{-1} γ_l)",
      "ours": "N = 389, a_389 = 1 (split multiplicative), so (α, β) = (1, 0) and 𝒫_389 = 1 - 389^{-1} gamma_389; at the trivial character 𝒫_389(1) = 388/389 with v_11 = 0 (a unit)",
      "kind": "element of H^1",
      "status": "canonical (Kato's zeta element with the S-Euler factor)"
    },
    {
      "object": "the statement",
      "paper": "there exists a Lambda-BASIS 𝔷_{Q_inf} of det^{-1}_Lambda RΓ(Z_S, 𝐓) that maps to z_inf^{(S)} under (2.9)",
      "ours": "a line-level statement (the lattice det^{-1}RΓ equals Lambda times the preimage of z^{(S)}) together with an element-level one (that preimage, unique by injectivity, is the distinguished basis)",
      "kind": "basis statement"
    }
  ],
  "levels": {
    "L1_ideal": "(2.12): char_Lambda(H^1(Z_S,𝐓)/Lambda z^{(S)}) = char_Lambda(H^2(Z_S,𝐓)) - an equality of ideals; by itself it names no element (an ideal of Lambda has as many generators as Lambda has units)",
    "L2_line": "det^{-1}_Lambda RΓ(Z_S,𝐓) = Lambda · (2.9)^{-1}(z^{(S)}) inside Q(Lambda) (x) H^1 - an equality of lattices (lines)",
    "L3_element": "𝔷_{Q_inf} := the unique preimage of z^{(S)} under the injective integral form of (2.9); it is a basis iff L2 holds",
    "Prop_2_2_3": "L1 ⟺ (L2 and L3), proved in CS through Kato93 I.2.1.5 / KS24 Prop 3.10",
    "Remark_2_2_4": "the divisibility ⊂ in (2.12) alone ⟺ 𝔷_{Q_inf} is integral (Lambda·𝔷 ⊂ det^{-1}RΓ); the reverse divisibility makes it a basis",
    "which_drive_A": [
      "L2_line",
      "L3_element"
    ],
    "which_do_not": [
      "L1_ideal alone"
    ],
    "uniqueness_needs": "injectivity of det^{-1}RΓ → H^1, i.e. H^2 torsion (Remark 2.2.1)"
  },
  "a_389_computed": 1,
  "split_multiplicative": true,
  "euler_factor_at_389": "1 - 389^{-1} gamma_389",
  "euler_factor_at_trivial_character": "388/389",
  "distinguished_element_obtained_from": "the injective map (2.9) applied to z^{(S)}: unique preimage",
  "what_the_word_canonical_covers": "the one element determined by z^{(S)} through (2.9)",
  "agrees": true
}