One formula for the forces of the 1/r type

Our research on the directional product, from a directional product to Dirac energies, derives theenergies of hydrogen-like ions from one product, (1 − ε)(1 + ε) = β2 with β = αZ/n, and its ellipticorbits reproduce the fine structure. Its binding energy, B = mv2/(1 + ε), is the energy reading ofthe directional product, the reading that the rotation curves of disk galaxies select in our research onthe dark-mass ladder (λ = 1/(1 + ε) = 0.49–0.50), so that one reading describes bound orbits fromions to galaxies. The formula needs only a potential of the form 1/r and a dimensionless coupling, soit can be written for every force that has one. We write it for the four interactions with their couplingsα = 0.4925, αW = 0.0316, αs(mb) = 0.212 and αG = 5.906 · 10−39 (two protons), and wetest it against measurement. For electromagnetism it holds to 0.0427% up to Fe25+, the rest beingquantum electrodynamics. For gravity the same root that gives the fine structure gives a perihelionprecession of 7.16′′ per century for Mercury, exactly one sixth of the observed 42.98′′: the sixth of therelativistic mass, the other five being the curvature produced by the Sun. Below a0 = 1.144 × 10−10m s−2the product gives the ladder law of disk galaxies; with it the gravitational atom of two protonshas its first orbit at 56.1 km instead of 7.5 · 106light years, with radii linear in n, and the NewtonianBohr radius equals the transition radius exactly at the minimum quantum M∗ = (¯h4a0/G3)1/7. Thedark acceleration of the law makes the perihelia regress; the ephemeris of Saturn bounds the transitionconstant of the law to Kt ≤ 47 and predicts a supplementary precession of −0.812 to −1.27 mas percentury for Saturn. For the strong force the 1/r part alone, fitted to the Υ(2S)−Υ(1S) spacing, needsαs = 0.5369 and gives 89.3 MeV for Υ(3S) − Υ(2S) against 331.82 MeV measured; the confiningterm σr restores it (345.2 MeV), and a coupling that runs with the radius of the orbit produces thatterm: fitted to 1P − 1S, it predicts 1D − 1P = 228.2 MeV on the circular orbits of the model and250.8 MeV in the radial wave equation, against 263.8 MeV measured, with Λ = 0.2573 GeV, while arunning mass is excluded. Written as E = xy + n Cx, the form in which the potential energy of theelectron was derived, the same orbit gives for two light quarks M2 = 8σn, and the string tension ofbottomonium predicts the slope of the Regge trajectories of the light mesons within 11%. For the weakforce the Yukawa potential binds only above the threshold 2µαW /MW = 1.6798, and not at all below1 (Bargmann), while electrons give 2.01 · 10−7and top quarks 6.78 · 10−2: the formula never applies;the orbit of the model is stable only below the golden ratio, Mr < φ. One formula therefore links theforces in form; it unifies them only if it predicts their couplings. A pre-registered test shows that theeight offsets of the wheel, alone or on the drawn vectors at their angles, do not predict them: their hitsare no more frequent than those of random structures, and the couplings and their slopes never fall onthe same vectors.