The Factorial Barrier: Where Prime 5 Enters Mathematics

A positive integer is 3-smooth if its only prime factors are 2 and 3. I prove that n! is 3-smooth if and only if n ≤ 4, and show that this single fact, the factorial barrier, controls arithmetic thresholds in at least fourteen independent mathematical domains: unit ball volumes, sphere packing densities, Weyl group orders of simple Lie algebras, Runge–Kutta order conditions, Newton–Cotes quadrature weights, Halley root-extraction iterations, even zeta values, Catalan numbers, Narayana numbers, symmetric group representations, Fibonacci numbers, Platonic solids, kissing numbers, and Virasoro minimal model central charges. In every case except one, prime 5 is the obstruction that forces the first departure from 3-smoothness. The sole exception, Motzkin numbers where prime 7 appears first, is explained by the mod-3 residue structure of the generating function, following Deutsch–Sagan (2006) and Burns (2017). The paper requires no background beyond elementary number theory. Its argument is that the simultaneous appearance of the same arithmetic boundary across domains with no mutual logical dependence constitutes a structural phenomenon deserving of recognition.