Energy Pressed into the Corners: The Proton's Core Mass Radius from the Consolidation Law of the Tetrahedral-Octahedral Lattice, r_p/√3 ≈ 0.485 fm

In the Theory of Space (TOS) program the proton is the octahedral cavity of level 66 of the doubling ladder of the tetrahedral-octahedral lattice (TOT): an octahedral void enclosed by eight tetrahedral cells of the same level, with octaradius set equal (the TOS identification assumed here) to the proton charge radius r_p, and with mass given by the closed-form relation m_pc² = 4ħc/r_p (Cpr = 4) [2]. We first remove an apparent conflict between two mass laws — mass ∝ 1/size between levels of the ladder and mass ∝ volume within one cell — by noting that both follow from a single mean energy density ρ̄ ∝ ħc/r⁴; for the octahedron it is exact, ρ̄_p = 3ħc/r_p⁴ ≈ 1185 MeV/fm³, and gives an MIT bag constant B^1/4 = 218 MeV. We then ask where inside the cavity the energy resides. The consolidation law of the lattice divides every octahedron into six corner octahedra and eight face tetrahedra of half the edge. If the energy is pressed away from the tetrahedral walls into the octahedral corners — the narrow crevices of the cavity wedged between the surrounding tetrahedra — and this rule is applied recursively, the root-mean-square radius of the energy distribution is, with no free parameter,

r_m(n) = r_p·√(1/3 − 4⁻ⁿ/30),

i.e. 0.4605 fm for a uniform cell (n = 0), 0.4793 fm at the first level of consolidation and 0.4854 fm = r_p/√3 in the limit. The result is insensitive to the depth: every n ≥ 1 lies in 0.479–0.485 fm. An independent construction gives the same number: if the cavity is read as a hollow shell — the octahedron of level 66 with its central node star (an octahedron of level 65, 1/8 of the volume) left empty — a uniform energy density in the shell yields r_m = r_p·√((3/10)(31/28)) = 0.4845 fm. It agrees with the nucleon core radius 0.48 ± 0.01 fm and the gluonic mass-core radius 0.51 ± 0.05 fm summarized in [8], In a two-zone model with uniform density in each zone, an rms radius above ≈ 0.56 fm is impossible inside the cell, which suggests that larger extractions of the full mass radius include mass outside it (the meson cloud); for an arbitrary distribution the only geometric bound is the octaradius itself. The limiting support of the recursion is a self-similar six-branch set, the octahedron flake, which contains the three axes but is not reduced to them. We state an explicit falsification criterion.

Version 3: the limit of the corner recursion is the octahedron flake (dimension log 6/log 2), not three lines; the 0.564 fm bound is restricted to the two-zone model; the octaradius–charge-radius equality is the assumed TOS identification; a GlueX value is added; the falsification criterion is statistical. The relation r_p = 4ħ/(m_p c) appears in Haramein's journal article (2013) and was derived independently by Trinhammer and Bohr (2019). Numerical results unchanged. Contact: stakhoff@gmail.com · https://theoryof.space