Rest energy and relativistic energy are not separate quantities related by a formula. They are the same measurement — the rate of wavefronts crossing a point — taken in two different frames.
Energy is not introduced as an abstract quantity. It is defined as the rate at which wavefronts of the particle's wave cross a fixed point, multiplied by ℏ — the same single measurement in every observer's description, taken with respect to that observer's own proper time.
In its own proper frame, a particle's wave propagates entirely around the internal circle. The wavefront rate measured there is the proper frequency of that internal rotation. Multiplying by ℏ gives the ontological energy EGEOM = m0c² — rest mass is nothing but this internal vibration rate, divided by c².
An observer who sees the particle translating measures a different wavefront rate — not because new energy was added, but because translation and internal rotation draw from the same propagation budget. The measured rate becomes γ times the proper rate, giving E = γm0c², the familiar relativistic energy.
Plotted on axes (Ex, Eθ), the ontological energy vector has fixed length m0c² and simply rotates as velocity changes — all internal at rest, all translational as v → c. Nothing is created or destroyed; the same fixed quantity is partitioned differently. The physical, observed energy is this vector's length scaled by γ.
m0c² is invariant — the same for every observer describing the particle in its own terms, and what lets the particle keep a coherent identity across descriptions. γm0c² is what is actually exchanged in any interaction — a fast-moving projectile deposits γm0c² on impact, not m0c². Both are physically real; they are simply two different things to ask about the same vibration.
This is exactly the circle-and-secant construction used throughout this site: a fixed radius (length 1, the rest-energy vibration) and a secant extended to the same angle (length γ) are two sections of one underlying object — the same dispersion cone seen from a circular cut and a hyperbolic cut. See Theory for the full derivation.