Local coprimality masks modulo n CRT, AND-convolution and zeta/Möbius diagonalization and a generalization to gcd profiles

We define local coprimality masks in a length-k window modulo n and the associated histogramvector Fk(n) ∈ N2k . We show that for gcd(n,m) = 1 the following multiplicativity relation holds:Fk(nm) = Fk(n) ⊡ Fk(m),where ⊡ is the convolution induced by bitwise AND (equivalently, the intersection/meet convolutionon the Boolean lattice P([k])). Next we introduce the zeta transform on P([k]) in the “sumover supersets” convention and prove the diagonalizationZ(A ⊡ B) = (ZA) · (ZB),which reduces the CRT arithmetic to pointwise multiplication of the coordinates ZFk(n). Finallywe present a generalization (Upgrade A) where, instead of the binary signal χn(x) = 1(gcd(x,n)=1),we consider the full window profile gcd(x + r, n), leading to an analogous theory on the divisorlattice.