Exists cross Block signed scaled error

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Int.GCD
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

section
open scoped BigOperators NumberField
open Module

namespace D1Bound5


theorem _root_.D1Bound5.exists_crossBlock_signed_scaled_error
    {R : Type*} [CommRing R] (f : R →+* ℂ)
    (a b : Fin 3 → R) (M : Fin 3 → Fin 3 → R)
    (hroot : ∀ i j, (M i j)^2 = a i*b j+1)
    (hlarge : ∀ i j, 9 ≤ ‖f (a i)*f (b j)‖) :
    ∃ (u v : Fin 3 → ℂ) (S : Fin 3 → Fin 3 → R),
      (∀ i, (u i)^2 = f (a i)) ∧
      (∀ j, (v j)^2 = f (b j)) ∧
      (∀ i j, (S i j)^2 = a i*b j+1) ∧
      (∀ i j, ‖(f (a i)*f (b j))*(u i*f (S i j)*v j)-
        (f (a i)*f (b j))^2-(1/2 : ℂ)*(f (a i)*f (b j))‖ < (2/15 : ℝ)) := by sorry









end D1Bound5
end