Rank-Two Bailey Pairs at Prime Bases: Exact Connected Coupling and Prime-Ratio Cancellation

We study the rank-two (C₂) Bailey pairs of Milne and Lilly with the variables placed at prime bases, x_k = p_k^(−s). For the unit pair we prove an exact closed form for the connected ratio, which is the part of the coupling that does not factor into one-prime marginals. It depends on the two primes only through their product u = x₁x₂, so it carries no prime-ratio information at any order. Its leading term is q^(N₁N₂). The cancellation is not special to the unit pair. It holds for every column of the inverse matrix M*, and at every rank ℓ the difference-root factors cancel from each connected quotient while the sum-root factors carry the coupling. We give the exact finite form of the weak Bailey lemma limit, as a corollary of Lilly–Milne Theorem 4.12, and a closed form for every column of the chain operator. Over all primes, the corresponding infinite product is proved not to converge. As a check, the chain built on the corrected kernel (the correction is due to Bartlett and Warnaar, 2015) reproduces Milne's 1994 modulus-6 C₂ Rogers–Ramanujan identity and the Griffin–Ono–Warnaar D₃⁽²⁾ family. We also explain why prime points lie outside the regime where such product evaluations occur. Quadratic cross-terms of this kind are standard and are not claimed as new. All identities are checked by the included Python scripts.