Non-existence of stabilizer absolutely maximally entangled states across infinitely many configurations

We prove a general reduction theorem for stabilizer absolutely maximally entangled states in composite local dimension. If a stabilizer $\mathrm{A}\mathrm{M}\mathrm{E}(n,D)$ state exists and $D={\prod }_{i=1}^m{q}_i$ is the prime-power factorization of $D$, then for every nonempty subset of factors there exists a stabilizer $\mathrm{A}\mathrm{M}\mathrm{E}(n,{\prod }_{i\in M}{q}_i)$ state. Thus any obstruction at a prime-power factor immediately obstructs stabilizer AME states in the composite dimension.