Obstructions and kernel transport for Hecke lifts of partition q-brackets

Abstract

We study lifts of the level-one Hecke action on quasimodular forms through the partition q-bracket. We prove two obstruction theorems: no exact lift on the genuine shifted symmetric algebra $\mathbb{Q}[Q_2,Q_3,\ldots]$ is multiplicative, and no exact lift satisfies a strict $Q_2$-tower condition. We classify fixed-weight exact lifts by kernel actions and kernel-valued Hecke cocycles, construct transported scalar lifts under Hecke stability of the q-bracket image, and derive kernel-transport and spectral-divisibility consequences from the injectivity of Zagier's lowering operator $B=\frac12(D-\partial^2)$ on the genuine homogeneous subspace. Exact rational rank computations show q-bracket surjectivity in weights at most $16$, yielding explicit Hecke lifts and kernel data in those weights.