In the Standard Model, fermion masses are free parameters encoded by Yukawa couplings to the Higgs field. This paper develops an alternative ontology of mass within Recognition Science (RS), a framework in which all physical structure is derived from a single functional equation—the Recognition Composition Law. We show that mass emerges as a geometric property of recognition boundaries: self-sustaining patterns on a discrete ledger whose persistence is governed by cost minimization. The unique cost functional $J(x) = \frac{1}{2}(x+x^{-1})-1$, forced by the Recognition Composition Law together with normalization and calibration, selects the golden ratio $\phi = (1+\sqrt{5})/2$ as the unique self-similar scaling base. Mass hierarchies are encoded by integer positions on a $\phi$-ladder, while sector-level scales are fixed by cube combinatorics (D=3). We derive the recognition operator $\hat{R}$ that replaces the Hamiltonian, show how the eight-tick closure cycle ($2^{3} = 8$) provides a canonical period, and demonstrate that interactions between recognition boundaries reduce to cost-weighted adjacency moves on the cubic ledger. The Higgs mechanism is reinterpreted as the low-energy effective description of a fundamentally discrete process. Companion papers develop phenomenological predictions (II), the neutrino sector (III), transport discipline (IV), the fine-structure constant (V), and the generation problem (VI).