Coefficient Transfer from Real-Axis Singular Scales for Strongly Irreducible Partitions

This paper is the coefficient-transfer sequel to the real-axis singular theory for strongly irreducible partitions. Its purpose is deliberately narrower than a formal announcement of a Hardy–Ramanujan formula. The preceding papers provide, under their stated hypotheses, the real-axis first-order singular estimate log P(e⁻ᵗ)~ π²/121/tlog(1/t)(t→0⁺). Such an estimate alone does not determine a pointwise coefficient asymptotic: minor-arc control and local Gaussian behavior are additional analytic inputs. We therefore prove the exact coefficient consequences that genuinely follow from the real-axis estimate, formulate a noncircular saddle-point transfer criterion, and compute the constants forced by that criterion. In particular, the real-axis estimate implies the sharp logarithmic upper bound log p(n)≤ (2√π²/6+o(1))√n/log n, and, under explicit Hayman-type locality and decay hypotheses, it yields the logarithmic coefficient asymptotic log p(n)~ 2√π²/6√n/log n. A full equivalent with a numerical prefactor is isolated as a stronger conditional theorem requiring a second-order singular expansion. The paper also explains why the frequently guessed expression C n-3/4exp(c√n/log n), with a constant independent of log n, is not justified by first-order singular information and has the wrong saddle scale unless further terms are supplied. Keywordsinteger partitions; pairwise coprime partitions; strongly irreducible partitions; saddle-point method; Hayman admissibility; coefficient asymptotics