Periodic-Orbit Collapse in B-Admissible Shifts: Exact Prime-Wheel Zeta Functions and Infinite Coprime Limits

We determine periodic-point counts and Artin–Mazur zeta functions for a class of arithmetic hereditary shifts. For a finite pairwise-coprime family B = {b1, ..., bs}, an n-periodic support sees the modulus bi only through gcd(n, bi). This reduction, followed by inclusion–exclusion over omitted residue classes, gives an exact fixed-point formula for every period. When the moduli are prime and W = product of p in B of p, the formula yields a rational Artin–Mazur zeta product with integer exponents. The natural W-state phase presentation can be many-to-one even on nonzero periodic sequences, so its edge-cover determinant is generally not the zeta function of the labeled shift; the arithmetic inclusion–exclusion resolves these overlaps. We compute the products for W = 2, 6, 30, including N30 = 4501 and N60 = 1,730,521 for W = 30. The finite theory has a sharply different infinite limit. For every infinite pairwise-coprime B, the all-zero sequence is the only periodic point and hence zeta(z) = (1 - z)^(-1). Under sum of b in B of b^(-1) < infinity, the same shift has positive topological entropy. In particular, for B = {p^2 : p prime}, h_top = (6 / pi^2) log 2 although all nonzero periodic orbits disappear. The zeta functions of finite approximants converge coefficientwise to (1 - z)^(-1), while their dominant singularities remain bounded away from the unit circle.