Unifying Ramanujan Congruences via Hecke Operators on Level-5 Modular Forms — E8 Intelligence Research

FINDING: Ramanujan-type partition congruences modulo 5, 7, 11 are unified via Hecke operators on level-5 modular forms; new restricted partition functions (cubic, c-colored, elongated diamond) yield congruences modulo 7 and 11, extending the classical framework. | MATH: Classical: p(5n+4)≡0 (mod 5), p(7n+5)≡0 (mod 7), p(11n+6)≡0 (mod 11). Hecke operator T_p acts on modular forms of weight k and level N; for level 5, the space S_k(Γ₀(5)) splits into oldforms (from level 1) and newforms. The generating function Σ p(n)q^n = 1/Π(1−q^n) = q^{1/24}/η(τ), with η(τ) the Dedekind eta. For the restricted functions: a_c(n) (c-colored even parts) and d_c(n) (c-elongated plane partition diamonds), the arXiv paper (2508.18286v3) proves explicit congruences: e.g., a_c(7n+5)≡0 (mod 7) and d_c(11n+6)≡0 (mod 11) under specific c-conditions (likely c≡0 mod 7 or 11 respectively, per standard patterns). | CONNECTION: The modulus 5, 7, 11 are primes p where p−1 divides 12 (5−1=4, 7−1=6, 11−1=10 — all diviso Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com