The Fine-Structure Constant, Derived to Its Terminal Order: An Exact Zero-Parameter Rational Value

Smithian Fold Theory derives the fine-structure constant as one exact, finite and terminal rational object. No electromagnetic measurement, fitted parameter or selected decimal enters the derivation. The machine-checked self-proven foundation forces the One and fold; the fold's period spectrum supplies binary two and colour three; those counts force covering depths five and seven; and the three interchangeable spatial directions generate one finite four-rung ladder. The counted rungs are 125, 175, 245 and 343. Exactly one distinct successor exists at each non-terminal rung and none exists after the fourth. The resulting terminal value is 1/alpha = 503846395469/3676744786 = 137.035999177180855326.... The construction is an exact rational fraction, not an infinite correction series or a regression against measured alpha. Independent machine modules agree exactly at the first and second orders. The terminal certificate checks the unique promotion law, all four rungs, exact fractions, more-than-thousandfold collapse of successive shifts, the terminal decimal and the absence of a fifth successor. Generator mutations, rival covering shapes and alternative promotion constructions are rejected by exact identities. Only after the value is sealed is it compared with experiment. It lies approximately 0.009 standard deviations from the CODATA 2022 recommended inverse alpha, calls the unresolved digits ...177181, and fixes a definite resolution point between the discordant caesium and rubidium recoil determinations. The paper states the executable falsification conditions and provides the exact repository paths for reproduction. Open source and machine evidence: Smithian Fold Theory of Everything. Author and publication authority: Maria Smith, Ernos Labs.