Density-Driven Growth Rates in Restricted Partition Asymptotics — E8 Intelligence Research

FINDING: Gafni's restricted partition asymptotics connect density of allowed parts to exponential growth rates, with Dedekind eta function as the modular backbone. | MATH: For A with counting function A(x)=α x + O(x^β), β<1, Gafni's result (arXiv:1407.4443) gives log pA(n) ~ π√3√α n + 1/4log n + O(1) — the leading term scales as √α n, generalizing Hardy–Ramanujan (α=1). The generating function ∑ pA(n)qⁿ = ∏ₐ(1-qᵃ)⁻¹ and modularity enters via η(τ) = q1/24∏ₙ₌₁^∞(1-qⁿ), where q=e2π iτ. The constant π/√3 ≈ 1.8138 is the saddle-point residue from the Dedekind eta transformation η(-1/τ)=√-iτ\,η(τ). | CONNECTION: The exponent α in the density A(x)~α x directly multiplies n inside the square root — this is a *linear scaling* of the modular Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com