Minimality of B-free systems in number fields

The paper states and proves: (a) with D := {d: dC ⊆ B for some infinite pairwise coprime C} and B* := (B ∪ D)prim, the array η* = 1_{F_{B*}} is Toeplitz, satisfies η* ≤ η, and X_{η*} is the unique minimal subset of X_η (Theorem A); and (b) for primitive B, the following are equivalent: minimality of (X_η,(S_g)), Toeplitzness of η (η ≠ 0), D = ∅, and X_η ⊆ Y where each x misses exactly one residue class modulo each b ∈ B (Theorem B) . The paper also proves Theorem C, giving D = ∅ ⇔ B = B* ⇔ η is Toeplitz, and that in all cases η* is Toeplitz (D* = ∅) . It supplies the key decomposition FB* = FB ∩ FD and a characterization of nonperiodic positions via Proposition D, used to build the period structure and the proximality/containment X_{η*} ⊆ X_η . In the proof of Theorem B, the paper establishes (i)⇒(ii) by showing X_{η*} = X_η then η = η* (hence Toeplitz), (ii)⇒(iv) using Lemma 4.13 (Toeplitz ⇒ η ∈ Y) and minimality, and (iv)⇒(ii) via Theorem A and a residue-class count argument . The candidate’s solution repeats these steps with the same definitions and CRT-based ideas and correctly notes η* ≤ η, Toeplitzness of η*, uniqueness of the minimal subset, and the equivalences in Theorem B. Small gaps in the candidate’s direct (i)⇒(iv) justification are covered explicitly in the paper via Lemma 4.13 and Theorem A. Overall, both are correct and employ substantially the same proof strategy grounded in the D–B* augmentation, CRT constructions, and Toeplitz minimality.