The Proton as a Stella Octangula: a Tetrahedron Jammed in the Octahedral Cavity, and the Three Radii of the Nucleon

In the Theory of Space (TOS) the proton is an octahedral cell of the tetrahedral–octahedral lattice (TOT) whose bare octaradius is related to the charge radius rp through a candidate correction (1 + α/3) [1–3, 21, 22]. In the lattice every face of this octahedron is shared with a tetrahedron, so the cell together with its eight face tetrahedra forms Kepler's stella octangula; inside the cavity sits one tetrahedron of the same edge, jammed with its base on one face and its apex at the centre of the opposite face [4, 6]. We read the proton as this whole figure. Its geometry fixes three concentric spheres in the exact ratio 1 : √3 : 3 — rp/√3 = 0.4854 fm, rp = 0.8408 fm and √3·rp = 1.4562 fm — to which the model assigns, by postulate, the roles of mass core, charge and range of the interaction carried by the face tetrahedra. Treating the cavity as soft and the jammed tetrahedron as hard, we compute the contact of two nucleons for every class of face-to-face contact, with any rotation about the common normal. The distance between the centres of the two cavities depends on orientation (0.32–0.97 fm), but the distance between the centres of the two hard bodies is the same, 2·rin(Tet) = rp/√3 = 0.4854 fm, for every face-to-face contact, and no contact of any kind brings them closer. This orientation-independent length is close to the single hard-core radius that the Hamada–Johnston potential uses in all states, 0.343 ƛ(π±) = 0.4849 fm [9, 10]; identifying the radial coordinate of the potential with the distance of the hard bodies is a hypothesis of the model. The outer sphere lies 0.39% below the reduced Compton wavelength of π⁰; equivalently, with rp = 4ħ/(mpc), mp/(4√3) = 135.43 MeV against 134.98 MeV (+0.33%). The mass-core sphere agrees with the mass-core estimate 0.48 ± 0.05 fm [8] within its large uncertainty. The picture joins two views put forward separately — the proton as an octahedral void and as a tetrahedral defect — as the cavity and the jammed body of one figure.