Two Lengths of the Proton on One Planck Ladder: the Reduced Compton Wavelength and the Charge Radius as Octaradii of Levels 64 and 66 of the Tetrahedral-Octahedral Lattice

The tetrahedral-octahedral lattice (TOT) reproduces itself when its edge is halved, which defines a ladder of levels whose edges are an = lP·2ⁿ, counted from the Planck length. The radius of the octahedral cell of level n — its octaradius — is one Hippasus step below the edge, Rn = an/√2. We show that two lengths of the proton sit on this ladder at integer addresses: the octaradius of level 64, lP·2⁶⁴/√2 = 0.210821 fm, gives the reduced Compton wavelength ƛp = 0.210309 fm, and the octaradius of level 66, lP·2⁶⁶/√2 = 0.843285 fm, gives the charge radius rp = 4ƛp = 0.841236 fm. Both deviate by the same +0.2437%, which lies within 0.2% (relative) of α/3. The ladder thus places the mass scale and the size of the proton on one structure, two doubling steps apart, with no free parameter other than the two integers. Three consequences follow: the four in rp = 4ƛp is the square of the doubling step (2² = 4); the charge radius fixes Newton's constant, rp = K₆₆·√G; and the α/3 reading of the common residual predicts G = 6.67424×10⁻¹¹ m³kg⁻¹s⁻², 8 ppm below CODATA 2022. We set out the priorities explicitly: the relation rp = 4ƛp was first written down by Haramein (2012) and derived independently by Trinhammer and Bohr (2019); the dimensionless counterpart of address 64, ħc/(Gmp²) ≈ 2¹²⁷, belongs to the combinatorial hierarchy (Parker-Rhodes 1961–62, Bastin 1966). What is new here is the derivation of both proton lengths as multiples of the Planck length on a geometric lattice, the address 66 of the charge radius, the Hippasus step, and the α/3 correction. Contact: Yevhenii Stakhov · stakhoff@gmail.com · https://theoryof.space