Rationality and realizability of the WLP binary sequence b_X

Investigate whether, for a finite set of points X ⊂ P^r, the real number b_X = 0.b1 b2 b3 …—where b_d = 1 if R/A_{X,d} has the Weak Lefschetz Property and b_d = 0 otherwise—is always rational; and classify which real numbers with binary expansions arising from 0–1 sequences can be realized as b_X for some X ⊂ P^r.

Background

The authors propose encoding the WLP behavior across all powers d by a binary sequence B_X and the associated real number b_X in [0,1]. In P^2, b_X corresponds to the constant-1 sequence due to known WLP results, while in P^3 more varied behaviors occur.

This question asks about eventual stability and periodicity (rationality corresponds to ultimately periodic binary expansions) and seeks a characterization of which numbers arise from geometric configurations X in projective space.