Modular Holonomy and Group Structure in Möbius Dynamics of Primes

We extend the Möbius–holonomy framework by embedding it into a group-theoretic and modular structure. Each covering word corresponds to an element of a discrete subgroup of PSL (2, C) PSL (2, C) PSL (2, C), while the associated meromorphic connection defines a holonomy representation into SU (2) SU (2) SU (2). In this setting, the signed residue sum acts as a group-valued charge that determines whether the holonomy lies in the identity class. Composite-like configurations arise when the total holonomy reduces to the identity element, corresponding to trivial classes in the representation space. In contrast, prime-like configurations correspond to nontrivial conjugacy classes, interpreted as irreducible defects that cannot be decomposed within the group structure. From a modular perspective, the iterative Möbius action induces a symbolic dynamics analogous to modular transformations, where primitive words correspond to irreducible orbits. The Möbius inversion formula then plays the role of a projection operator extracting primitive (prime-like) group elements from composite compositions. This unified picture suggests that primes emerge as stable, nontrivial elements in a modular–group dynamical system, while composites correspond to reducible configurations whose holonomy cancels under group composition.