{
  "proof_locators": [
    "RA.1",
    "RA.5",
    "RA.10",
    "RA.54",
    "RA.55"
  ],
  "human_source": "https://www.math.utah.edu/~milicic/Eprints/book.pdf",
  "human_locators": "Chapter2 section2; Chapter5 section1; convention comparison only",
  "group": "SL3; actual dominant lambda=omega1",
  "exact_cartan_coordinates": "(zeta1,zeta2,zeta3), sum=0",
  "render_coordinates": "x=(zeta1-zeta2)/sqrt(2), y=(zeta1+zeta2-2*zeta3)/sqrt(6)",
  "rho": [
    "1",
    "0",
    "-1"
  ],
  "lambda": [
    "2/3",
    "-1/3",
    "-1/3"
  ],
  "gamma": [
    "5/3",
    "-1/3",
    "-4/3"
  ],
  "minus_gamma": [
    "-5/3",
    "1/3",
    "4/3"
  ],
  "gamma_orbit": [
    [
      "-4/3",
      "-1/3",
      "5/3"
    ],
    [
      "-4/3",
      "5/3",
      "-1/3"
    ],
    [
      "-1/3",
      "-4/3",
      "5/3"
    ],
    [
      "-1/3",
      "5/3",
      "-4/3"
    ],
    [
      "5/3",
      "-4/3",
      "-1/3"
    ],
    [
      "5/3",
      "-1/3",
      "-4/3"
    ]
  ],
  "minus_gamma_orbit": [
    [
      "-5/3",
      "1/3",
      "4/3"
    ],
    [
      "-5/3",
      "4/3",
      "1/3"
    ],
    [
      "1/3",
      "-5/3",
      "4/3"
    ],
    [
      "1/3",
      "4/3",
      "-5/3"
    ],
    [
      "4/3",
      "-5/3",
      "1/3"
    ],
    [
      "4/3",
      "1/3",
      "-5/3"
    ]
  ],
  "cubic_gamma": "20/27",
  "cubic_minus_gamma": "-20/27",
  "w": "s1*s2; apply s2 first; w(a1,a2,a3)=(a3,a1,a2)",
  "Sigma_w": [
    "alpha1",
    "alpha1+alpha2"
  ],
  "Gamma_w": [
    "-alpha2"
  ],
  "mu0": [
    "0",
    "-1",
    "1"
  ],
  "line_fibre_minus_wlambda": [
    "1/3",
    "-2/3",
    "1/3"
  ],
  "mulambda": [
    "1/3",
    "-5/3",
    "4/3"
  ],
  "incorrect_mu0_minus_lambda": [
    "-2/3",
    "-2/3",
    "4/3"
  ],
  "PBW_negative_directions": [
    [
      "-1",
      "1",
      "0"
    ],
    [
      "0",
      "-1",
      "1"
    ],
    [
      "-1",
      "0",
      "1"
    ]
  ],
  "translation_hypothesis": "Every dominant actual lambda in X(T), including zero and non-ample cases",
  "forward": {
    "finite_module": "F_{-w0 lambda}",
    "source_parameter": "tau0=-rho",
    "selected_weight": "-lambda",
    "target_parameter": "tau0-lambda=-gamma",
    "sheaf_map": "p_A:(A tensor Fprime)^[chi_-gamma] -> L tensor A",
    "endpoint": "E.5/E.20; regularity belongs to original tau0"
  },
  "reverse": {
    "finite_module": "F_lambda",
    "source_parameter": "tau0-lambda=-gamma",
    "selected_weight": "lambda",
    "target_parameter": "tau0=-rho",
    "sheaf_map": "k_B:L^-1 tensor B -> (B tensor F)^[chi_-rho]",
    "endpoint": "E.4/E.18; target comparison uses tau=tau0"
  },
  "projector_linearity": "C-linear and U-linear; no O_X-linearity assertion",
  "factor_structure": "shifted TDO action transported through the selected endpoint isomorphism",
  "geometric_inverse_maps": "RA.42; BB counits then inverse-line evaluation then inverse BB unit",
  "scope_exclusions": "generalized, singular, parabolic, KL, affine, critical, factorization and analytic claims"
}