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 < s₁ < ⋯ < sₐ \\ mᵢ ≥ 1,\; ∑ mᵢ sᵢ = n 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 Mₖ₊₁ as its highest-weight component. They conjectured that any non-negative prime-vanishing expression in Q[n] ⊗ ₐ\ is a Q[n]-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 aₘₐₓ ≥ 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) = ∑n ≥ 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 Δ(q) = ∑ τ(n) qⁿ (Ramanujan's delta function). The resulting τ(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 Q[n]-span of E₁–E₄ at $d=2$; this direction is E₅.