The Complete Table: The Entire Architecture of the Periodic Table, Forced from Three Dimensions and Spin

One axiom. One operation. Zero free parameters. The periodic table has a peculiar shape — rows of 2, 8, 8, 18, 18, 32, 32 — with noble gases at 2, 10, 18, 36, 54, 86, 118. This paper forces the whole architecture from three spatial dimensions and spin. Three dimensions give each angular kind l its 2l+1 orientations; spin (the fold base two) doubles each orbital, so a subshell holds 2(2l+1): s=2, p=6, d=10, f=14, g=18, and a shell holds 2n². The periodic recurrence closes a row each time the covering pattern returns to unison, and the row lengths follow the doubled-square pattern 2, 8, 8, 18, 18, 32, 32, giving the noble-gas closures 2, 10, 18, 36, 54, 86, 118, 168. The table does not reach 168: it ends at element 137 = 1/α, cutting the eighth period partway, and the elements 119-137 open a brand-new g-block (l=4), never before entered. The entire scaffolding of chemistry is three dimensions and a doubling. 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