Bailey-Lattice Transfer of Generating Functions, Prime Euler Factors, and the von Mangoldt Measure

Let (α_n, β_n) be a Bailey pair relative to a with base q, and let G_P(a,q) = Σ_{n≥0} a^n q^{n²} α_n be its α-side generating function. Lovejoy's Bailey-lattice step carries a pair relative to a to a pair P′ relative to aq and depends on an auxiliary parameter b. We make explicit a transfer law implicit in Lovejoy's work: G_{P′}(aq,q) = (aq;q)_∞/(1−aq) · Σ_{n≥0} a^n q^{n²} [q^n(1−b)/(1−bq^n)] β_n, so that G_{P′}(aq,q) = (1−aq)^{−1} T_b(P;a,q) G_P(a,q) with a pair-dependent factor T_b that is identically 1 exactly when b = ∞. The weights q^n(1−b)/(1−bq^n) are the β-side of Lovejoy's same-parameter companion pair P*, so that T_b(P) G_P(a,q) is the α-generating function of P*; the b = ∞ identity is a summation by parts valid for arbitrary sequences, and the b = ∞ step is the inverse of McLaughlin's parameter-lowering lemma, whose prefactor (1−a) is the Euler factor in plain view. The hypotheses are minimal: Σ_n |a|^n |q|^{n²} |α_n| < ∞, a condition preserved by the lattice step. For the Rogers–Ramanujan pair the finite-b deformation interpolates between the two Rogers–Ramanujan functions (b = ∞ and b = 0) and generates the tower Σ_n q^{n²+jn}/(q;q)_n. Specializing a = 1, q = p^{−s} identifies the inverse local Euler factor with the b = ∞ transfer, identifies the m-th step along the unit-pair orbit with the Euler factor at ms, and, since d/ds = −(log p) q ∂_q, recovers the von Mangoldt measure on m log p as the logarithmic derivative of the first step. A separability statement then bounds what the mechanism can produce: the prime enters only through the Euler factor and the evaluation point p^{−s} of one fixed function of q, so one lattice step at one base yields, beyond ζ(s), only Dirichlet series of prime-independent multiplicative functions. The operator form on ℓ²(P) is given rigorously. The endpoint number theory is classical and the transfer law is Lovejoy's; the contribution of this note is to make the mechanism, its attribution, and its limits explicit.