Part I of the RLDS series. This paper establishes the mathematical prototype of the RLDS program. It studies a class of one-dimensional autonomous systems on a bounded interval with singular restoring force and proves existence, uniqueness in the admissible regime, global asymptotic stability, and structural stability of the attractor. The key analytical object is the auxiliary function H(x), whose geometry determines the equilibrium structure and long-time dynamics. Paper I (this work): singular-attractor prototype mathematics. Paper II: canonical RLDS framework and reduction criteria. Paper III: paired observational signature. Paper IV: cosmological consequences under RLDS closure. Paper V: representative microphysical descent to RLDS.