Numbers as Snap Geometry

Numbers are not quantities. They are discrete coherence states produced by threshold transitions in a continuous field. This paper develops a structural ontology of number grounded in Snap Geometry — the domain‑level expression of the Foundational Architecture. Under this view, numbers arise from Snap events: non‑linear threshold transitions that reorganize a continuous coherence field into stable, countable basins. Arithmetic, algebra, calculus, and topology emerge as transformations of Snap sequences and the gradients between them. Numbers become indices of coherence rather than objects; operations become reorganizations of boundaries rather than symbolic manipulations. This framework reframes the foundations of mathematics through the lens of coherence physics, revealing number as a structural phenomenon — a geometry of coherence emerging from threshold transitions.