jl-lemma-bound-tight-up-to-constants

Summary

The Johnson-Lindenstrauss bound is not just a convenient trick; it is essentially the best you can do. There are configurations of points that genuinely cannot be embedded into a much smaller space while preserving pairwise distances, so any system relying on dimensionality reduction should treat the log-N / epsilon-squared threshold as a hard wall rather than something that might be beaten by a smarter algorithm.