Featured publication · Liju James · 2026

An exact census of prime-detecting MacMahon combinations

A self-contained certified-computation artifact classifying prime-vanishing MacMahon combinations within K ≤ 8, degree ≤ 10, validated through n ≤ 3000.

Publication & artifacts ↗Research index

5Primitive generators

2, 3, 4, 5, 7Top levels

No level 6Gap proved directly

Exact arithmeticIntegers and rationals

The question

Which polynomial combinations of MacMahon's partition functions vanish precisely in the prime-detecting setting, and how much of that structure can be classified rather than merely sampled?

The result

The ℚ[n]-module of combinations in the declared census has exactly five primitive generators and generates the same module as Craig–van Ittersum–Ono Table 1. The work also computes exact cuspidal components for M₆, M₇, and M₈; M₈ alone carries the weight-16 cusp form ΔE₄.

Proved beyond the finite census: the dimension law dim V(K,D) = (K−1)(D−1) for 2 ≤ K ≤ 5 at every degree, completeness for K ≤ 5, the level-6 gap without invoking the classification theorem, and the degree-threshold rule 2(K−K′).

Why the evidence matters

Every result, figure, and the paper itself regenerates bit-identically. The supplied verifier checks the SHA-256 manifest and each certified claim. Floating-point approximation is not part of the proof path.

Included evidence

Standard-library Python regeneration code
Exact integer and rational computations
Machine-readable result files and figures
A one-command verification entry point

Claim boundary

The exact census is stated for K ≤ 8, degree ≤ 10, and validation through n ≤ 3000. Broader theorems are identified separately in the artifact so the finite computation and proved general statements are not conflated.