A Polynomial Recovery Criterion for Forced Commutativity in the Matrix Square-Root Fiber of a Square-Free Cubic Polynomial

Let $k$ be an algebraically closed field of characteristic different from $2$ and $3$, and let $f(t)=t^3+at+b\in k[t]$ be square-free. For $X\in M_m(k)$ set $A=f(X)$. This paper studies the full matrix equation $Y^2=A$, without imposing $[X,Y]=0$. Existence is the classical square-root problem for the fixed matrix $A$. The only obstruction is the nilpotent square-root pairing criterion on the zero-primary component of $A$. The main result is a fiberwise commutativity classification: if the fiber $\{Y\in M_m(k):Y^2=A\}$ is nonempty, then every such $Y$ commutes with $X$ if and only if $X\in k[A]$. The forward direction restricts the classical centralizer criterion of Thompson (1969) to the square root fiber. The new content is the converse: when $X\mathrm{\notin }k[A]$, an explicit square root $Y$ of $A$ with $[X,Y]\mathrm{\ne }0$ is exhibited. The witnesses arise from nonzero spectral collision under $f$, critical nilpotent collapse, or a multicolor zero-primary fiber.