Part III of the RLDS series. This paper derives a paired observational signature of the RLDS dark-sector attractor by systematic numerical integration of the reduced RLDS ODE. It identifies a stable ratio linking the dark-energy equation-of-state dip and the suppression of structure growth, and examines its current observational status in light of DESI DR2. The paired coefficient is presented as a numerical result with structural motivation, not as a closed-form analytic theorem. The expansion-sector half is qualitatively consistent with DESI DR2, while the growth-sector half remains the key open near-term test. Paper I: singular-attractor prototype mathematics. Paper II: canonical RLDS framework and reduction criteria. Paper III (this work): paired observational signature. Paper IV: cosmological consequences under RLDS closure. Paper V: representative microphysical descent to RLDS.