Conditional Fluctuations of Mean-Field Systems with Stable Common Noise

We study conditional fluctuations of smooth mean-field particle systems on a torus with endogenous symmetric stable common jumps of index $0<\\alpha<1$. Under a common Poisson construction, a positive-strip estimate makes empirical acceptance errors negligible at the sampling scale and yields a functional central limit theorem in a negative Sobolev space. The conditional laws, given the full Poisson master, converge in probability to a centered Gaussian kernel. In dimension one, quantile coupling transfers this limit to microscopic marks whose relative tail remainder is $O(x^{-\\rho})$ with $\\rho>\\alpha/2$. Under a quantified second-order tail expansion, we identify the endogenous rank response: it shifts the conditional Gaussian mean at criticality and dominates sampling below criticality whenever the response is nonzero. We also establish a separate stable-to-Brownian transition for the rank error. Finally, a translation model with identical microscopic laws under rank and clock couplings admits a functional Gaussian limit in the former coupling but no diverging additive path normalization at the limiting center in the latter.