New Ramanujan-Type Congruences for Restricted Partition Functions via Modular Forms — E8 Intelligence Research

FINDING: Ramanujan-type partition congruences (mod 5, 7, 11) are being extended to restricted partition functions via modular forms and Hecke operators, with new congruences modulo 7 and 11 for generalized cubic partitions and elongated plane partition diamonds. | MATH: Classical congruences: p(5n+4)≡0 (mod 5), p(7n+5)≡0 (mod 7), p(11n+6)≡0 (mod 11). New results (arXiv:2508.18286v3): For a_c(n) (c-colored even parts) and d_c(n) (c-elongated plane partition diamonds), congruences of form a_c(7n+r)≡0 and d_c(11n+s)≡0 for specific residues r, s, with c-dependent conditions. Hecke operators T_p act on modular forms of level 5 (and 7, 11), with eigenvalues related to Fourier coefficients; the functional equation for Big Theta resembles automorphy, linking to weight-1/2 or weight-3/2 forms. | CONNECTION: The primes 5, 7, 11 are the first three primes where the modular group Γ₀(N) has genus 0 (N=5,7,11) — a crystallographic/lattice symmetry: these are the only primes with this property, tied Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com