Full ring unit pair even same sign 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.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

section
open scoped NumberField
open Module

namespace D1Bound5










theorem _root_.D1Bound5.full_ring_unit_pair_even_same_sign_impossible
    {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]
    (hK : finrank ℚ K = 2) (m n : ℤ)
    (hm : 0 < m) (hn : 0 < n) (hm4 : m % 4 = 3) (hn4 : n % 4 = 3)
    (hs : ∃ r : 𝓞 K, r * r = ((m * n + 1 : ℤ) : 𝓞 K)) : False := by sorry

end D1Bound5
end