Energy as Counted Volume: The Factor 4 in r_p = 4ħ/(m_p c) from the Tetrahedron Jammed in the Proton Cavity

The relation rp = 4ħ/(mpc) between the proton charge radius and mass is an empirical observation: with CODATA 2022 the index Cpr = mpc·rp/ħ is 3.9977 ± 0.0030, 0.76σ from 4. Earlier lattice readings obtained the 4 as two doubling steps between addresses 64 and 66 of a Planck ladder, which required rounding measured addresses (63.996, 65.996) to integers. We give a derivation of the 4 that uses no rounding and no ladder. Proved (geometry): for a common edge a, a regular octahedron has exactly four times the volume of a regular tetrahedron, and the width of the tetrahedron (the distance between its opposite edges) equals the octaradius of the octahedron, a/√2. Postulate: the energy of a cell is the number of elementary-tetrahedron volumes it contains times the quantum ħc/w, where w is the width of the elementary tetrahedron; equivalently, the energy density is 3ħc/w⁴. Consequence: the octahedral cavity of the proton carries 4ħc/rp, so rp = 4ħ/(mpc) exactly, and the tetrahedron jammed in the cavity carries one quantum, ħc/rp: the cavity radius is the reduced Compton wavelength of a quarter of the proton mass. The same rule gives both mass laws of an earlier note (mass proportional to volume within a cell, inversely proportional to size between levels) and makes the address 64 of the Compton wavelength a consequence rather than an input. The postulate is stated, not derived. For other baryons described by the same single cavity it predicts r = 4ħ/(mc), listed for comparison with measured radii.