From Nothing to Fold: A Premise-Free, Parameter-Free and Machine-Closed Foundation for Smithian Fold Theory

From Nothing to Fold reports the completed Foundation branch of the third clean-room reconstruction of Smithian Fold Theory (SFT). Beginning with the premise-free operational theorem there is no nothing, it derives in dependency order the structural One, exact positive finite count, exact positive part coordinates, cross-partition equivalence, the minimal structural Fold, complete finite Fold-word support, the complete recursive foundational form grammar, canonical form identity and the one-way derivation-to-measurement boundary. No conventional number axioms, numerical zero, negative quantity, irrational or imaginary proof value, floating-point proof, completed infinity, fitted parameter, trained weight, target answer or previous SFT conclusion is admitted as a premise. The ten formal claims are processed through one fail-closed admission engine. Their complete declared candidate grammars contain 2,450 generated classes. Every class is decided; every admitted claim has exactly one survivor, minimality and named-shape uniqueness, a depth-independent base/successor or structural-induction certificate, four adverse-control classes, a cryptographic derivation seal and implementation-distinct recomputation. The accompanying open evidence archive contains the paper source, frozen branch inventory, complete claim packages, candidate censuses, elimination receipts, controls, certificates, executable scientific modules, engine receipts, independent validators, publication evidence map and verification tests. The definitive repository run reports 96 unit, adverse-control and end-to-end tests, 1,264 of 1,264 core-engine executable lines observed across 15 modules, and fresh independent execution of all ten derivations. The Foundation branch is formally closed within its declared generated finite boundary. This record does not claim closure of downstream mathematics, information science, computation, physics or empirical natural science. It establishes the exact evidence objects and one-way measurement discipline those later branches must cite.