Mixed-Support Criteria and First-Order Asymptotic Transfer for Strongly Irreducible Partitions

This paper is a rigorous continuation of the analytic framework developedfor strongly irreducible partitions. The first two papers establish the com-binatorial core decomposition, the prime-power Euler-product lower model,and the benchmark-scale asymptoticlog Q◦(e−t) ∼ π212 B(t), B(t) = 1t log(1/t), t → 0+,under the classical prime number theorem. The remaining obstruction to thecorresponding first-order formula for the full ones-free series is the mixed-support logarithmic correctionLms(t) = log( P ◦(e−t)Q◦(e−t)).The purpose of the present manuscript is to formulate, with no circular defi-nitions and no hidden asymptotic assumptions, the exact estimates that aresufficient for proving Lms(t) = o(B(t)). We give a complete least-mixed-partdecomposition, pass to support-grouped weights, isolate the explicit prime-power baseline, and formulate a baseline–excess criterion whose hypothesesimply the desired transfer theorem. The paper deliberately does not assertan unconditional proof of mixed-support negligibility: the estimates neededfor the excess term are stated as precise verifiable hypotheses. This avoidsthe common erroneous arguments based on divergent support sums or on thefalse comparison 1/t = o(B(t))