The Spectral Phase Transition Theorem for State Spaces of Multi-Base Power Mapping Networks over the Multiplicative Group of a Finite Field

This paper investigates the spectral properties and phase transition mechanisms of state space reachability structures for coupled random walks with multi-base power mappings over the finite field multiplicative group $\mathbb{F}_p^\times$. By introducing the discrete logarithm isomorphism, the nonlinear modular exponentiation is transformed into a linear operator on the additive ring $\mathbb{Z}_{p-1}$; furthermore, utilizing the Chinese Remainder Theorem (CRT), the global transition matrix is decomposed into a tensor product of local operators over prime power components. Based on the invariant subspace structure of local operators on $q$-adic valuation layers, the second-largest eigenvalue $\lambda_2$ of the transition matrix is exactly computed. We strictly prove a hard phase transition theorem for multi-base coupled networks: the Markov chain converges exponentially to a unique absorbing state (i.e., $\lambda_2 < 1$, where the stationary distribution degenerates to a point mass $\delta_1$) if and only if $N=p-1$ is a $P(S)$-smooth number. Moreover, we derive the exact analytical expression $\lambda_2(T) = \max_{q \mid N} \frac{|S_q|}{|S|}$. This discovery reveals the "spectral rigidity" where the network convergence rate is strictly determined by local divisibility features, providing a rigorous algebraic characterization for state space fragmentation and valuation collapse mechanisms in arithmetic dynamical systems.