The Table Has a Last Door: The Periodic Table Ends at Element 137, Forced by the Fine-Structure Constant

One axiom. One operation. Zero free parameters. Chemistry has never said where its own table ends. This paper answers it exactly: the periodic table is finite and ends at element 137, the integer part of the inverse fine-structure constant. The innermost (1s) electron's binding coupling is Z·α. The fold forbids any bound part from exceeding the whole, so the coupling reaches its ceiling at Z·α = 1, i.e. Z = 1/α = 137.036. The last whole-charge element is 137, and element 138 cannot exist (it would need a bound coupling 34500/34259 = 1.0070… > 1). Because the same theory forces 1/α = 34259/250 = 137.036 with zero free parameters, it forces the end of the elements at the very number that sets the strength of light: the coupling reaching the One, read once as a strength and once as a limit. The consensus ~173 is a finite-nuclear-size dressing of the structural threshold, not a replacement for it. A standalone result within the Smithian Fold Theory of Everything (SFTOE), building on The Counted Constant (the fine-structure constant). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory