Successor Maps as Rigid Solenoidal Translations: Finite-State Rigidity, Cohomology, and Cylinder Laws

This paper develops a second-stage theory for successor Hamming complexity in canonical numeration systems, building on the finite-state and exact counting results established in the companion manuscript Successor Hamming Complexity in Metallic Numeration Systems. Our aim is to place those results inside a broader arithmetic-dynamical framework. We formulate a finite-state successor classification problem for sofic numeration systems, reinterpret successor-cost identities as cohomological equations over odometer-type translations on solenoidal models, and organize both finite-dimensional and transfer-operator routes to intrinsic cylinder laws. The main contribution of the present manuscript is structural. First, we isolate a finite-state successor package for canonical numeration systems with regular admissible languages and bounded carry lookback. Second, we formulate a rigid cohomological framework in which the successor Hamming cost is expressed as a coboundary plus a locally constant finite-state correction. Third, we organize the intrinsic cylinder-law problem into a defect-space formalism that already yields a purely finite-dimensional quasi-spectral completion criterion at fixed order and that, beyond that, admits a transfer-operator refinement designed to capture asymptotic Gibbs-type behavior. For quadratic Ostrowski systems with eventually periodic continued fractions, and in particular for the period-two benchmark =1;2, 1, we show how the previous explicit formulas fit this architecture. The present paper therefore turns the structural picture into a theorem-level fixed-order theory: it proves a general finite-state/cohomological formalism, proves a finite-dimensional fixed-order completion route under nondegeneracy and unit-circle spectral exclusion, and identifies the rigid-solenoidal mechanism that governs the proved transfer reductions used throughout the manuscript. Conceptually, the paper isolates what should remain as a reusable reference architecture for the subject. It shows that successor Hamming complexity in rigid numeration systems is not an ad hoc collection of counting identities: once the finite-state package and canonical transport package are present, every fixed-order limiting question factors through one common three-layer mechanism---finite symbolic data, odometer/cohomological realization, and centered defect dynamics. In that sense, the manuscript is intended not merely as another example paper, but as a foundational reduction theorem for the arithmetic-dynamical study of successor observables.