{
  "R": "H_Gamma",
  "localization": "R_S = H_Gamma[1/X^n, 1/L_E, 1/L_p(Ad g)] (RUN-098's S; nothing more inverted)",
  "inverted_only_temporarily": [
    "X^n",
    "L_p(g x psi)",
    "L_p(Ad g)"
  ],
  "why": {
    "X^n and L_p(g x psi) = L_E": "Psi_1^{-1} (RUN-098: injective, not surjective, inverse over H_Gamma[1/L_E][1/X^n])",
    "L_p(Ad g)": "Psi_3's fraction as written (RUN-098)"
  },
  "cancel_in_composite": [
    "L_p(g x psi)",
    "L_p(g x tau)",
    "L_p(Ad g)",
    "C_d"
  ],
  "named_extractions_remaining": {
    "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_as_written": "log^{[r+1]} / X^n",
  "final_category": "rank-1 projective R_S-modules in cohomological degree 0, both ends included",
  "K1_lives_in": "Aut(L_aux (x) R_S') = R_S'^x for the S' at which both routes invert - not needed: one route, h = id",
  "non_unit_divisors_are_not_K1_defects": [
    "L_E's zero (jet / L-slot)",
    "X^n (jet order (14.1))",
    "log^{[r+1]} (C1.8 extraction)",
    "lattice indices (L)"
  ],
  "same_localization_both_ends": true,
  "nothing_more_inverted": true,
  "agrees": true
}