Bound State Energy from the Euler Oscillator Geometry: Charge as Rotation Direction, Dark Matter as Spacetime, and the Absence of Dark Energy

We derive the exact bound state energy of the Euler Oscillator System (EOS) purely from the geometry of the compacted circle in the Euler complex plane, requiring no Coulomb potential, no electric charge as a source of force, and no perturbativecorrections. The bound state energy E_B(n) = − {hf/(1−β)}{1 − β^2/n^2}^1/2 emerges directly from Pythagorean decomposition of the radial and tangential velocities on the contracted circle, with β = 1/m = εm/ε. We show that electric charge is not afundamental property of particles but a perspective — the rotation direction on the compacted circle — encoded in Euler’s identity since 1748 as e(±ix) = cos x ± i sin x, with charge magnitude ±e ∝ ±√α. Charge vanishes for free particles (α = 0) andexists only as a property of interaction between two Prince waves. We extend this framework to distinguish mass and energy as independent properties with separate frequencies, m = hf_m/c^2 and E = hf_E, with equivalence E = mc2 holding onlywhen f_m = f_E. This naturally identifies dark matter as stationary spacetime vortices with f_m ̸= 0, f_E = 0 — gravity itself — and shows that dark energy does not exist: the vacuum oscillation ±hf/2 has average zero at every spatial point, resolvingthe vacuum catastrophe. The Casimir effect is shown to arise from asymmetry in vacuum fluctuations, not from net positive vacuum energy.