The square-norm integers are covered by their integer-translation fibers. Fibers with even generated-subfield index are absorbed into the small-subfield set when there is no index-two intermediate field.

theorem CCDN.exists_norm_curve_exception_split (K : Type u_1) [Field K] [NumberField K] (hnoTwo : ∀ (L : IntermediateField ℚ K), Module.finrank (↥L) K ≠ 2) (B : ℝ) : ∃ (β : ↥(boundedSquareNormTranslationClasses K B) → NumberField.RingOfIntegers K), Function.Injective β ∧ (∀ (q : ↥(boundedSquareNormTranslationClasses K B)) (w : NumberField.InfinitePlace K), w ↑(β q) ≤ B) ∧ (boundedSquareNormIntegers K B).ncard ≤ (boundedSmallSubfieldIntegers K B).ncard + ∑ q : ↥(boundedSquareNormTranslationClasses K B), if Odd (Module.finrank (↥ℚ⟮↑(β q)⟯) K) then count (signedNormCurveLocus K (β q) 1) (max (2 * B) √(B ^ Module.finrank ℚ K)) + count (signedNormCurveLocus K (β q) (-1)) (max (2 * B) √(B ^ Module.finrank ℚ K)) else 0

For a number field K without an intermediate field of index two, there is an injective family of algebraic integers β q bounded by B, indexed by the bounded square-norm translation classes, such that the number of bounded square-norm integers is at most the number of bounded integers in small subfields plus the number of points of height at most max (2 * B) (√(B ^ n)) on the signed norm curves for those β q with [K : ℚ(β q)] odd.