The proton as Kepler's star — and a point the lattice never reaches

Two preprints went on Zenodo, and they belong together.

1. The unreachable face centre — doi:10.5281/zenodo.23767792. The TOT lattice refines into itself by halving every edge. The whole refinement reduces to halving two segments — the edge and the octahedron's diagonal — so every node of every level has coordinates k/2ⁿ. Edge midpoints and octahedron centres become nodes at level 1, tetrahedron centres at level 2; the centre of a triangular face never does: 1/3 is not a dyadic fraction. In 3-adic terms the nodes stay at a fixed distance from it — a 3-adic gap in a dyadic hierarchy. Yet a regular tetrahedron of the same edge, set on one face inside an octahedron, has its apex exactly at the centre of the opposite face. The point the lattice cannot reach is reached only by a body inserted from outside.

2. The proton as a stella octangula — doi:10.5281/zenodo.23774478. In TOS the proton is an octahedral cell. In the lattice each of its eight faces is shared with a tetrahedron, so the cell and its neighbours form Kepler's star; inside sits the jammed tetrahedron from (1). We read the proton as this whole figure. Its geometry fixes three concentric spheres in the exact ratio 1 : √3 : 3:

• r_p/√3 = 0.4854 fm — mass core;
• r_p = 0.8408 fm — charge radius;
• √3·r_p = 1.4562 fm — range of the interaction.

The roles are a postulate; the numbers are what make it interesting. Bring two such protons into face-to-face contact: the distance between the cavities depends on orientation, but the distance between the two hard tetrahedra is the same for every contact, 0.4854 fm. The Hamada–Johnston nucleon potential (1962) uses a single hard-core radius in all states: 0.4849 fm. The outer sphere lies 0.39% below the reduced Compton wavelength of the π⁰; equivalently m_p/(4√3) = 135.43 MeV against the measured 134.98 MeV.

Status: the face-centre results are proven geometry. The proton reading is a hypothesis: identifying the potential's radial coordinate with the distance between the hard bodies is an assumption of the model, and the matches are coincidences until a dynamical derivation exists. Scripts for every number are in the Zenodo records.