Spectral Integrality and Jordan Structure of Q-Digraphs Over Finite Cyclic Q-Algebras via Block Matrix Reduction

We study the ordinary adjacency and out-Laplacian spectra of the loop-suppressed doubling Q-digraphs associated with (Zn,−,0). Writing n=2sm with m odd, a block-matrix reduction in natural residue order gives χΓ2N(λ)=λNχΓN(λ). This yields an explicit arithmetic specialization of the established characteristic-polynomial factorization for functional digraphs. Our principal family-specific result is the exact rank sequence rank(Ank)=n/2k for 1≤k≤s and rank(Ank)=m−1 for k≥s+1. These ranks determine the multiplicity of every Jordan block at the zero eigenvalue, rather than only its algebraic multiplicity. We also obtain the minimal polynomial, the adjacency- and out-Laplacian-integrality criterion m∈{1,3}, and explicit spectral-energy formulas. Finally, the transpose adjacency matrix provides an exactly solvable homogeneous linear-propagation operator whose arithmetic parameters give sharp graph-induced rank ceilings, a finite transient horizon and, when m>1, a recurrent period. No claim about predictive or training performance is made.