Arithmetic Inverse Limits, Clopen Observable Modules, and Cohomological Transport for Finite-State Successor Systems

This paper is the third installment in a program that studies successor Hamming complexity in canonical numeration systems. Building on the exact arithmetic formulas and finite-state automata established in the two companion manuscripts for metallic and quadratic Ostrowski systems, the present work isolates the inverse‑limit and cohomological layer as an independent, reusable theorem package. We prove five kinds of results. 1. The arithmetic inverse‑limit space of a canonical scale system is compact, totally disconnected, and generated by residue cylinders; the coherent arithmetic image of the non‑negative integers is dense, and the successor map acts as a translation on this image. 2. Residue‑compatible finite‑state observables extend uniquely to finite‑dimensional translation‑invariant subspaces of locally constant functions, and the union of these subspaces is dense in the space of continuous functions. 3. The affine successor law transports to a genuine cohomological equation on the compact phase space, admits a grouped form by telescoping, and yields exact invariant‑mean identities. 4. Exact arithmetic‑to‑Haar averaging theorems and asymptotic mean formulas for successor costs are obtained from the residue‑compatible structure. 5. A complete operator‑theoretic reduction of the uniform spectral‑gap conjecture is carried out. The reduction isolates four theorem‑sized inputs — a Banach‑space realization, a Lasota–Yorke/quasi‑compactness mechanism, Perron isolation, and quotient compatibility — and proves that, once these four inputs are supplied, all fixed‑order intrinsic cylinder laws exist, are unique, and converge exponentially along grouped canonical intervals, with constants independent of the order. The corresponding weakened form of the conjecture is therefore established in the present paper. Moreover, for the period‑two quadratic benchmark =1;2, 1, we verify a concrete canonical cylinder‑induced ambient operator model and close, inside that model, all the remaining benchmark verification problems (branch multiplicity, cylinder image, weight distortion, peripheral shadow, and Perron template). The explicit numerical choice =1/10 turns the critical quotient‑scale inequality into 51/100<1. Thus the full operator‑level spectral package is completely realized for the canonical period‑two route. The gap between this weakened theorem and the fully unconditional conjecture is compressed to a single, sharply defined identification step: proving that the canonical cylinder‑induced realization coincides with the natural grouped return operator of the arithmetic dynamics. What remains is therefore no longer a diffuse analytic obstacle but a concrete arithmetic‑operator identification theorem, whose formulation is now rendered precise inside the architecture built here.