The structure of $k$-potents and mixed Jordan-power preservers on matrix algebras

Let $M_n(\mathbb{F})$ denote the algebra of $n\times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic different from $2$. For $n\ge 2$, we classify all maps $\phi :M_n(\mathbb{F})\to M_n(\mathbb{F})$ satisfying the mixed Jordan-power identity $$\phi (A^k\circ B)=\phi (A{)}^k\circ \phi (B),\text{for all }A,B\in M_n(\mathbb{F}),$$ where $\circ$ denotes the (normalized) Jordan product $A\circ B:=\frac{1}{2}(AB+BA)$ and $k\in \mathbb{N}$. We show that every such map is either constant, taking a fixed $(k+1)$-potent value, or there exist an invertible matrix $T\in M_n(\mathbb{F})$, a ring monomorphism $\omega :\mathbb{F}\to \mathbb{F}$, and a $k$-th root of unity $\epsilon \in \mathbb{F}$ such that $\phi$ takes one of the forms $$\phi (X)=\epsilon T\omega (X)T^{-1}\text{ or }\phi (X)=\epsilon T\omega (X{)}^t{T}^{-1},$$ where $\omega (X)$ denotes the matrix obtained by applying $\omega$ entrywise to $X$, and $(\cdot {)}^t$ denotes matrix transposition. In particular, every nonconstant solution is necessarily additive. The classification relies fundamentally on the preservation of $(k+1)$-potents and their intrinsic structural properties.