Phase Transition in Partition Asymptotics via Eta-Function Modularity — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: 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 $\mathcal A$ with counting function $A(x) \sim \alpha x$ (density α), Gafni's result (arXiv:1407.2918) gives: $p_{\mathcal A}(n) \sim \frac{1}{2\pi\sqrt{2\alpha n}} \exp\left( \pi \sqrt{\frac{2\alpha n}{3}} \right)$ for $0<\alpha<1$ (with error terms via saddle-point on the Dedekind eta $\eta(\tau)$). For α=1 (all integers), this reduces to Hardy–Ramanujan: $p(n) \sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{2n/3}}$. The generating function is $\sum p_{\mathcal A}(n)q^n = \prod_{a\in\mathcal A}(1-q^a)^{-1} = \frac{1}{\eta(\tau)} \cdot \text{(modular correction)}$ when $\mathcal A$ is arithmetic progression-like. Key constant: the exponent $\pi\sqrt{2\alpha/3}$ — linear in $\sqrt{\alpha}$, not α. | CONNECTION: The factor $\sqrt{2/3}$ in the exponent — note $\sqrt{2/3} \approx 0.8165$, and the ratio $\frac{ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222813619
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