Let M_C1(m,n) and M_C5(m,n) be the coefficients defined by the corresponding two-variable generating functions in the source, with m∈Z and n∈N, where N denotes the positive integers. Garvan–Jennings-Shaffer nonnegativity conjecture. For every m∈Z and n∈N, both M_C1(m,n) and M_C5(m,n) are nonnegative. The paper's title states that this conjecture is proved, so its status is solved; the claim concerns the nonnegativity of coefficients associated with the rank-like functions M_C1 and M_C5.

Garvan–Jennings-Shaffer nonnegativity conjecture for M_C1 and M_C5

Garvan and Jennings-Shaffer proposed the conjecture in 2016: the coefficient functions $M_{C1}$ and $M_{C5}$ should be nonnegative for every $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ .

Known results

Jang and Kim proved positivity for each fixed $m$ and sufficiently large $n$ using Wright’s circle method.

September 16, 2025 claimed proof

On September 16, 2025, Bing He and Shuming Liu’s preprint Proof of a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of $M_{C1}(m,n)$ and $M_{C5}(m,n)$ claimed a complete proof. Their lattice-point argument reduces the two cases to auxiliary coefficient sequences and includes finite Mathematica checks; a later preprint also cites their result, but no independent verification or published correction is recorded.

Current status (as of September 2026): A preprint claims the conjecture is proved for all $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ , but the proof remains unverified in this record.

Garvan–Jennings-Shaffer nonnegativity conjecture for M_C1 and M_C5

Primary source

Progress summary

Solutions 0