{
  "items": {
    "LR C1.9": {
      "statement": "the Coleman-map image of the Beilinson-Flach class factors with a factor c_f(k), a meromorphic function on V_1 ALONE, regular and non-vanishing at every integer weight k >= -1 EXCEPT possibly k = r, where it is still regular",
      "what_D_takes": "the factorisation and the partner-independence of c_f",
      "what_D_may_NOT_take": "the non-vanishing at our point: k = r is exactly the weight the clause excepts",
      "covers_k_equals_r": false,
      "caveat_source": "src97 (RUN-095), field c19_nonvanishing_excepts_k_equals_r",
      "hypotheses_discharged_at": "RUN-094/095"
    },
    "LR C1.13": {
      "statement": "the critical degeneration: the specialisation of the BF class at f_beta equals [C C_d log^{[r+1]} L_p(g x tau)/L_p(Ad g)] times Kato's class, with C = c^*_f(r) in L^x",
      "what_D_takes": "c^*_f(r) in L^x IS the nonvanishing at our point, asserted by the paper in the construction; and X^n kappa-hat_d is well-defined and nonzero modulo X",
      "proof_shape": "equal regulator images, then equality of CLASSES by Proposition C1.1 — scalar-first, lifted",
      "hypotheses_discharged_at": "RUN-094 (kappa_d exists, C1.4 with (d,6S) = 1; f_beta non-critical) and RUN-095 (g noble, ordinary, residually irreducible)"
    },
    "LR Prop C1.1": {
      "statement": "the Greenberg Selmer group ker( H^1(Q, V(g)^*(chi)(n) (x) H_Gamma(-j)) -> H^1(Q_p, F^- V(g)^*(chi)(n) (x) H_Gamma(-j)) ) VANISHES",
      "proof_content_D_uses": "at each classical specialisation H^1(Q, V(g_l)^* (x) H_Gamma(-j)) is FREE OF RANK 1 over H_Gamma and contains Kato's Euler system as a canonical element, whose localisation at p maps under the Perrin-Riou regulator for F^- V(g_l)^* to the p-adic L-function, which is NOT A ZERO-DIVISOR",
      "the_statement_and_the_proof_give_different_things": "the STATEMENT gives injectivity of loc_p (its kernel is the Greenberg Selmer group, which vanishes); the PROOF gives rank one and the non-zero-divisor image, hence the regulator's nonvanishing. D-NV2 needs both.",
      "what_D_takes": "the regulator is injective on a rank-one module — i.e. D-NV2, and the lifting of C1.13 to classes",
      "what_D_may_NOT_take": "surjectivity, or an integral isomorphism",
      "surjectivity_claimed": false,
      "partner_hypothesis_is_only": [
        "ordinary"
      ],
      "over_strong_additions": [],
      "hypotheses_discharged_at": "RUN-095 (ordinary on both curves, computed); residual irreducibility needed by C1.13's Kato class, discharged by semistability + Mazur"
    },
    "Kato reciprocity": {
      "statement": "localisation at p followed by the Perrin-Riou regulator for F^- V(g)^* lands on L_p(g x psi); Kato's L_p agrees with the Mazur-Tate-Teitelbaum L_p of modular symbols up to the period frame",
      "what_D_takes": "the arrow Psi_4 exists and is injective",
      "period_convention": {
        "audited_at": "RUN-080 (calibrator plus = Neron) and RUN-096 C8 (indices 2, 2)",
        "frozen_by": "C6 = PASS / canonical, eta_pdR = 1",
        "reopened_here": false
      },
      "hypotheses_discharged_at": "RUN-095 (ordinary, residually irreducible, noble, prime-to-p level, p-distinguished)"
    }
  },
  "inherited_hypothesis_list_389a1": {
    "ordinary at p (a_p not divisible by p)": true,
    "unit root mod p": 7,
    "prime-to-p level (p does not divide N)": true,
    "Hecke polynomial X^2 - a_p X + p has distinct roots (disc nonzero)": true,
    "p-distinguished (the two roots differ mod p)": true,
    "noble (prime-to-p level + distinct Hecke roots), Assumption B1.1": true,
    "non-critical slope (ordinary means slope 0 < r_1 + 1)": true,
    "Hida (ordinary), as section B2 restricts to": true,
    "newform of its level, as section B2 assumes": true,
    "weight 2, trivial nebentypus": true,
    "minimal model (Delta squarefree, and 1728 Delta = c_4^3 - c_6^2)": true,
    "semistable (no additive reduction: every prime dividing Delta fails to divide c_4)": true,
    "residually irreducible at p = 11 (semistable + Mazur: a rational l-isogeny forces l <= 7)": true
  },
  "all_inherited_hold": true,
  "all_hypotheses_point_back_to_an_actual_PASS": true,
  "no_new_assumption_introduced": true,
  "c1_9_not_stretched_over_k_equals_r": true,
  "c1_1_not_read_as_surjective": true,
  "normalization_not_reopened": true,
  "agrees": true
}