Phase Transition in Partition Asymptotics via Eta-Function Modularity — E8 Intelligence Research

FINDING: Gafni's restricted partition asymptotics connect density α of allowed parts to Dedekind eta-function modular behavior, yielding a phase transition in growth rate. | MATH: For A with counting function A(x) ~ α x (density α), Gafni's result (arXiv:1407.2918) gives: pA(n) ~ 12π√2α n exp( π √2α n/3 ) for 0<α<1 (with error terms via saddle-point on the Dedekind eta η(τ)). For α=1 (all integers), this reduces to Hardy–Ramanujan: p(n) ~ 14n√3 eπ√2n/3. The generating function is ∑ pA(n)qⁿ = ∏a∈ A(1-qᵃ)⁻¹ = 1/η(τ) · (modular correction) when A is arithmetic progression-like. Key constant: the exponent π√2α/3 — linear in √α, not α. | CONNECTION: The factor √2/3 in the exponent — note √2/3 ≈ 0.8165, and the ratio ${ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com