Quantitative non-Gaussian and Gaussian approximation for exponential random graphs in the single-phase critical regime

We establish quantitative non-Gaussian and Gaussian approximations for edge-count fluctuations in dense exponential random graphs at single-phase criticality of any fixed even order $2m_*\ge 4$. The Hamiltonians use injective homomorphism counts of fixed simple graphs, with nonnegative leading coefficients for the higher-order terms. In bounded calibrated windows, edge-density fluctuations on the scale $n^{-1/m_*}$ admit an explicit non-Gaussian approximation with error $O(n^{-1/m_*})$. For parameter perturbations by fixed multiples of $n^{-1}$ with a nonzero first-order drift correction at the critical density, the deterministically centered edge count admits a Gaussian approximation on the scale $n^{(3m_*-2)/(2m_*-1)}$, with error $O(n^{-1/(2m_*-1)})$. Both rates hold in Wasserstein and Kolmogorov distances. The key step is to construct auxiliary kernels for exterior graphs outside an event of exponentially small probability. These kernels contract uniformly on the full row state space and agree with the conditional Gibbs updates on typical row configurations, yielding the conditional cumulant estimates used in Stein's method.