Roots of Unity as Covering-Space Invariants: Holonomic Closure in Coset Partitions

In prior work we showed that integer invariants arise wherever transport around a topological obstruction must return to an equivalent value after projection. This holonomic closure condition produces quantization in quantum phase transport, rotor holonomy in spin dynamics, and linking numbers in circular DNA without invoking quantum mechanics or chemistry. Here we examine the same mechanism in a purely algebraic setting: coset partitions of free groups. Using Chouraqui's cyclotomic formulation of coset partitions, we show that the appearance of complex phases arises from cycle-return conditions in Schreier graphs. The cycle structure encodes winding behavior, and the requirement of equivalence after transport forces these windings to manifest as roots of unity. Inverting the exponential map on μ_k recovers the exponent modulo k. A computational analysis reveals an exceptional class of index-6 subgroups whose generator cycle structure avoids producing primitive sixth roots of unity. This "mimic" class explains why naive index-based arguments fail and isolates the true obstruction: the feasibility of a cyclotomic closure condition. The existence of a partition with indices {2,3,6} is shown to reduce to a finite feasibility problem in ℤ[ζ₃].