{
  "round": "RUN-104",
  "gate": "src106",
  "A": "PASS",
  "route": "CITED_THEOREM",
  "sub_statuses": {
    "kato_determinant_imc": "PASS/CITED_THEOREM",
    "canonical_determinant_line": "PASS",
    "distinguished_element": "PASS (unique by Remark 2.2.1, a basis, nonzero)",
    "core_rank": 1,
    "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]; conditional on the inherited Sha[11] = 0 for the numbers, unconditional in shape)",
    "stark_image": "PASS (basis of SS_1(E[11], F_can, P_1); Theorem 3.4 CITED, its hypotheses verified)",
    "A_char0": "PASS/CITED_THEOREM",
    "A_mod11_compat": "UNMEASURED"
  },
  "the_element": "Delta_A^{char 0} = 𝔷_{Q_inf}(1) ∈ det^{-1}_{Z_11} RΓ(Z_S,T) = Z_11 (P∧Q) (x) Sel_str(T); Delta_A^{mod 11} = epsilon_1 ∈ SS_1(E[11], F_can)",
  "outcome_read": "A PASS by cited theorem: the determinantal Kato main conjecture holds at (389.a1, 11) by [Wan15, BCS25] through CS Thm B and Prop 2.2.3; the determinant line is canonical; the distinguished element is unique, a basis, nonzero; its placement at the trivial character is computed, not forced",
  "objects_of_Conjecture_2_2_2": [
    {
      "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)"
  },
  "main_conjecture": {
    "hypotheses": {
      "p > 3": {
        "holds": true,
        "label": "COMPUTED_HERE",
        "evidence": "p = 11"
      },
      "(sur) rho-bar_11 surjective": {
        "holds": true,
        "label": "DERIVED_FROM_VERIFIED_INPUTS",
        "evidence": "RUN-069 (src69/src71): six maximal-subgroup witnesses"
      },
      "good ordinary at 11": {
        "holds": true,
        "label": "COMPUTED_HERE",
        "evidence": "#E(F_11) = 16, a_11 = -4, 11 ∤ a_11"
      },
      "(tor_Q) E(Q)[11] = 0": {
        "holds": true,
        "label": "DERIVED_FROM_VERIFIED_INPUTS",
        "evidence": "RUN-011 (src12): torsion trivial; also implied by (sur)"
      },
      "H^2(Z_S, 𝐓) Lambda-torsion": {
        "holds": true,
        "label": "CITED_THEOREM",
        "evidence": "Kato Thm 12.4(1), as cited in CS section 2.2; not re-read"
      },
      "the MSD main conjecture at (389.a1, 11)": {
        "holds": true,
        "label": "CITED_THEOREM",
        "evidence": "CS section 1.1.4, proof of Theorem B: the MSD main conjecture in the good ordinary case is proved in [48, 7] = [Wan15, BCS25]; equivalent to [24, Conj. 12.10] = Kato, hence to Conjecture 2.2.2 (Proposition 2.2.3); the internal hypotheses of Wan15 and BCS25 are NOT re-read by this line (UNMEASURED beyond CS's list)"
      },
      "Kato Conj 12.10 ⟺ the MSD conjecture": {
        "holds": true,
        "label": "CITED_THEOREM",
        "evidence": "CS: 'known to be equivalent' (Kato section 17); not re-read"
      },
      "Prop 2.2.3: Conj 12.10 ⟺ Conj 2.2.2": {
        "holds": true,
        "label": "CITED_THEOREM",
        "evidence": "proved in CS section 2.2 through Kato93 I.2.1.5"
      }
    },
    "route": "[Wan15, BCS25] ⟹ MSD main conjecture ⟹ (Kato) Conj 12.10 ⟹ (Prop 2.2.3) Conjecture 2.2.2 ⟹ 𝔷_{Q_inf} exists and is a basis",
    "status": "PASS/CITED_THEOREM",
    "unmeasured": [
      "the internal hypotheses of Wan15 and BCS25",
      "Kato Thm 12.4(1) (read through CS only)",
      "optimality of 389.a1 / the Manin constant (not needed here)"
    ]
  },
  "trivial_character": {
    "ranks": {
      "H1": 2,
      "H2": 1
    },
    "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
    },
    "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
    },
    "conditional": {
      "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"
    }
  },
  "core_rank": {
    "relaxed": 1,
    "ordinary": 0,
    "vertex": {
      "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"
    }
  },
  "stark": {
    "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"
      }
    },
    "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"
      }
    ],
    "fitting": {
      "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"
    },
    "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
    }
  },
  "controls": {
    "A": {
      "complex": "C = [Z^2 --(2, 3)--> Z] in degrees 0, 1, standard bases",
      "H0": "Z · (3, -2)",
      "H1": "0",
      "canonical_basis_of_det^{-1}(C)": "e1 ∧ e2 (x) f^{-1}",
      "canonical_isomorphism": "det^{-1}(C) ≅ det H^0 (x) det^{-1} H^1 from 0 → H^0 → Z^2 → Z → H^1 → 0",
      "lift_of_f": [
        -1,
        1
      ],
      "det[k; s]": 1,
      "image_of_the_basis": "+(3, -2), a generator of det H^0",
      "accepted_as_a_canonical_nonzero_element": true,
      "rule": "a basis of the determinant of a perfect complex, carried by a canonical isomorphism, is accepted at once; no explicit vector is demanded"
    },
    "B": {
      "data": "only the equality of characteristic ideals (6) = (6) between Z/6 and a module with the same Fitting ideal; no map, no line",
      "generators_of_the_ideal": [
        -6,
        6
      ],
      "a_distinguished_generator_exists": false,
      "PASS_granted": false,
      "rule": "an ideal equality names no element: over Lambda the ambiguity is Lambda^x; PASS is refused until a map such as (2.9) singles one out"
    },
    "agrees": true
  },
  "not_claimed": [
    "the value of the unit epsilon (𝔷_{Q_inf}(1) = epsilon · (P∧Q) (x) R_str) - not computed",
    "epsilon_1 = [det loc_{397*991}] or any equality between the Stark line and B_N's line",
    "the main conjecture re-proved here (cited: Wan15, BCS25, Kato)",
    "Sha(E/Q)[11] = 0 verified here (inherited)",
    "the internal hypotheses of Wan15 / BCS25 / Kato 12.4(1) / MR04-BSS19 core-vertex existence",
    "the Manin constant or optimality",
    "a rank-2 Stark module (the core rank is 1)",
    "any of B, C, D, E, F, G",
    "BSD"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "#E(F_11) = 16, a_11 = -4; a_389 = +1; the discriminant 389",
      "E(Q_11)[11^inf] = E(Q_389)[11^inf] = 0",
      "the formal-group depth of 16P, 16Q: index 11^0, H^2 torsion-free, strict line F_11(P + 4Q)",
      "the core rank 1 by two formulas; the ordinary core rank 0",
      "Hypothesis 2.11 (i)-(iv); the acyclicity at 389; 397, 991 in P_1; the core-vertex conditions at 397*991"
    ],
    "CITED_THEOREM": [
      "the MSD main conjecture at (389.a1, 11): Wan15, BCS25 via CS Thm B",
      "Kato Thm 12.4(1), Conj 12.10 and its equivalence with MSD",
      "CS Prop 2.2.3, (2.9), (2.13), (2.14)",
      "MCS Prop 2.4, 2.18, 3.3, Theorems 2.20, 3.4, Cor 3.5",
      "MR04 / BSS19 core-vertex existence in general"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "RUN-069's surjectivity witnesses",
      "RUN-011's torsion and generators",
      "RUN-022's Selmer cube (det M_loc = 2, Sel_both = 0)",
      "RUN-067's point counts and c_389 = 1",
      "RUN-068's Kurihara number"
    ],
    "INHERITED": [
      "Sha(E/Q)[11] = 0"
    ],
    "UNMEASURED": [
      "the value of the unit epsilon (𝔷_{Q_inf}(1) = epsilon · (P∧Q) (x) R_str) - not computed",
      "epsilon_1 = [det loc_{397*991}] or any equality between the Stark line and B_N's line",
      "the main conjecture re-proved here (cited: Wan15, BCS25, Kato)",
      "Sha(E/Q)[11] = 0 verified here (inherited)",
      "the internal hypotheses of Wan15 / BCS25 / Kato 12.4(1) / MR04-BSS19 core-vertex existence",
      "the Manin constant or optimality",
      "a rank-2 Stark module (the core rank is 1)",
      "any of B, C, D, E, F, G",
      "BSD"
    ],
    "FAILED": []
  },
  "status": {
    "normalization_certificate": "CLOSED (architecture), frozen",
    "DAG": {
      "A": "PASS (cited-theorem route; A-mod11-compat UNMEASURED)",
      "B-G": "untouched"
    },
    "BSD": "OPEN"
  }
}