A Geometric Visualization Framework for Dynamical Dark Energy: The C-Coordinate and Seven Parameterizations Applied to DESI DR2

We introduce the C-coordinate, a linear transformation of the dark energy equation of state defined as C(a) = (1 − w(a))/2, which maps the w0–wa parameter degeneracy into a temporal trajectory on a bounded geometric plane. Applying this framework to DESI DR2 data under the CPL parameterization, we find that dark energy evolves from the phantom regime at early times (C(0) = 1.298 ± 0.081) toward the ΛCDM limit at present (C(1) = 0.944 ± 0.039), with the phantom crossing geometrically evident as a crossing of the C = 1 boundary. We extend this mapping to six additional parameterizations—JBP, Hyperbolic, BA, Logarithmic, Oscillating, and Power Law—demonstrating that each maps to a geometrically distinct trajectory type: linear, parabolic, divergent, quadratic, logarithm-divergent, oscillatory, and monotone-decay. A model discrimination metric (RMS distance from ΛCDM in C-space) is introduced, providing a unified quantitative comparison. This framework offers a model-agnostic visualization tool that makes phantom crossing, temporal evolution, and inter-model distinctions geometrically transparent without modifying the underlying physics.