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PreprintCC BY 4.0Zenodo

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

  • Andrew Stewart Caldin1
  1. 1Independent Researcher, United Kingdom
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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 𝒜 with counting function A(x) ∼ α x (density α), Gafni's result (arXiv:1407.2918) gives: p𝒜(n) ∼ 1/(2π√(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) ∼ 1/4n√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 $\frac{

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Cite

BibTeXRIS
BibTeX
@misc{caldin2026phase,
  title = {{Phase Transition in Partition Asymptotics via Eta-Function Modularity — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23430998},
  url = {https://zenodo.org/records/23430998},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Phase Transition in Partition Asymptotics via Eta-Function Modularity — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/17
PB  - Zenodo
DO  - 10.5281/zenodo.23430998
UR  - https://zenodo.org/records/23430998
AB  - 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: p_𝒜(n) ∼ 1/(2π√(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) ∼ 1/4n√3 e^π√(2n/3). The generating function is ∑ p_𝒜(n)q^n = ∏_a∈𝒜(1−q^a)^−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 $\frac{  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -