Residue Holonomy and Prime Duality

We introduce a geometric framework in which Möbius iterations generate a meromorphic su(2)su(2)su(2)-valued connection whose holonomy is governed by a signed residue sum.Within this model, cancellation of residues produces trivial holonomy (composite-like behavior), while non-cancellation yields persistent defects (prime-like behavior). We analyze the associated dynamical system, derive a closed-form description of holonomy growth in the constant-defect regime, and show that primitive loop density exhibits logarithmic thinning analogous to the prime number theorem. The framework provides a unified perspective linking Möbius dynamics, residue geometry, and loop proliferation, while leaving the direct arithmetic correspondence as an open problem.