The quadratic critical value has two local preimages

import Definitions.Def_ErdosProblems_Erdos1041_Counterexample_Defs
import Definitions.Def_ErdosProblems_Erdos1041_Counterexample_Bottleneck
import Mathlib
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.EMetricSpace.BoundedVariation

open Erdos1041
open Erdos1041.Counterexample
noncomputable section
open Topology

open Erdos1041.Counterexample

theorem Erdos1041.Counterexample.bottleneck_two_local_preimages (p : Polynomial ℂ) (cc : ℂ)
    (hcrit : (Polynomial.derivative p).IsRoot cc)
    (aHat : ℂ) (haHat : aHat ≠ 0) (h s δ : ℝ) (hh : 0 < h) (hs : 0 < s)
    (hsh : 5 / 4 * s < h) (hsδ : ‖aHat‖ * s ^ 2 ≤ δ) (hδ : δ = 1 - ‖p.eval cc‖)
    (hdisk : ∀ z : ℂ, ‖z‖ ≤ h → ‖(shiftQuad p cc).eval z / aHat - 1‖ ≤ 1 / 4)
    (η : ℂ) (hη : η ≠ 0) (hηs : ‖η‖ < s) :
    ∃ x₁ ∈ connectedComponentIn (Omega p) cc, ∃ x₂ ∈ connectedComponentIn (Omega p) cc,
      x₁ ≠ x₂ ∧ p.eval x₁ = p.eval cc + aHat * η ^ 2 ∧
        p.eval x₂ = p.eval cc + aHat * η ^ 2 ∧
        ‖x₁ - cc‖ ≤ 5 / 4 * ‖η‖ ∧ ‖x₂ - cc‖ ≤ 5 / 4 * ‖η‖ := by sorry