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)\\equiv 0 \\pmod{5}\\), \\(p(7k+5)\\equiv 0 \\pmod{7}\\), \\(p(11k+6)\\equiv 0 \\pmod{11}\\). Hecke operator \\(T_p\\) acts on modular forms of weight \\(k\\) and level \\(N\\): \\((T_p f)(z) = p^{k-1} f(pz) + \\frac{1}{p}\\sum_{b=0}^{p-1} f\\left(\\frac{z+b}{p}\\right)\\). For the new results (arXiv:2508.18286v3): \\(a_c(n)\\) and \\(d_c(n)\\) satisfy congruences of form \\(a_c(An+B)\\equiv 0 \\pmod{7}\\) and \\(d_c(An+B)\\equiv 0 \\pmod{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 \\(\\prod_{m=1}^\\infty (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