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

$\mathcal{Q}^{\circ}(q)=\prod_p\left(1+\sum_{a\ge 1} q^{p^a}\right)
\qquad (|q|<1)$.

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

$\log \mathcal{Q}^{\circ}(e^{-t})=\sum_p e^{-tp}-D(t)+H(t)$.

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

$D(t)\sim \left(1-\frac{\pi^2}{12}\right)\frac{1}{t\log(1/t)}
\qquad\text{and}\qquad
\log \mathcal{Q}^{\circ}(e^{-t})\sim \frac{\pi^2}{12}\,\frac{1}{t\log(1/t)}$.

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.