{
  "round": "RUN-098",
  "gate": "src100",
  "R0.5": "C",
  "why": "localization at explicit divisors: L_p(Ad g) (inverted), L_p(g_E x psi), X^n, log^{[r+1]}; the composite's residual is exactly the named extractions",
  "divisors": [
    "L_p(Ad g) (inverted)",
    "L_p(g_E x psi)",
    "X^n",
    "log^{[r+1]}"
  ],
  "correction_to_RUN_097": "'R0 PASS over H_Gamma-lines' -> 'R0 PASS over H_Gamma-lines localized at X^n and at the divisor of L_p^g, the composite's residual log^{[r+1]}/X^n being the two named extractions'",
  "composite": {
    "cancel_exactly": [
      "L_p(g x psi)",
      "L_p(g x tau)",
      "L_p(Ad g)",
      "C_d"
    ],
    "residual": "log^{[r+1]} / X^n",
    "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)"
    }
  },
  "arrows": [
    {
      "name": "Psi_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)",
      "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",
      "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",
      "scalar": "C * C_d * log^{[r+1]} * L_p(g x tau) / L_p(Ad g)",
      "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",
      "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"
    }
  ],
  "data": {
    "C_d": {
      "value": 26,
      "status": "unit",
      "why": "26 mod 11 = 4"
    },
    "L_p(g_E x psi) = L_E": {
      "first_coefficients_mod_p": [
        0,
        0,
        2,
        2,
        2,
        0,
        5,
        6,
        10,
        7,
        0
      ],
      "order_at_t_0": 2,
      "status": "non-zero-divisor, non-unit"
    },
    "L_p(g_0 x psi) = L_0": {
      "first_coefficients_mod_p": [
        9,
        3,
        10,
        3,
        1,
        6,
        9,
        6,
        2,
        0,
        6
      ],
      "order_at_t_0": 0,
      "status": "unit"
    },
    "L_p(g_E x tau) = lambda_E": {
      "first_coefficients_mod_p": [
        5,
        9,
        10,
        6,
        6,
        2,
        0,
        8,
        8,
        9,
        10
      ],
      "order_at_t_0": 0,
      "status": "unit"
    },
    "L_p(g_0 x tau) = lambda_0": {
      "first_coefficients_mod_p": [
        4,
        3,
        10,
        9,
        4,
        10,
        2,
        7,
        3,
        8,
        3
      ],
      "order_at_t_0": 0,
      "status": "unit"
    },
    "E_adj": {
      "E": 3,
      "0": 7,
      "status": "units (nonzero mod 11)"
    },
    "Pi_i": {
      "status": "unit",
      "source": "RUN-091 Attack B (DERIVED); value unknown"
    },
    "c_f": {
      "form": "X^n * c_f*",
      "c_f*_status": "unit (C1.13: c*_f(r) in L^x)",
      "n": null,
      "status": "unit iff n = 0; n unknown (C1.10)"
    },
    "log^{[r+1]}": {
      "status": "non-zero-divisor, non-unit",
      "why": "zeros at the characters of degree <= r"
    }
  },
  "C5": {
    "divisors_accounted": {
      "L_p(g x psi)": true,
      "L_p(g x tau)": true,
      "L_p(Ad g)": true,
      "C_d": true,
      "log^{[r+1]}": true,
      "X^n": "the weight-jet order m = n of Round 030 (14.1), not a delta-factor"
    },
    "new_L_factors": [],
    "verdict": "PASS kept"
  },
  "positive_control": {
    "input": {
      "scalar_status": "unit",
      "module": "free rank 1",
      "expected": "isomorphism, inverse over the base ring"
    },
    "classified": {
      "injective": true,
      "surjective": true,
      "cokernel": "0",
      "inverse": "over the base ring",
      "status": "unit"
    },
    "accepted": true
  },
  "claim_labels": {
    "COMPUTED_HERE": [
      "C_d a unit mod 11",
      "the unit / non-unit status of L_E, L_0, lambda_E, lambda_0 from their constant terms (RUN-081/090 series)",
      "the composite's exponents and residual"
    ],
    "CITED_THEOREM": [
      "LR Eq. 11 (the factorization of L_p^g)",
      "C1.8 (division by log)",
      "C1.10 (n), C1.13 (c*_f(r) in L^x)"
    ],
    "DERIVED_FROM_VERIFIED_INPUTS": [
      "Pi_i an 11-unit (RUN-091)",
      "RUN-097's slot table"
    ],
    "UNMEASURED": [
      "n (C1.10)",
      "the value of Pi_i",
      "the values of eta_pdR, kappa_det"
    ],
    "FAILED": []
  },
  "status": {
    "C1-C4": "PASS",
    "R0": "PASS (category corrected: C)",
    "C5": "PASS (kept; no new L-factor)",
    "C8": "PASS",
    "C6, C7, C9, C10": "UNMEASURED",
    "BSD": "OPEN"
  }
}