The Coincidence at Ten to the Fortieth: Dirac's Large Numbers, Priced

In 1937 Paul Dirac noticed that two dimensionless numbers built from completely different physics — the ratio of electric to gravitational force between a proton and an electron, and the age of the universe measured in atomic time units — are both enormous and, at the present epoch, roughly comparable. I credit the notice at full strength: the numbers really are there, my own bench returns them below, and Dirac's insistence that such a proximity should not be waved away as accident seeded varying-constant cosmology, scalar–tensor gravity, and the modern experimental discipline of bounding how fast the constants may drift — a discipline that exists because he made the question sharp. Nothing in this paper falsifies the arithmetic of the coincidence, because the arithmetic is right. What I evaluate is the coincidence itself, stated as arithmetic, and the invoice a mechanism would have to meet to explain it. On declared inputs only, my bench returns the force ratio N_F = 2.268661e+39 (log10 = 39.355770) and the age ratio N_t = 4.633103e+40 (log10 = 40.665872): the two numbers stand 1.310102 dex apart, a factor of 20.422188 — "equal" only in the logarithmic sense in which anything within a factor of twenty counts as the same size. The third large number, the particle count in a Hubble volume at the declared critical density, is N_p = 5.524834e+79 (log10 = 79.742319), against N_F squared at log10 = 78.711539: the celebrated square relation misses by 1.030780 dex, a factor of 10.734452. And the coincidence is epoch-stamped: with the force ratio held fixed, the age ratio equalled it exactly at 0.675736 Gyr, and the two stand within half a dex of each other only between 0.213686 Gyr and 2.136864 Gyr of a 13.8 Gyr history. A mechanism that explains the coincidence must therefore produce three relations at once, hold the first of them at all epochs against a natural drift of 7.246377e-11 per year, and say why the relations are good to about a dex rather than exact. No mechanism in 1937 or 1938 attempts any of the three. Grade for this census entry, printed at full size: REASON FOUND / STILL TURNING — PREMISE-ONLY, explicitly not a kill and not a falsification. The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.