{
  "objects": {
    "K_aux": {
      "is": "C_aux = D_cris(F+ D(f)^*(-1-k)) framed by the unique omega_f (RUN-097 L0)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_mid": {
      "is": "the Beilinson-Flach family line in H^1(Q, V(f,g)^* (x) H_Gamma) (L1)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_lead": {
      "is": "its leading line at X = 0, image in H^1(Q, V(g)^*(psi^{-1}) (x) H_Gamma(-j)) (L2)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_kato": {
      "is": "the Kato line: H^1(Q, V(g)^*(psi^{-1}) (x) H_Gamma(-j)), free of rank 1 (Prop C1.1), generated by kappa(g x psi) (L3)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    },
    "K_arith": {
      "is": "C_arith = the L_p(g x psi) line inside H_Gamma (L4)",
      "kind": "rank-1 projective module",
      "degree": 0,
      "det_exponent": 1,
      "det": "det_R(L) = (L, 1): the identity on lines (Knudsen-Mumford)"
    }
  },
  "all_rank_one_modules_no_complex": true,
  "same_degree_on_both_sides": true,
  "det_exponent_matches_parity": true,
  "no_hidden_shift": "the route never passes through det R-Gamma; read off it, H^1 would carry exponent -1 on BOTH sides (BF's H^1 and Kato's H^1 are both degree-1 cohomology): the same parity, so no shift can hide",
  "agrees": true
}