The Closed Census: Why the Inventory of Everything That Can Exist Is Finite

One axiom. One operation. Zero free parameters. Is the list of things that can exist finite or endless? Physics leaves it open. This paper closes it. The fold has no continuum and no infinity: every magnitude lies between the floor and the One, so every ladder, nucleus, and spectrum is finite and countable. Three bounds: the tower height — the deepest depth is seven, so no excitation ladder exceeds 2⁷ = 128 rungs; the element cap — the table ends at 137 = 1/α; the species count — the force sectors are sealed at four (primes 2,3,5,7), so the fundamental species are a finite list (gauge carriers, photon/graviton/Higgs, the Standard-Model fermions, the twelve Smithions). A finite set of species, each with a finite tower of at most 128 rungs, combining to a bounded mass: the inventory of everything is a finite count, not an endless list. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory