Central Limit Theorem Breakdown: Heavy-Tailed Convergence

Most Central Limit Theorem demos only show the theorem succeeding: any skewed source, given enough samples, produces sample means that pile up into a bell curve. This simulation asks the more interesting question — when does that promise actually fail? It draws sample means from five sources spanning a spectrum of tail weight, from a plain Gaussian through Exponential and two Pareto tail indices (α=3, with finite variance, and α=1.5, without) to a Cauchy distribution with no finite mean at all, and standardizes every one by the theorem's own √n·(X̄_n−μ)/σ formula in real time. For the finite-variance sources the live 3D histogram tightens around the standard normal curve exactly as advertised. For the infinite-variance sources it visibly refuses to — the off-axis rate climbs instead of falling, because the correct scaling for a heavy tail with index α<2 grows like n^(1/α), not √n, so the classical normalization under-corrects and the spread keeps growing with n.