{
  "rows": [
    {
      "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"
    }
  ],
  "statuses": {
    "BKS 1910.07404 (2.3.2)-(2.3.3): the Bock": "FAIL (as an identification)",
    "BKS Theorem 1.3 (order of vanishing, Con": "NOT-APPLICABLE",
    "BKS Conjecture 1.1(iii) (Generalized Per": "UNMEASURED (conjectural)",
    "Kim 2203.12159, the semi-local theorem (": "PASS (for what it states)",
    "Kim Theorem 3.11 = Castella-Sano Theorem": "PASS (up to the unit u)",
    "Castella-Sano 2601.14504 Conjecture B / ": "NOT-APPLICABLE",
    "Macias Castillo-Sano 2603.23978: det of ": "UNMEASURED"
  },
  "P5_CANON_BocID": "FAIL as posed",
  "BKS_hypothesis_d_fails_at_kurihara_primes": true,
  "E_Q_l_has_11_torsion": {
    "397": true,
    "991": true
  },
  "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)",
  "agrees": true
}