A Theorem-Safe Algebraic-Geometric Reinterpretation of Modular Periods for Weighted Lucas Sums

This paper gives a theorem-safe algebraic-geometric reinterpretation of the modular-period theory for weighted Lucas sums developed in the earlier papers on multiplicative, affine, polynomial, and cyclic finite-state weights. The goal is conceptual clarification rather than a new prime-level divisor theorem. We show that the finite-state state space used in the prime-level theory can be modeled canonically as a finite étale zero-dimensional scheme over Fₚ, and that the state-update rule is an automorphism of that scheme. In this language, the least period of a weighted Lucas sum is exactly the orbit length of a closed point under a scheme automorphism. We then isolate the exact geometric meaning of period collapse in a theorem-safe form: the reduced source period ₚ is the minimal cycle quotient through which the reduced output morphism factors. Thus collapse is not described here by speculative flatness language, but by an explicit factorization phenomenon on finite cycle schemes. Next we reinterpret the Fourier-support criterion from the earlier prime-level multiplicative paper as the diagonalization of the cyclic shift operator on the group algebra of a finite cycle. This gives a precise linear-algebraic shadow of the geometric character picture, while avoiding unsupported cohomological claims. We also formulate a rigorous fiber-translation model for the one-step lifting dichotomy and prove the rationality of the associated finite dynamical zeta function. Throughout the paper every object is defined before use, all geometric statements are reduced to concrete finite-scheme calculations, and conceptual remarks are clearly separated from theorem-level claims.