Integrality of the coefficients α_i in S^z decomposition

Ascertain whether the coefficients α_i in the decomposition S^z = p + Σ_{i=1}^N α_i h_i are positive integers for general N (except for the two-site case), thereby clarifying the arithmetic nature of these weights.

Background

After presenting explicit examples for N = 2, 3, and 4, the paper notes a pattern in the coefficients α_i appearing in S^z = p + Σ α_i h_i and raises the conjectural possibility that they are positive integers (with the two-site case singled out as special).

This observation invites a more general determination of the arithmetic nature of α_i across N, potentially connecting to combinatorial structures, though the paper does not commit to a specific interpretation.