{
  "enlarged_module": {
    "object": "H^1(Q, (1/X)V_c(f,g)^*) — the 1/X-enlarged Beilinson-Flach family module",
    "ring": "O(V_1 x V_2 x W)",
    "rank": 1,
    "role": "where kappa_d(f,g) is constructed; NOT the ambient of the quotient D uses"
  },
  "tau_quotient": {
    "object": "the tau-quotient (1/X)V_c(f,g)^*/V(f,g)^* = Q_p(tau^-1)(1+r) (x) V(g)^* (x) H_Gamma(-j)",
    "what_happens_there": "C1.6: the image of kappa_d is ZERO",
    "role": "the direction the class DIES in — the reason it descends into the ambient, not the transverse quotient"
  },
  "H": {
    "object": "H^1(Q, V(f,g)^*) — the Beilinson-Flach FAMILY module itself",
    "ring": "O(V_1 x V_2 x W)",
    "rank": 1,
    "role": "the AMBIENT. Not the transverse quotient.",
    "why_the_class_is_here": "C1.6 kills the tau-quotient image, so kappa_d descends; C1.7: that lift is UNIQUE",
    "source": "src99's type record for L1 (RUN-097); LR C1.6, C1.7"
  },
  "L": {
    "object": "ker( H -> the psi^{-1}-quotient ) — the image of H^1(Q, V(g)^* (x) H_Gamma(-j)) under the inclusion 1 -> V(f_beta)^*",
    "why_the_1_part_is_the_sub": "dualising RUN-093's ordering 0 -> psi -> V(f_beta) -> 1 -> 0 gives 0 -> 1 -> V(f_beta)^* -> psi^-1 -> 0, so the 1-part is a SUB of V(f,g)^* and psi^-1 is its quotient",
    "role": "the reference subobject L of PC-001/Round 030 — the direction the leading term is taken transverse to"
  },
  "Q": {
    "object": "H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j))",
    "ring": "H_Gamma",
    "rank": 1,
    "freeness": "free of rank 1 over H_Gamma (Prop C1.1's proof)",
    "why_this_is_the_quotient": "C1.7: the unique lift's image in the psi^{-1}-quotient is the object that is taken",
    "role": "D's DOMAIN"
  },
  "first_nonzero_projective_class": {
    "object": "X^n kappa-hat_d(f,g) mod X",
    "assertion": "well-defined and NON-ZERO modulo X",
    "source": "LR, the construction inside C1.13 (quoted in the tree's EXTRACTS)",
    "label": "CITED_THEOREM",
    "which_n": "the weight-jet order n of C1.10, equal to Round 030's m (RUN-098's Psi_2 localisation at X^n)",
    "is_the_first_nonzero_graded_class": true
  },
  "round_030_language": "ambient H, reference subobject L, quotient Q = H/L, the first nonzero projective class",
  "Q_is_the_quotient_of_H_by_L": true,
  "the_class_survives_in": "the psi^{-1}-quotient",
  "the_class_dies_in": "the tau-quotient",
  "quotients_the_right_way_round": true,
  "whole_BF_family_line_called_the_transverse_quotient": false,
  "H_is_not_Q": true,
  "H_is_not_the_enlarged_module": true,
  "the_leading_object_sent_into_Q": {
    "object": "X^n kappa-hat_d(f,g) mod X",
    "in": "H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j))",
    "nonzero": true,
    "by": "LR's own assertion, lifted to a class statement by Prop C1.1 (C1.13's proof)"
  },
  "two_variables_kept_apart": {
    "weight direction X": "where the leading-term extraction happens, order n",
    "cyclotomic direction t": "where the target A_p lives; its vanishing order is E's"
  },
  "agrees": true
}