Håstad's general-graph inapproximability result supplies the hardness theorem used by the reduction. We use its rational promise-gap form. For every integer $q>2$, a bounded-error polynomial-step algorithm distinguishing $n$-vertex graphs $H$ with $\alpha(H)\leq n^{1/q}$ from those with $n^{1-1/q}\leq\alpha(H)$ would imply $NP\subseteq BPP$.

This formulation is proved here from the registered PCP-to-clique reduction in lax-253009. The proof complements the graph, guards the relative threshold, handles graphs below the solver's cutoff by bounded exhaustive search, and certifies the encoding conversion and randomized composition in the finite Turing-machine models.