One wall threshold controls all three core cone errors

import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.Real.Sqrt

namespace BianchiIX
end BianchiIX

noncomputable section
open BianchiIX

theorem BianchiIX.exists_core_cone_threshold {R A B : ℝ} (hR : 0 ≤ R) (hA : 0 ≤ A) (hB : 0 ≤ B) :
    ∃ ε > 0, ε ≤ 1 / 10000 ∧ 7 * R * Real.sqrt ε ≤ 1 / 32 ∧
      38400 * A * Real.sqrt ε ≤ 1 / 32 ∧ 3 * B * Real.sqrt ε ≤ 1 / 32 := by sorry