F02 Descartes' Cardioid and the Nuclear Charge Radius

In 1637, René Descartes published the polar curve \(r = a(1 - \cos\theta)\) in *La Géométrie*. For 389 years, it was read as a heart-shaped mathematical curiosity — the cardioid. It was not. Through four precise substitutions — replacing \(a\) with the Compton wavelength \(\lambda_☉\), \(\theta\) with the vortex phase difference \(\Delta\theta\), the single curve with a dual-vortex projection, and the static form with the phase perturbation \(\sin\Delta\theta \cdot \cos\Delta\theta\) — the cardioid becomes the nuclear charge radius formula: \(r_R = \lambda(1 + \cos\Delta\theta)\sin\Delta\theta\cos\Delta\theta\). The proton charge radius is predicted as \(r_p = 0.857\) fm (Stage I, \(\Delta\theta = 60^\circ\), error 1.9%) and \(r_p = 0.835\) fm (Stage II, \(\Delta\theta = 60.85^\circ\), error 0.7%). The magnetic radius formula \(r_I = \lambda(1 - \cos\Delta\theta)/8\) gives the I³ vortex core scale. For the neutron, \(r_I = \lambda_☉/4 = r_☉ = 0.330\) fm. The neutron mean-square charge radius emerges as \(\langle r_n^2 \rangle = (i \cdot r_☉)^2 = -r_☉^2 = -0.109\) fm², matching the experimental value of \(-0.116\) fm² within 6.2%. The negative sign arises automatically from \(i^2 = -1\) in the I³ → R³ projection. The nuclear radius has two independent geometric components: the electric radius \(r_R\) from the R³ cardioid projection, and the magnetic radius \(r_I\) from the I³ vortex core. Multi-nucleon radii for A = 2–18 are derived from three fixed parameters — the proton effective radius \(r_p^{\text{eff}} = r_R - r_I = 0.750\) fm, the neutron effective radius \(r_n^{\text{eff}} = r_☉ = 0.330\) fm, and the magnetic projection \(r_☉^2 = 0.1089\) fm² — combined with interface projection rules derived from vortex chirality. Average error across 19 nuclei is < 3%. Zero free parameters. All predictions follow from the geometric structure of the ☉ Solo Unit within the R³I³ Framework.