From Fold to Mathematics: An Exact, Parameter-Free and Machine-Closed Derivation of Mathematical Foundations from Smithian Fold Theory

From Fold to Mathematics is the standalone paper for the completed Mathematics branch of the third clean-room reconstruction of Smithian Fold Theory (SFT). Starting only from the ten admitted Foundation receipts, it derives in dependency order exact arithmetic and number structure; discrete mathematics; combinatorics; graph and network theory; algebraic structures; order and lattice structure; finite computational geometry and topology; exact probability and statistics; optimization; finite dynamical systems; logic and proof theory; and category, type and compositional structure. No conventional mathematical axiom system, semantic numerical zero, negative quantity, irrational or imaginary proof value, floating-point proof quantity, completed infinity, ungenerated continuum, stochastic cause, fitted parameter, pretrained model or application result is admitted as a premise. The twelve complete claim grammars execute 7,424 generated candidate structures. Every structure is decided, each grammar has exactly one all-preserving survivor, and every claim passes minimality, named-shape uniqueness, a depth-independent base/successor certificate, four adverse-control classes, cryptographic sealing and implementation-distinct recomputation. The 41-page paper documents every derivation in full: dependency chain, exact theorem, complete candidate axes, every rejection class, unique survivor, operational laws, unfavorable witnesses, induction certificate, scientific meaning, correspondence boundary, exact exclusions and source, validator, seal and receipt identities. The formal probability theorem remains superdeterministic: uncertainty is the exact relation between complete deterministic support and distinctions closed by observation, not an imported random cause. The accompanying evidence archive contains the Markdown source, archival PDF, frozen inventory, all twelve complete claim packages, 7,424 candidate records and decisions, controls, certificates, executable law modules, independent validators, engine receipts, tests, publication evidence map and checksum ledger. The definitive cross-platform verification passes 111 unit and end-to-end tests, observes all 1,264 executable lines of the 15-module core engine and freshly reruns all 22 admitted Foundation and Mathematics derivations in dependency order. The Mathematics branch is formally closed within its exact generated finite boundary and contains no unclassified or frontier obligation. The next branch is Information Science. Named special structures and natural empirical claims still require their own generated witnesses and, where observation is involved, preregistered blind validation.