{
  "title": "A modular gap, its density orientation and two support repairs",
  "base_rotation": "(sqrt(5)-1)/2",
  "c": "log(2)",
  "rho": "1/4 on [0,1/2), 1/2 on [1/2,1)",
  "left_coefficient": "theta(rho)(eta)=rho(eta-b mod1)",
  "epsilon": "1/20",
  "special_modular_time": "pi/log(2)",
  "on_I_right_phase": 1,
  "on_TI_wrong_left_phase": -1,
  "gap_coordinate": "-log(a)/log(2)",
  "attained_gap_points": [
    -1,
    0,
    1
  ],
  "allowed_outer_spectrum": "(-infinity,-1] union [1,infinity); not all points asserted",
  "proper_isometry": "S e_n=e_(n+1), S*S=1, SS*=1-e_00",
  "residual_initial": "w_k f; f=(w_1...w_k)*(w_1...w_k)",
  "residual_final": "f_prime w_1; f_prime=(w_1...w_k)(w_1...w_k)*",
  "proof_locators": [
    "lac-models/LACM2-LACM6",
    "lac-unitary/LAC26-LAC27",
    "lac-converse/LAC3"
  ],
  "figure_boundary": "Exact scalar density functions and spectral bounds; carrier drawing is a finite schematic of an infinite isometry, not finite-rank equality."
}