Extending Partition-Theoretic Prime Detection: The Cusp Form Barrier and the Derivation of E₅

The study of integer partitions and prime numbers has historically occupied distinct branches of number theory—additive and multiplicative, respectively. However, recent results by Craig, van Ittersum, and Ono 1 establish a direct algebraic bridge between the two. They proved that for n 2, specific non-negative polynomial combinations of MacMahon partition functions vanish if and only if n is prime. The MacMahon functions, Mₐ (n), represent weighted sums over strict a-part partitions of n, defined as: Mₐ (n) = ₀ < ₒ䃑 < < ₒ䂯 \\ ₌㶁 ₁, \; ₌㶁 ₒ㶁 = ₍ m₁ m₂ mₐ By leveraging the **quasi-shuffle algebra** of these functions—an algebra structure on formal power series of MacMahon partition functions, governed by the quasi-shuffle product and Ramanujan's differential equations (see 1, Section 3) —and their connection to quasimodular forms, Craig–van Ittersum–Ono constructed a sequence of prime-detecting expressions E₁ (n) through E₄ (n), where each Eₖ introduces M₊+₁ as its highest-weight component. They conjectured that any non-negative prime-vanishing expression in Qn \Mₐ\ is a Qn-linear combination of these foundational entries. In this paper we develop an exact computational framework in Julia to test and extend this conjecture. A computational sweep reveals that the conjecture fails for a_ 5 unless a fifth expression E₅ (n) is added. We derive E₅ explicitly. A natural extrapolation of the established pattern suggests E₅ should incorporate M₆ (n). However, we demonstrate that this extrapolation fails due to modular arithmetic. The generating function U₆ (q) = ₍ ₁ M₆ (n) qⁿ is a quasimodular form of weight 12. At weight 12, the space of modular forms on SL₂ (Z) becomes two-dimensional, spanned by the Eisenstein series E₁₂ and the unique cusp form (q) = (n) qⁿ (Ramanujan's delta function). The resulting (n) component in M₆ forces the M₆ columns to be pivot columns in the prime evaluation matrix: no prime-vanishing expression can involve M₆ (n). Confined to M₁ through M₅, we perform a degree sweep. Contrary to the initial expectation that E₅ would appear at polynomial degree d=3 (matching the pattern of E₁, , E₄), we find that E₅ first appears already at d=2. This is the minimal-degree canonical form of E₅, and it is what we record and verify. The extraction procedure yields one unique new direction outside the Qn-span of E₁–E₄ at d=2; this direction is E₅.