Spherical Number Algebras: An Invariant-First Framework with Quotient Semantics and Symbolic-Numeric Translation

This work introduces Spherical Number Algebras (SNA) as a foundational invariant-first framework in pure mathematics. Rather than taking numbers, objects, or metric continua as primitive, the theory begins with admissible assembly-events endowed with invariant packages from which observable, symbolic, and numerical descriptions are derived. The paper is divided into three parts: a conceptual architecture of admissibility and invariant primacy; a point-free pre-numeric formal core based on normalization, quotient semantics, and rewriting-theoretic convergence; and a translation layer in which symbolic codings and numeric valuations appear as tertiary projections. The framework thereby establishes a structurally coherent basis for future realizations involving stabilization theory, derived observables, and richer algebraic forms.