{
  "arithmetic": {
    "P_arith": "H^1_B(Y_1(N))[g]^+- with a chosen Q-rational basis gamma_g^+- (B2.1); in-tree: the Z-primitive modular-symbol vectors of 19.a1 (RUN-080) and 389.a1 (RUN-096, content 1) - an admissible choice (B2.3 needs Q(f)-rational)",
    "D_dR_arith": "D_dR(V(g)) with the newform differential omega_g; the tree's series use the Neron differential omega_E, and the ratio is MEASURED, not assumed",
    "epsilon_arith": "the Eichler-Shimura / comparison pairing <gamma, omega>; Omega^+- IS its value on gamma^+- (B2.1)",
    "measured_frame_data": {
      "lambda (modular-symbol period / Neron period), mod 11": {
        "19.a1 plus": 1,
        "389.a1 minus": 6
      },
      "tag": "G",
      "tree line inside the Z-primitive Manin lattice (C8)": {
        "plus": {
          "index": 2,
          "v_11": 0
        },
        "minus": {
          "index": 2,
          "v_11": 0
        }
      },
      "source": "RUN-076, RUN-080, RUN-096"
    },
    "theorems": [
      "B2.1 (periods from a chosen gamma)",
      "B2.2 (Kitagawa-Mazur, exact given gamma)",
      "B2.3 (adjoint, Q(f)-rational gamma)",
      "Kato [Kat04] Thm 16.6, cited through LR C1.13 - not re-read this session"
    ]
  },
  "auxiliary": {
    "P_aux": "the f-side line X^{-n} O(V_1)",
    "D_dR_aux": "D_cris(F^+ D(f)^*(-1-k))",
    "epsilon_aux": "omega_f (C1.10): the unique p-adic interpolation of the same comparison pairing, specialising to pairing with omega_f_k",
    "canonical": true,
    "theorems": [
      "C1.10 (unique omega_f)",
      "Remark C1.11 (determined by f)"
    ]
  },
  "same_construction": "both are the comparison pairing between a Betti class and a differential: canonical on the f-side by C1.10; on the g-side stated against a chosen gamma_g^+- that cancels along the route",
  "different_differentials_not_conflated": "omega_f (aux) and omega_g (arith) belong to different motives; omega_g and omega_E differ by the measured lambda^+-, tagged G, never set to 1",
  "the_u_asked_for": {
    "name": "q = [gamma_Kato : gamma_tree]",
    "tag": "L",
    "bucket": "q_D",
    "value": "a rational lattice index, v_11 UNMEASURED",
    "reading": "Kato's class is linear in gamma: z_{gamma_Kato} = q z_{gamma_tree}; Col(z_{gamma_tree}) is exactly the tree's series. q is a lattice index - Round 030 (10.2): magnitude; (16): the q_D bucket (lattice / finite Euler rational), not a_D (explicit convention), not eta_D"
  },
  "bucket_of_tag": {
    "G": "a_D",
    "L": "q_D",
    "E": "eta_D"
  },
  "declared_period_unit_between_Kato_and_tree": null,
  "agrees": true
}