From Fold to Mathematics: An Exact, Parameter-Free and Machine-Closed Complete-Field Derivation from Smithian Fold Theory

Mathematics Branch Paper 001, version 1.5.0, executes the complete-field roadmap: 323 of 323 frozen obligations across 24 families, 97,280 generated candidates, 323 unique survivors and 1,292 passed controls. The paper preserves the complete foundation, prior-corpus and scientific-calculator record and adds claim-by-claim derivation, observation, boundary, certificate and receipt detail for arithmetic, algebra, combinatorics, graphs, geometry, topology, analysis, probability, optimization, dynamics, logic, category, numerical, symbolic, validation and handoff families. The branch is complete to its dated census and explicitly open to lawful extension. No axiom, free or fitted parameter, semantic numerical zero, negative, irrational, imaginary or floating proof scalar, completed infinity, ungenerated continuum or stochastic cause is admitted as a premise. The engine and verification authority remain unchanged.