We study stochastic fixed-point iterations $x_{n+1}=θ_nx_0+(1-θ_n)\widehat T(x_n)$ for nonexpansive and contractive operators on Hilbert spaces, with a single-point unbiased oracle of bounded variance. Deterministically, every anchor schedule $θ_n=c_n/(n+2)$ whose density $c_n\in(0,1]$ does not oscillate between scales is either polynomially suboptimal on contractions or super-polynomially slow on rotations; for constant densities $θ_n=c/(n+2)$ the tradeoff is exact, with $\norm{x_n}\asympΓ(c+1)(n\varphi)^{-c}$ on rotations by angle $\varphi$ ($\varphi\to0$, $n\varphi\to\infty$) and, for the classical schedule, $\norm{x_n}=\frac{2|\sin(n\varphi/2)|}{n\varphi}+O(\frac1n)$. The dichotomy fails exactly under lacunary anchor concentration: an explicit $γ$-oblivious schedule, one per target accuracy, is simultaneously contraction-optimal and rotation-polynomial with exponent $1/α_0$, $α_0=\log_2(3/2)$, and, for each fixed target accuracy, the matching lower bound holds for Lebesgue-a.e.\ angle. For stochastic contractions, the minimax rate is $Θ(σ^2\eps^{-2}(1-γ)^{-2})$ on affine maps, and within the anchored class $Θ(σ^2\eps^{-2}(1-γ)^{-2}+(1-γ)^{-1}\ln(D/\eps))$ for known modulus via geometric batching; without a certified modulus bound no sound distance certificate exists, and a certified ceiling is the exact boundary. For nonexpansive maps we prove $O(Kσ^2D^2\eps^{-4})$ in every $2$-uniformly smooth Banach space and the Hilbert rate $\tildeΘ(σ^2\eps^{-2}+D\eps^{-1})$ by reduction to monotone inclusions. Numerical experiments are consistent with every scaling prediction.

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