Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Dirac noticed startling numerical coincidences between very large dimensionless numbers built from fundamental constants and cosmological quantities. Typical examples: ===
# The ratio of the electric to gravitational force between proton and electron:

e24πε0Gmpme∼1039–40.\frac{e^2}{4\pi\varepsilon_0 G m_p m_e} \sim 10^{39\text{--}40}.4πε0Gmpmee2∼1039–40.
# The ratio of the Hubble length (or age) to the classical electron radius (or other microscopic length):

ct0re∼1040.\frac{c t_0}{r_e} \sim 10^{40}.rect0∼1040.
# The particle number in the observable universe:

N∼Mump∼1080∼(1040)2.N \sim \frac{M_u}{m_p} \sim 10^{80} \sim (10^{40})^2.N∼mpMu∼1080∼(1040)2.
Dirac hypothesized that these are not coincidences but reflect laws like G∝1/tG \propto 1/tG∝1/t (or equivalent scaling), or that the mass of the Universe scales with time: Mu∝t2M_u \propto t^2Mu∝t2. Any QAT connection should either reproduce these scalings, offer a mechanism for them, or explain the coincidences geometrically.