{
  "round": "RUN-099",
  "gate": "src101",
  "C6": "PASS / canonical",
  "eta_pdR": 1,
  "basis": "one construction (the comparison pairing) on both sides: omega_f canonical (C1.10); every g-side object normalised against the one chosen gamma_g^+- with exponents read off B2.2/B2.3 or forced by C1.9/C1.13 and by Kato's exact reciprocity, the two forcings agreeing; L_p^g, eta_g and Psi_4's scalar frame-independent; the g-frame cancelling in the composite (RUN-097)",
  "pairings": {
    "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",
    "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"
    }
  },
  "frame_laws": {
    "L_p(g x psi)": [
      -1,
      0
    ],
    "L_p(g x tau)": [
      0,
      -1
    ],
    "L_p(Ad g)": [
      -1,
      -1
    ],
    "L_p^g": [
      0,
      0
    ],
    "c_f": [
      0,
      0
    ],
    "Col": [
      0,
      0
    ],
    "eta_g": [
      0,
      0
    ],
    "kappa from C1.13": [
      -1,
      0
    ],
    "kappa from Kato": [
      -1,
      0
    ],
    "C1.13 RHS with Kato's kappa": [
      0,
      0
    ],
    "Psi_4 scalar": [
      0,
      0
    ]
  },
  "read_off": {
    "L_p(g x chi)": "B2.2, exponent -1 in the sign's period, sign = (-1)^j chi(-1)",
    "L_p(Ad g)": "B2.3, exponents (-1, -1)",
    "c_f": "C1.9, (0, 0)"
  },
  "forced": {
    "eta_g": "Col linear in eta_g, kappa_d canonical, Col = c_f C_d L_p^g (C1.9)",
    "kappa (i)": "C1.13's left side canonical; C, C_d, log frame-free",
    "kappa (ii)": "Kato's reciprocity Col(kappa) = L_p(g x psi), exact given gamma (B2.2)"
  },
  "slot_update": {
    "eta_pdR": {
      "tag": "E",
      "net_exponent": 1,
      "value_now": 1,
      "status": "PASS / canonical"
    },
    "id(Kato class, modular symbol)": {
      "tag_before": "E",
      "tag_after": "L",
      "bucket": "q_D",
      "net_exponent": 1,
      "value_now": "q = [gamma_Kato : gamma_tree]",
      "v_11": "UNMEASURED"
    },
    "E_slots_before": [
      "eta_pdR",
      "id(Kato class, modular symbol)",
      "id(LR class, PC-001 projected class)",
      "kappa_det"
    ],
    "E_slots_after": [
      "eta_pdR",
      "id(LR class, PC-001 projected class)",
      "kappa_det"
    ],
    "moved_E_to_L": 1,
    "moved_L_or_G_to_E": 0,
    "net_exponents_unchanged": true,
    "records": 23,
    "g_side_slots_cancel": true,
    "C5": "kept"
  },
  "positive_control_accepted": true,
  "not_claimed": [
    "v_11(q) = 0 for q = [gamma_Kato : gamma_tree] (C8 measured the tree's line inside the Manin lattice, not the Manin lattice against V_Z(g))",
    "the values of Pi_i (the gamma-choice moves L_p(Ad g), hence Pi_i)",
    "kappa_det (C7)",
    "Kato's Thm 16.6 re-read this session"
  ],
  "claim_labels": {
    "COMPUTED_HERE": [
      "the frame laws on exact instances from B2.2/B2.3 and the invariances; the agreement of the two forcings of kappa"
    ],
    "CITED_THEOREM": [
      "LR B2.1, B2.2, B2.3, Eq. 11, C1.9, C1.10, C1.13",
      "Kato [Kat04] reciprocity (through LR; not re-read)"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "lambda_0^+ = 1 (RUN-080), lambda_E^- = 6 (RUN-076), C8 indices (RUN-096)",
      "RUN-097's slot table"
    ],
    "UNMEASURED": [
      "v_11(q) = 0 for q = [gamma_Kato : gamma_tree] (C8 measured the tree's line inside the Manin lattice, not the Manin lattice against V_Z(g))",
      "the values of Pi_i (the gamma-choice moves L_p(Ad g), hence Pi_i)",
      "kappa_det (C7)",
      "Kato's Thm 16.6 re-read this session"
    ],
    "FAILED": []
  },
  "status": {
    "C1-C5": "PASS",
    "R0": "PASS (category C)",
    "C6": "PASS / canonical",
    "C7": "UNMEASURED",
    "C8": "PASS",
    "C9-C10": "UNMEASURED",
    "BSD": "OPEN"
  }
}