{
  "round": "RUN-103",
  "gate": "src105",
  "A": "PARTIAL",
  "route": "D (derived), but not through B_N: through Kato's Kolyvagin derivative kappa_{397*991}^Kato, which the literature ties to B_N's canonical content by theorem",
  "P5_CANON_BocID": "closed NEGATIVELY as posed (B_N is not BKS's Boc_F); re-posed as the smallest missing theorem",
  "source": "kappa_{397*991}^Kato in H^1(Q, E[11]) (Kato's Euler system, Kolyvagin derivative); its canonical shadow det(loc_n) != 0",
  "theorem_identification": "Kim's semi-local theorem (nonvanishing equivalence) + Kim 3.11 (up to a unit u); no element-level identification exists in the four papers read",
  "mod_11_determinant": "nonzero: det(M_loc) = 2, delta-tilde_n != 0",
  "characteristic_zero_lift": "UNMEASURED",
  "rank_2_independence": "PASS by the determinant",
  "local_conditions": "PASS",
  "Delta_A_nonzero": {
    "mod 11": true,
    "over Q_11": "UNMEASURED"
  },
  "smallest_missing_theorem": "an element-level identification: the image of Kato's Kolyvagin/Stark system at n = 397*991 under MCS's canonical isomorphism, reduced mod 11, equals det(loc_n) with the unit u of Kim 3.11 computed - equivalently BKS Conjecture 1.1(iii) at F = Q(397*991)^(11), or a Kurihara-type formula delta_n = +-det(loc_n) * (explicit unit)",
  "typing_of_B_N": {
    "domain": "V = Sel_11(E/Q) = E(Q)/11 (Sha[11] = 0 certified; torsion trivial), basis P = (0,0), Q = (1,0) - the Mordell-Weil generators (P5 v1.2; RUN-011)",
    "codomain": "W (x) I/I^2 with W = (+)_l E(F_l) (x) F_11 = (+)_l H^1_f(Q_l, E[11]) (Kummer), basis e_l a chosen generator; E(F_397), E(F_991) cyclic of orders 374, 1045 with v_11 = 1 (RUN-067)",
    "base_ring": "A = F_11[G], G = G_397 x G_991 = C_11 x C_11 = the 11-part of Gal(Q(zeta_397 zeta_991)/Q); I its augmentation ideal; X_l = gamma_l - 1 for CHOSEN generators gamma_l",
    "cohomological_origin": "the restriction map H^1_f(Q, E[11]) -> (+)_l H^1_f(Q_l, E[11]) - canonical, no choice - composed with the isomorphism E(Q_l)(x)F_11 = E(F_l)(x)F_11 (good reduction) and then tensored with X_l",
    "connecting_morphism": "none: B_N is not the boundary of any exact triangle; the tensor with X_l is the canonical identification I_l/I_l^2 = G_l (x) F_11 evaluated on a chosen generator",
    "bases": "P, Q (Mordell-Weil, canonical up to GL_2(Z)); e_397, e_991 (generators of cyclic groups, a unit choice each); gamma_397, gamma_991 (generators of C_11, a unit choice each)",
    "verdict_hand_built_or_derived": "hand-built: the canonical localization matrix M_loc dressed with augmentation generators; not a Bockstein morphism induced from a Selmer / Euler-system complex",
    "M_loc": [
      [
        1,
        2
      ],
      [
        1,
        4
      ]
    ],
    "det_M_loc_mod_11": 2
  },
  "theorem_map": [
    {
      "theorem": "BKS 1910.07404 (2.3.2)-(2.3.3): the Bockstein regulator map Boc_F",
      "source_object": "^r H^1(Z_S, T_11(E)) over Z_11",
      "target_object": "H^1(Z_S,T) (x) ^{r-1} H^2(Z_S,T)_tf (x) I_F^{r-1}/I_F^r",
      "construction": "the connecting morphism beta_F of the exact triangle of RGamma(O_{F,S},T) (x)^L over Z_11[G]/I^2 -> Z_11: a derived Bockstein",
      "hypotheses": "p odd; F a finite real abelian extension unramified outside S with [F:Q] a p-power",
      "ours": {
        "F = Q(397*991)^(11), G = C_11 x C_11, unramified outside S = {11, 389, 397, 991, inf}": true
      },
      "identification_with_B_N": "different object: Z_11 vs F_11; augmentation degree r-1 = 1 vs r = 2; target H^1 (x) H^2_tf (x) Q^1 vs ^2 W (x) Q^2; a derived connecting morphism vs the localization map dressed with generators",
      "status": "FAIL (as an identification)"
    },
    {
      "theorem": "BKS Theorem 1.3 (order of vanishing, Conjecture 1.1(i))",
      "source_object": "N_{F/Q}(z_F) in H^1(O_{F,S},T) (x) Z_11[G]",
      "target_object": "membership in H^1 (x) I^{r-1}",
      "hypotheses": "(a) p > 3; (b) Sha(E/F)[p^inf], Sha(E/Q)[p^inf] finite; (c) image contains SL_2(Z_p); (d) E(Q_l)[p] = 0 for every l in S",
      "ours": {
        "(a)": true,
        "(b) Sha(E/F)[11^inf] finite for the degree-121 field": "UNMEASURED",
        "(c)": true,
        "(d) E(Q_l)[11] = 0 at l = 397, 991": {
          "397": false,
          "991": false
        }
      },
      "identification_with_B_N": "none; and the theorem does not apply at this F: (d) fails at exactly the Kurihara primes, whose E(F_l)[11] = Z/11 is what makes them witnesses",
      "status": "NOT-APPLICABLE"
    },
    {
      "theorem": "BKS Conjecture 1.1(iii) (Generalized Perrin-Riou): [N(z_F)] = Boc_F(eta^BSD)",
      "source_object": "eta^BSD in C_11 (x) ^r H^1(Z_S,T)",
      "target_object": "H^1 (x) I^{r-1}/I^r",
      "hypotheses": "a conjecture; proved equivalent to Perrin-Riou's in analytic rank one (Thm 1.4)",
      "ours": {
        "rank": 2
      },
      "identification_with_B_N": "would give the element-level leading term, in BKS's shape, not B_N's",
      "status": "UNMEASURED (conjectural)"
    },
    {
      "theorem": "Kim 2203.12159, the semi-local theorem (delta-tilde^(1)_n != 0 at minimal nu(n) <=> Kato's mod-p Kolyvagin system non-trivial <=> IMC; then loc_n : Sel(Q,E[p]) -> (+) E(F_l)(x)Z/p is an isomorphism; with Sha[p] = 0, rank = ord = nu(n))",
      "source_object": "delta-tilde_{397*991} in F_11 (computed: RUN-068 = 5 in the disclosed normalisation, a unit under root changes RUN-069)",
      "target_object": "loc_n an isomorphism <=> det(M_loc) != 0 (computed: 2, RUN-011/067)",
      "hypotheses": "p >= 5; rho-bar surjective; Manin constant prime to p; E(Q_p)[p] = 0; Tamagawa factors prime to p",
      "ours": {
        "p >= 5 / p > 3": true,
        "rho-bar surjective (image contains SL_2(Z_11) for p >= 5)": true,
        "Manin constant prime to 11": "EXTERNAL - LMFDB/Cremona record c = 1; not computed in this tree (RUN-069)",
        "E(Q_11)[11] = 0": true,
        "Tamagawa factors prime to 11 (c_389 = 1)": true,
        "Sha(E/Q)[11^inf] finite (= 0, Kurihara certificate R308)": true
      },
      "identification_with_B_N": "an EQUIVALENCE of nonvanishing between the Euler-system quantity and the canonical content of B_N (det loc_n) - not an identification of elements",
      "status": "PASS (for what it states)"
    },
    {
      "theorem": "Kim Theorem 3.11 = Castella-Sano Theorem 2.1.3: xi o exp*_{omega_E} o loc^s_p(kappa_n^Kato) = u * p^t * delta-tilde_n, u in (Z_p/I_n)^x, p^t = #E(Q_p)[p^inf]",
      "source_object": "kappa_{397*991}^Kato in H^1(Q, E[11]) - Kato's Kolyvagin derivative (I_n = 11 Z_11 here)",
      "target_object": "u * delta-tilde_n in F_11, t = 0 since E(Q_11)[11] = 0",
      "hypotheses": "p > 3, rho-bar surjective, Manin constant prime to p",
      "ours": {
        "p >= 5 / p > 3": true,
        "rho-bar surjective (image contains SL_2(Z_11) for p >= 5)": true,
        "Manin constant prime to 11": "EXTERNAL - LMFDB/Cremona record c = 1; not computed in this tree (RUN-069)"
      },
      "identification_with_B_N": "the derived Kato class's local image equals the Kurihara number UP TO A UNIT u the theorem does not pin; delta-tilde_n != 0 => kappa_n^Kato != 0",
      "status": "PASS (up to the unit u)"
    },
    {
      "theorem": "Castella-Sano 2601.14504 Conjecture B / Theorem A: M_inf(delta) = ord_p(Tam_E) <=> IMC",
      "source_object": "the p-divisibility invariant of the collection {delta_n}",
      "target_object": "ord_p of the Tamagawa product",
      "hypotheses": "p > 3, (sur), p not dividing the Manin constant",
      "ours": {
        "Tam_E = 1": true,
        "IMC_11": "CLOSED (Burungale-Castella-Skinner), cited, external - not checked by this line (RUN-021)"
      },
      "identification_with_B_N": "none: a valuation statement, not an identification",
      "status": "NOT-APPLICABLE"
    },
    {
      "theorem": "Macias Castillo-Sano 2603.23978: det of a Selmer complex canonically isomorphic to the module of Stark systems; Nekovar's Selmer complexes = Poitou-Tate complexes; BD and Nekovar derived heights coincide",
      "source_object": "det_R RGamma_f(Q, T) (Nekovar Selmer complex)",
      "target_object": "the module of Stark systems of rank r",
      "hypotheses": "mild (per the abstract); the statements are canonical isomorphisms of modules",
      "ours": {
        "read this round": "abstract and contents only; no statement in it specialises to a finite mod-p matrix"
      },
      "identification_with_B_N": "the right framework for a canonical determinant, but no theorem there identifies det(loc_n) (x) X with the image of Kato's system",
      "status": "UNMEASURED"
    }
  ],
  "kato_chain": [
    {
      "step": "Kato's zeta element z_{Q(zeta_n)}",
      "source": "Kato [Kat04]; Kim section 2.3",
      "status": "canonical"
    },
    {
      "step": "Kolyvagin derivative D_n z -> kappa_n^Kato in H^1(Q, E[11]), n = 397*991",
      "source": "Kim section 2.3 (Mazur-Rubin's construction)",
      "status": "canonical"
    },
    {
      "step": "res_p then exp*_{omega_E}: kappa_n^Kato -> u * delta-tilde_n",
      "source": "Kim Thm 3.11 / CS Thm 2.1.3",
      "status": "up to the unit u"
    },
    {
      "step": "delta-tilde_n != 0 (computed)",
      "source": "RUN-068 (src70): 5 in the disclosed normalisation; every unit reachable by root choice (RUN-069)",
      "status": "COMPUTED"
    },
    {
      "step": "=> kappa_n^Kato != 0, and loc_n : Sel(Q,E[11]) -> (+) E(F_l)(x)F_11 is an isomorphism",
      "source": "Kim's semi-local theorem",
      "status": "theorem, hypotheses instantiated"
    },
    {
      "step": "<=> det(M_loc) != 0 (computed: 2)",
      "source": "RUN-011 / RUN-067 (src12, src69)",
      "status": "COMPUTED"
    },
    {
      "step": "= the canonical content of B_N: [det loc_n] in P(det)",
      "source": "this round",
      "status": "identification of LINES; the coefficient is a gauge"
    }
  ],
  "char0": {
    "integral_lattice": "E(Q) (x) Z_11 is free of rank 2 (Sha[11^inf] = 0, torsion trivial)",
    "integral_target": "(+)_l E(F_l) (x) Z_11 = (+)_l Z/11 - TORSION: loc over Z_11 has no characteristic-zero determinant",
    "so": "B_N has no integral lift; a mod-11 nonvanishing of det(loc_n) is a statement about E(Q)/11 only",
    "char0_source_that_exists": "the Z_11-adic Kolyvagin system {kappa_n} (Kim section 2.5), non-trivial <=> IMC_11 (Kim); IMC_11 here: CLOSED (Burungale-Castella-Skinner), cited, external - not checked by this line (RUN-021)",
    "char0_element_that_would_be_canonical": "BKS's eta^BSD / N(z_F) leading term - conjectural (GPR)"
  },
  "controls": {
    "A": {
      "matrix": [
        [
          3,
          1
        ],
        [
          7,
          2
        ]
      ],
      "det": -1,
      "det_mod_11": 10,
      "accepted_for_char0_nonvanishing": true
    },
    "B": {
      "B": [
        [
          1,
          2
        ],
        [
          1,
          4
        ]
      ],
      "U": [
        [
          2,
          1
        ],
        [
          1,
          1
        ]
      ],
      "V": [
        [
          1,
          3
        ],
        [
          0,
          1
        ]
      ],
      "B_prime": [
        [
          3,
          17
        ],
        [
          2,
          12
        ]
      ],
      "det_B": 2,
      "det_B_prime": 2,
      "same_determinant_line": true,
      "accepted": true
    }
  },
  "not_claimed": [
    "B_N = the BKS / Nekovar Bockstein regulator (it is not)",
    "Delta_A != 0 over Q_11",
    "IMC_11 verified here (cited, external)",
    "the value of the unit u in Kim 3.11",
    "any of B-G"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "det(M_loc) and the coefficient sweep under generator changes",
      "the integral controls",
      "E(Q_l)[11] != 0 at 397, 991 from the point counts"
    ],
    "CITED_THEOREM": [
      "BKS (2.3.2)-(2.3.3), Conj 1.1, Thm 1.3",
      "Kim Thm 3.11, the semi-local theorem",
      "CS Thm A, Thm 2.1.3",
      "MCS abstract",
      "IMC_11 (Burungale-Castella-Skinner, external)"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "RUN-011/067's M_loc",
      "RUN-068's Kurihara number",
      "RUN-069's hypotheses and sweep"
    ],
    "UNMEASURED": [
      "B_N = the BKS / Nekovar Bockstein regulator (it is not)",
      "Delta_A != 0 over Q_11",
      "IMC_11 verified here (cited, external)",
      "the value of the unit u in Kim 3.11",
      "any of B-G"
    ],
    "FAILED": [
      "B_N = Boc_BKS, as an identification"
    ]
  },
  "status": {
    "normalization_certificate": "CLOSED (architecture), frozen",
    "DAG": {
      "A": "PARTIAL",
      "B-G": "untouched"
    },
    "BSD": "OPEN"
  }
}