Cassini Constraints and Symmetry Reductions in Fibonacci Carry Lifts

This paper investigates the arithmetic structure of Fibonacci carry sequences in base 32 through Cassini constraints, modular step vectors, and exact symmetry reductions. Starting from the stabilized Fibonacci lift representation, the study distinguishes 96 Cassini-admissible step vectors from the 48 vectors realized by Fibonacci arithmetic. A complement theorem establishes a precise relationship between Cassini-admissible carry words and Fibonacci carry words. The complete classification yields 48,336 admissible 32-bit Fibonacci carry words. Under cyclic rotation and reverse-complement symmetry, these words reduce to 796 equivalence classes. The 48 Fibonacci step vectors are further organized into seven arithmetic types, belonging to two divisibility families. The intersections between these types are reduced to two fundamental cases, each involving exactly 96 common carry words. The paper also establishes the separation of two arithmetic types through cyclic seven-bit block-frequency profiles and examines the structure of minimal forbidden binary factors up to length 32. The results combine algebraic proofs with exhaustive, reproducible finite enumerations. They reveal how a large family of apparently diverse Fibonacci carry patterns is governed by a remarkably small number of arithmetic structures and symmetry principles.