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

Mathematics Branch Paper 001, version 1.4.0, preserves the complete mathematical-foundations and Smithian Fold Scientific Calculator evidence while publishing the full-field Mathematics roadmap and keeping the exact discoveries, their meaning and the Ernos Labs public-science standard before artifact identities. The twenty-two foundational claims execute 9,984 generated candidate structures and reconstruct exact arithmetic, discrete mathematics, combinatorics, graph and network theory, algebra, order and lattices, finite geometry and topology, deterministic-support probability and statistics, optimization, finite dynamics, proof theory and exact composition. The complete 763-entry V1/V2 source surface remains reviewed: 71 atomic Mathematics obligations are assigned and closed, with none unclassified or open at the version boundary. The calculator lineage preserves five exact claim records and 11,520 further candidate structures, including superseded adverse evidence and its independently forced correction. It provides one proof-producing evaluator, an intuitive calculator surface, fail-closed invalid-form behavior, and local application access on macOS, Windows, Linux and same-network phones. Displayed 0 denotes structural empty One; exact decimals become rational Fold parts; prohibited outputs halt transactionally. Version 1.4 preserves every admitted result and registers the complete-field programme across arithmetic, discrete structures, algebra, order, geometry, topology, deterministic probability, optimization, dynamics, logic and compositional mathematics. A calculator output is exact computational evidence, not automatic theorem admission: every new claim must still pass registration, complete enumeration, uniqueness, controls, independent validation and the untouched engine.