Unit Pair Arithmetic Model — imaginary same axis impossible

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.Group.Int.Units
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Analysis.Complex.Norm
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Sqrt
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_UnitPairArithmeticModel_Definitions

section
open scoped NumberField QuadraticAlgebra
open Module

namespace D1Bound5












theorem _root_.D1Bound5.UnitPairArithmeticModel.imaginary_same_axis_impossible
    {R : Type*} [CommRing R] (M : UnitPairArithmeticModel R)
    (hd : 2 ≤ M.d) (m n : ℤ) (hmn : 0 < m*n)
    (hs : ∃ r : R, r*r = ((m : R)*M.w)*((n : R)*M.w)+1) : False := by sorry



end D1Bound5
end