Modular Congruences for Generalized Cubic Partitions and Plane Partition Diamonds — E8 Intelligence Research

FINDING: The arXiv paper (2508.18286v3) is the sole substantive hit — it establishes congruences modulo 7 and 11 for two restricted partition functions: generalized cubic partitions \\(a_c(n)\\) (even parts in \\(c\\) colors) and \\(d_c(n)\\) counting \\(c\\)-elongated plane partition diamonds. The generating functions are eta-quotients, and the congruences are proven via modular forms. MATH: - \\(a_c(n)\\): generating function \\(\\sum_{n\\ge0} a_c(n)q^n = \\prod_{n\\ge1} \\frac{1}{(1-q^{2n-1})(1-q^{2n})^c}\\) — equivalently \\(\\frac{(q^2;q^2)_\\infty^c}{(q;q)_\\infty^{c+1}}\\) (eta-quotient form). - \\(d_c(n)\\): generating function for \\(c\\)-elongated plane partition diamonds — standard form: \\(\\prod_{n\\ge1} \\frac{(1-q^{2n})^c}{(1-q^n)^{c+1}}\\) (up to shift; exact form in paper). - Congruences: for specific \\(c\\) (likely \\(c=1,2,3\\) or similar), \\(a_c(An+B)\\equiv 0 \\pmod{7}\\) and \\(d_c(Cn+D)\\equiv 0 \\pmod{11}\\) — exact moduli and arithmetic progressions in the paper. - Eta-quotient: \\(\\eta(z)^a \\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com