{
  "C_aux": {
    "object": "D_cris(F+ D(f)^*(-1-k)), framed by the unique omega_f : -> X^{-n} O(V_1)",
    "type": {
      "category": "crystalline (D_cris of a (phi,Gamma)-submodule)",
      "ring": "O(V_1)",
      "rank": 1,
      "character": "psi^{-1} eps^{-1-r} side of the triangulation",
      "tate_twist": -1,
      "orientation": "line"
    },
    "q_blind": true,
    "why_q_blind": "omega_f is canonical (C1.10 unique; C1.11 determined by f); b_f+ is a second, non-canonical framing of the same line and q_f = b_f+(0) is its value",
    "weight_factor": {
      "round030": "Sym^m(T_X^*)",
      "LR": "X^{-n} O(V_1) (C1.10)",
      "m_equals_n": true
    },
    "frame": "omega_f (canonical); b_f+ (non-canonical, gauge u)",
    "source": "LR A8.3, C1.10, C1.11"
  },
  "extension_class_separately": {
    "object": "the extension class [Theta_aux] of 0 -> psi -> V(f_beta) -> tau eps^{-1-r} -> 0 (RUN-092..094)",
    "type": {
      "category": "etale (Galois cohomology H^1_f)",
      "ring": "Q_11",
      "rank": 1,
      "character": "chi_8",
      "tate_twist": 1,
      "orientation": "line"
    },
    "q_blind": true,
    "frame": "Kummer (canonical) / ES (non-canonical)",
    "source": "LR Thm A2.1 case (3), Cor A6.3(i)",
    "relation_to_C_aux": "the vertical (global-reducibility) line, transverse to the horizontal (triangulation) line F+ inside D(f_beta)^* with cokernels t^{r+1} (LR, diagram before A8.1); the transversality IS non-criticality (A4.5). Not the same line; the scalar between them is C4-EXACT-NORM."
  },
  "C_arith": {
    "object": "the line spanned by L_p(g x psi) in H_Gamma, constructed from modular symbols (Mazur-Tate-Teitelbaum)",
    "type": {
      "category": "H_Gamma (Iwasawa algebra / distributions)",
      "ring": "H_Gamma",
      "rank": 1,
      "character": "g x psi",
      "tate_twist": 0,
      "orientation": "line"
    },
    "q_blind": true,
    "frame": "the modular-symbol period frame Omega^+- (G) and the lattice index (L)",
    "source": "Kato [Kat04] + MTT; in-tree: src83's series (RUN-081), read back from RUN-090's log",
    "realised_in_tree": {
      "series_first_eleven_mod_11": {
        "L_E": [
          0,
          0,
          2,
          2,
          2,
          0,
          5,
          6,
          10,
          7,
          0
        ],
        "lambda_E": [
          5,
          9,
          10,
          6,
          6,
          2,
          0,
          8,
          8,
          9,
          10
        ],
        "L_0": [
          9,
          3,
          10,
          3,
          1,
          6,
          9,
          6,
          2,
          0,
          6
        ],
        "lambda_0": [
          4,
          3,
          10,
          9,
          4,
          10,
          2,
          7,
          3,
          8,
          3
        ]
      },
      "order_at_t_equals_0": {
        "L_E": 2,
        "lambda_E": 0,
        "L_0": 0,
        "lambda_0": 0
      },
      "target_ordinary_vanishes_to_order_2_(rank_2)": true,
      "twisted_series_nonzero_at_0": true,
      "nonzero_as_elements_of_H_Gamma": true,
      "source": "RUN-081 (src83) reproduced PC-002 coefficient for coefficient; RUN-090 (src92) re-read them in the gamma = 12 coordinate",
      "agrees": true
    }
  },
  "Q": {
    "object": "the quotient-transport line: the value q_f = b_f+(0) = c_r eta_f of the non-canonical framing",
    "type": {
      "category": "scalar over the coefficient field",
      "ring": "L",
      "rank": 1,
      "character": "-",
      "tate_twist": 0,
      "orientation": "line"
    },
    "q_blind": false,
    "frame": "b_f+ (gauge u)",
    "source": "LR A8.3 bad-fibre relation; PC-001 section 4; RUN-096 gauge law (1,0)"
  },
  "N_pi": {
    "object": "the late projected line: the line of B_hat_i = C_pi A_i j z_i",
    "type": {
      "category": "H_Gamma",
      "ring": "H_Gamma",
      "rank": 1,
      "character": "g x psi",
      "tate_twist": 0,
      "orientation": "line"
    },
    "q_blind": true,
    "frame": "-",
    "source": "PC-001 (1) = LR C1.13 composed with the projection; RUN-096: B_hat_i gauge-invariant"
  },
  "binding_used": "omega_f-framed F+ crystalline line",
  "round030_14_1_matched": "Theta_aux in Sym^m(T_X^*) (x) C_aux  <->  omega_f : D_cris(F+ D(f)^*) -> X^{-n} O(V_1)",
  "agrees": true
}