Exists ring sign root perturbation

import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring

section
namespace D1Bound5


theorem _root_.D1Bound5.exists_ring_sign_root_perturbation {R : Type*} [CommRing R]
    (f : R →+* ℂ) (u : R) (t : ℂ) (c E : ℝ)
    (ht : t ≠ 0) (hc : 0 ≤ c) (hE : 0 < E)
    (herr : ‖(f u)^2-t^2‖ ≤ E) (hgap : E < (1-c^2)*‖t‖^2) :
    ∃ v : R, v^2 = u^2 ∧ (1+c)*‖t‖*‖f v-t‖ < E := by sorry



end D1Bound5
end