Sharp low-degree prediction exponents for noisy polynomial single-index models

Let $X$ be standard Gaussian in ${\mathbb{R}}^D$ and let $Y=f(u\cdot X)+sG$, where $u$ is an unknown unit vector, $G$ is independent standard Gaussian noise, and $f$ is a fixed centered polynomial with $\mathbb{E}f(Z{)}^2=1$ for standard Gaussian $Z$. Suppose its lowest nonzero Hermite order is $p\ge 3$. For $s=D^a$, $a>0$, we determine the sample exponent $B_f(a)$ for polynomial prediction of $f(u\cdot x)$ from finitely many transformed-moment indices. Below $B_f(a)$ by any fixed positive margin, expected minimax risk tends to one for label degree a positive power of $D$, even with arbitrary measurable design and query coefficients. Above it by any fixed positive margin, an explicit predictor of total degree $O(\log D)$ in the design, labels and query has expected risk tending to zero, with supplied link coefficients and no noise parameters. The upper constructions use known transformed-moment recovery ingredients. For pure odd Hermite links, the upper scale $\min \{D{s}^4,D^{p/2}{s}^2\}$ already follows for recovery, up to logarithms, from existing quantitative single-index bounds. The unrestricted fixed-accuracy sample requirement is ${\Theta }_P(D{s}^2)$ for link degree at most $P$. Hermite rank alone does not determine the polynomial exponent: for fixed odd rank-three links of arbitrary degree, the possible curves are exactly $B_f(a)=\min \{1+\kappa a,3/2+2a\}$ for even $\kappa \ge 4$. Thus every fixed $a>0$ admits a fixed link attaining $3/2+2a$, whose degree may depend on $a$. A degree-$21$ example gives $7/4$ at $a=1/8$, versus the pure cubic's $3/2$.