A Local Möbius Inversion Formula via Resolvent Operators on Posets

We analyze a one-parameter deformation of the incidence algebra of a locally finite poset with bounded degree, where the zeta function is weighted by a parameter q associated with chain lengths. By viewing incidence functions as linear operators on l^infty(P), we establish that the q-Zeta function corresponds to the resolvent of the adjacency operator. We prove that the associated q-Möbius function is strictly local, acting as a sparse difference operator supported solely on covering relations. This contrasts with the classical Möbius function, which depends on the global topology of intervals. We discuss connections to spectral graph theory, Katz centrality, and algorithmic implications for weighted enumeration.