Exact Arithmetic at Roots of Unity for the Parity-Shifted $q$-Series: The Even-Order Companion

We introduce a parity-shifted q-hypergeometric series whose Pochhammer symbol (q;-q)_r exhibits multiplicative periodicity at primitive k-th roots of unity for k ≡ 2 (mod 4). This periodicity forces the infinite series to collapse to a finite sum, producing exact cyclotomic integers S_k ∈ ℤ[ζ_k] whose field norms N(S_k) obey a splitting law: all prime factors p ∤ k satisfy p ≡ 1 (mod k). We tabulate the sequence for k ≤ 30, discuss logarithmic extension across the unit circle via Möbius interpolation, and identify the k ≡ 0 (mod 4) barrier as a fundamental singularity. The logical structure of the proofs is formalized in Lean 4. This work serves as the even-order companion to the author's odd-order radial limit theorem for Ramanujan's classical third-order mock theta function (Zenodo 20225375).