Prime-Power Euler Products and Singular Scales for Strongly Irreducible Partitions

This paper develops an analytic framework assuming only the classical prime number theorem for the ones-free generating series of strongly irreducible partitions, together with the prime-number-theoretic consequences that follow from that input. Our main object is the prime-power subfamily, whose generating function is shown to admit an exact absolutely convergent Euler product Q∘(q)=∏ₚ(1+∑a≥ 1 qpᵃ) (|q|<1). We prove finite-prime degree stability, monotone convergence, and explicit uniform tail bounds for this product. We then decompose the full ones-free series into the prime-power model plus an explicit mixed-support correction, and on the real axis near $q=1$ we obtain an exact logarithmic splitting of the form log Q∘(e⁻ᵗ)=∑ₚ e⁻ᵗᵖ-D(t)+H(t). Here the benchmark prime sum is explicit, the higher-power term $H(t)$ is rigorously controlled, and the logarithmic defect $D(t)$ is shown to have benchmark order with an explicit leading constant. Assuming only the classical prime number theorem, we prove D(t)~ (1-π²/12)1/tlog(1/t)log Q∘(e⁻ᵗ)~ π²/12\,1/tlog(1/t). We then recast the remaining mixed-support obstruction as an exact decomposition by the least mixed-support part together with four nested quantitative criteria for benchmark-negligibility: an m-indexed restricted-quotient criterion, a support-grouped criterion, a crude support-grouped criterion, and a baseline–excess criterion that isolates an explicit support-local main term from a residual excess. The paper contains no conjectural Hardy–Ramanujan or modular-form theorem: its contribution is to establish the exact analytic structure under the classical prime number theorem, the first explicit leading constant for the prime-power model on the real axis, and a sharpened reduction of the full ones-free problem to the mixed-support correction.