Quantitative conjugacy-class products in metric ultraproducts of finite simple groups

We prove a quantitative conjugacy-class product theorem for nondiscrete metric ultraproducts of finite nonabelian simple groups over a countable index set. For the normalised logarithmic conjugacy metric, every pair of nonidentity inputs admits a product separated from the entire conjugacy orbit of their original product by at least the square of the smaller input length divided by 73728. We also obtain an attained interval of natural-length values and exactly continuum many conjugacy classes in every such class product. The proof combines uniform finite support estimates, interpolation along generating words and comparison of the natural metrics with conjugacy length.