Symbolic Arithmetic of Physical Constants: Foundations of a New Branch — Complexity Hierarchies for Dimensionless Constants

We propose the foundations of a new branch of arithmetic — the Symbolic Arithmetic of Physical Constants — which asks whether the dimensionless constants of physics occupy specific, identifiable positions in a hierarchy of symbolic complexity, or whether they are distributed arbitrarily among the reals. We define a formal complexity metric S over a symbolic grammar G of physical and mathematical constants, establish a five-level complexity hierarchy for real numbers, and state the Foundational Conjecture: every physical constant admits a minimal symbolic expression at some finite depth. The first empirical result — a negative result at depth d=3 showing no statistically significant enhanced compressibility (Mann-Whitney p=0.2754) — is already established. The open-source tool SymSearch provides the reproducible experimental platform for all future investigations. Six open problems are stated for future researchers.