Light Trapped in an Octahedron: Maxwell Eigenmodes of the Octahedral Cavity and the Proton Relation r_p = 4ƛ_p

The Theory of Space reads the proton as an octahedral cell of a tetrahedral–octahedral lattice whose octaradius is set by the proton charge radius rp, with mass given by rp = 4ħ/(mpc) = 4ƛp. If the mass is the energy of light trapped in the cavity, the lowest cavity mode must satisfy k·R = R/ƛp: 4 if R = rp, and 4.0097 if R is the geometric octaradius of the companion papers, which exceeds the observed radius by a candidate electromagnetic correction 1 + α/3. We compute the electromagnetic eigenmodes of hollow cavities with perfectly conducting walls by Nédélec edge elements on a self-similar mesh of the lattice itself, validated on the cube and the ball. The regular octahedron has its lowest mode, a T1u triplet, at k·R = 4.087 ± 0.001 (R the circumradius): 2.2 % above 4 and 1.9 % above 4.0097, with no adjustable parameter. The regular tetrahedron gives 5.414, the truncated octahedron 4.195, the cuboctahedron 4.717. The frequency is nearly a function of volume alone (k·V1/3 ≈ 4.3–4.5 for all compact shapes; a ball of the same volume gives 4.018, nearer to 4 than the octahedron), so closeness to 4 is not specific to the octahedron and this calculation does not establish that it is; exactly 4 needs 6 % more volume than the octahedron has. We record three ways to it, all hypotheses with free parameters: lower-scale lattice pieces added on the faces (the deficit is Oct/16 = 2 Tet(a/2)); an octahedron inflated by a few per cent, possibly by the interstitial tetrahedron; and a weakly dielectric interstitial tetrahedron (refractive index ≈ 1.05), a phenomenological variant. The energy of the lowest mode lies at 0.482 R, not at the mass radius rp/√3 = 0.577 R of a related lattice model (the root-mean-square radius of the electromagnetic energy and the mass radius of that model are different quantities until a link between them is derived). Each claim is labelled result, observation or hypothesis. Contact: Yevhenii Stakhov · [email protected] · https://theoryof.space