Hecke Operators and New Partition Congruences Mod 7 and 11 — E8 Intelligence Research

FINDING: Ramanujan-type partition congruences (mod 5, 7, 11) are governed by Hecke operators on level-5 modular forms; new restricted partition functions (generalized cubic partitions, elongated plane partition diamonds) exhibit congruences mod 7 and 11, extending the classical framework. | MATH: Classical: \(p(5k+4)≡ 0 {5}\), \(p(7k+5)≡ 0 {7}\), \(p(11k+6)≡ 0 {11}\). Hecke operator \(T_p\) acts on modular forms of weight \(k\) and level \(N\): \((T_p f)(z) = pᵏ⁻¹ f(pz) + 1/p∑b=0ᵖ⁻¹ f(z+b/p)\). For the new results (arXiv:2508.18286v3): \(a_c(n)\) and \(d_c(n)\) satisfy congruences of form \(a_c(An+B)≡ 0 {7}\) and \(d_c(An+B)≡ 0 {11}\) for specific arithmetic progressions (explicit \(A,B\) depend on \(c\) and the generating function's modular form level). | CONNECTION: The primes 5, 7, 11 are exactly those where the partition generating function \(∏ₘ₌₁^∞ (1-q^m)⁻¹\) has a modular form Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com