{
  "arrows": [
    {
      "name": "Psi_2",
      "source": "H^1(Q, (1/X)V_c(f,g)^*), the 1/X-enlarged family module, rank 1 over O(V_1 x V_2 x W)",
      "target": "Q = H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j)), rank 1 over H_Gamma",
      "ring": "H_Gamma after the weight specialisation",
      "theorem": "C1.6 (the tau-quotient image is zero, so kappa_d descends into the ambient H = H^1(Q, V(f,g)^*)) + C1.7 (that lift is unique; its image in the psi^{-1}-quotient is taken); then the leading term X^n kappa-hat_d mod X",
      "role": "the ONE passage family -> leading term: the quotient that defines D's domain",
      "level": "integral over H_Gamma after the named extractions",
      "localised_at": [
        "X^n"
      ],
      "injective": true,
      "surjective": false,
      "status_target": "non-zero-divisor, non-unit",
      "status_calibrator": "non-zero-divisor, non-unit",
      "nonzero": "X^n kappa-hat_d is well-defined and non-zero modulo X (LR, C1.7 area)",
      "frame": {
        "b_f+": "none: a quotient projection of Galois modules"
      },
      "line_level": "class-level (quotient maps of Galois modules)",
      "from_log": "src99 (RUN-097) + src100 (RUN-098)"
    },
    {
      "name": "Psi_3",
      "source": "Q = the psi-quotient line (the BF leading line)",
      "target": "the Kato line, the same free rank-1 H_Gamma-module (C1.1)",
      "ring": "H_Gamma, as written an identity in the total quotient ring",
      "theorem": "C1.13: kappa-hat^*_d(f_beta,g) = [C C_d log^{[r+1]} L_p(g x tau)/L_p(Ad g)] kappa(g x psi), proved scalar-first and lifted to classes by Proposition C1.1",
      "role": "identifies D's leading object with Kato's class",
      "level": "rational: fractional as written, integral after clearing L_p(Ad g)",
      "localised_at": [
        "log^{[r+1]}",
        "L_p(Ad g) (inverted)"
      ],
      "injective": true,
      "surjective": false,
      "status_target": "fractional",
      "status_calibrator": "fractional",
      "nonzero": "both generators nonzero; the scalar C C_d L_p(g x tau)/L_p(Ad g) a non-zero-divisor",
      "frame": {
        "b_f+": "C = c*_f(r) scales by u; the class identity is frame-free"
      },
      "line_level": "scalar-first, lifted to classes by Prop C1.1",
      "from_log": "src99 (RUN-097) + src100 (RUN-098)"
    },
    {
      "name": "Psi_4",
      "source": "the Kato line",
      "target": "A_p = the H_Gamma-line spanned by L_p(g x psi)",
      "ring": "H_Gamma",
      "theorem": "Kato's explicit reciprocity law: loc_11 then the Perrin-Riou regulator for F^- V(g)^*, landing on L_p(g x psi); the Betti-period convention audited at RUN-080/096 and frozen by C6 (RUN-099)",
      "role": "THE reciprocity arrow: lambda_rec itself",
      "level": "integral over H_Gamma after the named extractions",
      "localised_at": [
        "L_p(g_E x psi)"
      ],
      "injective": true,
      "surjective": false,
      "status_target": "non-zero-divisor, non-unit",
      "status_calibrator": "unit",
      "nonzero": "injective on the line: the p-adic L-function is a non-zero-divisor (C1.1's proof); in-tree the series are nonzero",
      "frame": {
        "Omega^+-": "G (period frame)",
        "lattice index": "L (2 on both signs of the target, 2 and 1 for the calibrator)"
      },
      "line_level": "class-level; the equality of Kato's and MTT's L_p is an equality of elements of H_Gamma",
      "from_log": "src99 (RUN-097) + src100 (RUN-098)"
    }
  ],
  "family_arrow": {
    "name": "Psi_1",
    "role": "the Coleman/Perrin-Riou computation on the FAMILY line — the source of the scalars, not of lambda_rec",
    "theorem": "Perrin-Riou regulator L^PR (LZ16) on loc_p; image computed by LR C1.9",
    "image": "c_f(k) * C_d * L_p^g",
    "localised_at": [
      "X^n",
      "L_p(g_E x psi)",
      "L_p(Ad g) (inverted)"
    ],
    "note": "listed so the route is complete; D's map is Psi_4 after Psi_2 and Psi_3"
  },
  "composite": {
    "lambda_rec": "Psi_4 o Psi_3 o (the leading-term class of Psi_2)",
    "source": "Q = H/L, the psi-quotient line",
    "target": "A_p = the H_Gamma-line of L_p(g_E x psi)",
    "named_extractions": {
      "log^{[r+1]}": "C1.8 divides kappa-hat_d by it",
      "X^n": "the leading term at X = 0 (C1.7 area; C1.10's n)"
    },
    "residual": "log^{[r+1]} / X^n",
    "every_L_cancels_in_the_composite": [
      "L_p(g x psi)",
      "L_p(g x tau)",
      "L_p(Ad g)",
      "C_d"
    ]
  },
  "rebuilt_here": false,
  "read_from": [
    "src99-r0-comparison-route-realization.json",
    "src100-r05-localization-audit.json"
  ],
  "every_arrow_typed": true,
  "every_arrow_injective": true,
  "no_arrow_surjective_for_the_target": true,
  "kato_scalar_lifted_by_C1_1": true,
  "agrees": true
}