[
  {
    "name": "Psi_1",
    "base_ring": "H_Gamma (over O(V_1 x V_2))",
    "modules": "BF line -> C_aux (x) partner line (x) H_Gamma, free rank 1",
    "scalar": "c_f * C_d * L_p^g,  L_p^g = L_p(g x psi) L_p(g x tau) / L_p(Ad g)  (LR Eq. 11)",
    "factors": {
      "c_f": "unit iff n = 0; n unknown (C1.10)",
      "C_d": "unit",
      "L_p(g x psi)": {
        "target": "non-zero-divisor, non-unit",
        "calibrator": "unit"
      },
      "L_p(g x tau)": "unit (both)",
      "L_p(Ad g)": "inverted: E_i units, Pi_i unit"
    },
    "target": {
      "injective": true,
      "surjective": false,
      "cokernel": "R/(s), nonzero",
      "inverse": "on the image only; over R[1/s]",
      "status": "non-zero-divisor, non-unit"
    },
    "calibrator": {
      "injective": true,
      "surjective": false,
      "cokernel": "R/(s), nonzero",
      "inverse": "on the image only; over R[1/s]",
      "status": "non-zero-divisor, non-unit"
    },
    "divisors": [
      "X^n",
      "L_p(g_E x psi)",
      "L_p(Ad g) (inverted)"
    ],
    "note": "the family-level map; NOT the X^n-divided Psi_1*"
  },
  {
    "name": "Psi_2",
    "base_ring": "H_Gamma",
    "modules": "BF line -> BF leading line in H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j))",
    "scalar": "the projection (surjective); then the leading term: division by X^n",
    "target": {
      "injective": true,
      "surjective": false,
      "cokernel": "R/(s), nonzero",
      "inverse": "on the image only; over R[1/s]",
      "status": "non-zero-divisor, non-unit"
    },
    "calibrator": {
      "injective": true,
      "surjective": false,
      "cokernel": "R/(s), nonzero",
      "inverse": "on the image only; over R[1/s]",
      "status": "non-zero-divisor, non-unit"
    },
    "divisors": [
      "X^n"
    ],
    "note": "the ONE place the passage family -> leading term happens; a localization at X, named"
  },
  {
    "name": "Psi_3",
    "base_ring": "H_Gamma",
    "modules": "the same free rank-1 module (C1.1)",
    "scalar": "C * C_d * log^{[r+1]} * L_p(g x tau) / L_p(Ad g)",
    "factors": {
      "C": "unit",
      "C_d": "unit",
      "log": "non-zero-divisor, non-unit",
      "L_p(g x tau)": "unit",
      "1/L_p(Ad g)": "fractional"
    },
    "target": {
      "injective": true,
      "surjective": false,
      "cokernel": "not a module map over R",
      "inverse": "over the total quotient ring",
      "status": "fractional"
    },
    "calibrator": {
      "injective": true,
      "surjective": false,
      "cokernel": "not a module map over R",
      "inverse": "over the total quotient ring",
      "status": "fractional"
    },
    "divisors": [
      "log^{[r+1]}",
      "L_p(Ad g) (inverted)"
    ],
    "note": "an identity in the total quotient ring as written; integral after clearing L_p(Ad g); non-unit through log"
  },
  {
    "name": "Psi_4",
    "base_ring": "H_Gamma",
    "modules": "Kato line -> C_arith = the L_p(g x psi)-line",
    "scalar": "L_p(g x psi)",
    "target": {
      "injective": true,
      "surjective": false,
      "cokernel": "R/(s), nonzero",
      "inverse": "on the image only; over R[1/s]",
      "status": "non-zero-divisor, non-unit"
    },
    "calibrator": {
      "injective": true,
      "surjective": true,
      "cokernel": "0",
      "inverse": "over the base ring",
      "status": "unit"
    },
    "divisors": [
      "L_p(g_E x psi)"
    ],
    "note": "non-unit for the target (order 2 at t = 0), a Lambda-unit for the calibrator"
  }
]