We decompose the full-reptend condition (ord_a(p) = p−1) into independent local obstruction events indexed by the prime factors of p−1. For each prime q, the obstruction event is determined by the Frobenius class in a finite extension, with exact density 1/q (unconditional Chebotarev). The underlying fields are linearly disjoint, yielding exact independence for every finite block. Artin's constant emerges as the natural limit of the finite products, confirmed empirically to four significant digits. We prove that the passage from finite to infinite product reduces to the absence of infinitely many Siegel zeros in the ultra-sparse family {L(s, χ_{±q}) : q prime}, where the discriminants are ±q (squarefree by construction, single genus). Cross-domain verification in a deterministic PDE (the Ω-field simulator) shows that the analogous homogeneous state has empty basin of attraction. Companion code for full reproducibility is included.