Monomial Moving Zeros for the Heat Semigroup: An Explicit Finite-Order Differential Hierarchy

Let $P_t=\exp (t{\text{D}}^2\mathrm{/}2)$ be the heat semigroup. For every integer $k\ge 2$ and parameter $\rho \in \text{C}$, we construct an entire function ${\mathrm{\Gamma }}_{\rho ,k}$ whose heat evolution has the prescribed moving zero $x=\rho t^k$ and constant spatial flux there: $$(P_t{\mathrm{\Gamma }}_{\rho ,k})(\rho t^k)=0,{\mathrm{\partial }}_x(P_t{\mathrm{\Gamma }}_{\rho ,k})(\rho t^k)=1.$$ The initial function is explicit, $${\mathrm{\Gamma }}_{\rho ,k}(x)=x\sum _{q\ge 0}\frac{(-2^k\rho x^{2k-1}{)}^q}{q!}\frac{(kq)!}{((2k-1)q+1)!},$$ and is a generalized hypergeometric function whose entire-function order is strictly below two. Most importantly, it is annihilated by an explicit polynomial-coefficient differential operator of order $2k-1$. Thus every monomial moving boundary admits a finite-order stationary realization after a natural boundary-flux selection, even though the naive order-$k$ family ${\text{D}}^k+a{x}^{k-1}$ already fails to produce the exact boundary at $k=4$. No minimality claim is made: the result gives a universal finite-order hierarchy, while the least possible differential order remains open and depends on the choice made inside the inverse kernel.