Coefficient Transfer for the Prime-Power Model of Strongly Irreducible Partitions

This paper isolates the coefficient-transfer problem for the prime-power lower model of strongly irreducible partitions. The model has the exact Euler product (q)=∏ₚ(1+∑a≥1qpᵃ) (|q|<1), and the preceding real-axis analysis gives, under the prime number theorem, log (e⁻ᵗ)~ π²/121/tlog(1/t) (t→0⁺). The purpose of the present paper is deliberately not to convert this radial statement into an unconditional Hardy–Ramanujan formula by fiat. A radial first-order singular estimate does not determine local Gaussian behavior, minor-arc decay, or the bounded residual exponent needed for a numerical prefactor. We therefore prove exactly what follows without further complex input, formulate coefficient-free admissibility hypotheses for the prime-power Euler product, and compute the constants forced by those hypotheses. In particular, the real-axis estimate alone gives log (n)≤ (π√23+o(1))√n/log n, while first-order coefficient admissibility gives the matching logarithmic asymptotic. A full equivalent has the conditional form (n)~ 1√2π(π²/24)1/4 n-3/4(log n)-1/4 exp\!( π√23√n/log n+εₙ ), provided a second-order saddle exponent datum εnεn and the first-order arc estimates are supplied. The paper also records why the frequently guessed constants$ 2A2A$, $3A3A$, or $π2π2/9π2π2/9% are not the constants produced by the saddle equation for the scale$ A/(tlog⁡(1/t))A/(tlog(1/t))$.