more familiar context: the “infamous upper tail" [@Janson2] of the triangle count $T$ in the Erdős–Rényi random graph, $G(n,p)$. After many years of development of increasingly strong bounds, the first breakthrough was made by Kim and Vu [@Vu] and Janson [@Janson3] independently, who proved that, for any $\delta >0$ and whenever $p \gg (\log n)/n$, $$\exp[ - c(\delta) n^2 p^2 \log(1/p)] \leq \mathbb{P}[T > (1 + \delta)\mathbb{E}[T]] \leq \exp[- C(\delta) n^2 p^2] \, ,$$ where $c(\cdot)$ and $C(\cdot)$ depend only on $\delta$. Recently, there has been renewed interest in these type of tail estimates. In 2010, Chatterjee [@Chatterjee2] and Demarco and Kahn [@Demarco] (in independent works) established the correct order of the upper tail of triangles and other small cliques by adding the missing logarithmic term to the upper bound, without providing good control of the leading-order constants. The recent work of Chatterjee and Dembo [@Chatterjee3] on nonlinear large deviations allowed Lubetzky and Zhao [@Lubetzky] to calculate both the order and the leading-order constant for the upper tail question in a certain regime of sparse Erdős–Rényi random graphs. Unfortunately, the technique is based on the analysis of polynomials in Bernoulli random variables, and thus ill-suited to the problem we are 