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

Mathematics Branch Paper 001, version 1.3.0, preserves the complete version 1.2 mathematical-foundations and Smithian Fold Scientific Calculator evidence while placing 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 addition preserves five exact claim records and 11,520 further candidate structures. Claim 003 remains visible as superseded adverse evidence after a later audit found reversed alternating arctangent endpoint parity. Claim 004 independently corrects the exact enclosure law. Claim 005 forces the declared scientific expression language and stateful app; claim 006 completes the calculator-first interface, law explorer, fail-closed resources and full active-file coverage; claim 007 preserves the immutable evaluator while forcing a standards-rendered local application for macOS, Windows, Linux and same-network phones. This is not conventional floating arithmetic with SFT branding. Displayed 0 denotes structural empty One. A result crossing below that boundary halts transactionally and cannot enter answer, memory or history. Exact decimals become rational Fold parts; non-rational correspondence values remain certified rational intervals; orthogonal values remain typed Fold fibres; NaN, infinity, floating transcendental answers and silent complex promotion are excluded. The visible browser performs no arithmetic and every action reaches one admitted exact evaluator. The paper documents the complete candidate axes, rejected alternatives, unique survivors, witnesses, adverse controls, correction lineage, source and receipt identities, operation grammar, memory and history semantics, phone behavior, local-network boundary, portability, test scope and coverage. Claim 006 records 1,869 statements and 696 branches at 100 percent coverage; claim 007 records 293 statements and 86 branches at 100 percent coverage. Direct manual testing and its unfavorable blank-widget result are retained as part of the correction route. A calculator output is exact computational evidence, not an automatic theorem admission. New claims must still pass registration, complete enumeration, uniqueness, controls, independent validation and the untouched SFT admission engine. The PDF is the primary landing-page paper; the complete Markdown, evidence map, publication receipt, frozen version inventory and exact source/evidence archive are deposited beside it.