In quantum field theories, virtual particles cause quantities like the couplings to "run" with energy scale. Amateur physics numerologists find formulas for the low-energy values, but the professionals expect that these quantities will take their simplest form in some high-energy unified theory, so professional physics numerology involves the high-energy values.
The low-energy values - which are the quantities that are measured and listed in the physics databooks - are therefore regarded as being equal to "simple high-energy value + messy correction full of logarithms etc". However, there is the phenomenon of the "infrared fixed point". This occurs when the dynamics of the running (as described by a beta function) converges on the same low-energy value, for a range of starting values at high energy. In the language of dynamical systems theory, this means that the beta function enters an attractor at low energy.
This strikes me as one of the few ways in which amateur physics numerology might be realized within an actual quantum field theory: an attractor might dictate simple relations between the low-energy values. I have no examples of low-energy numerology being realized in this way, but it's a possibility.
It is therefore exceptionally intriguing to see some low-energy numerology which utilizes a famous constant from dynamical systems theory, Feigenbaum's constant. Mario Hieb has noticed that
(2 pi) times the fine-structure constant ~= 1 / (Feigenbaum's constant squared)
to 1 part in 1000.
It's a very attractive formula. It's simple, "2 pi" is a very "physical" factor, and the fine-structure constant is the epitome of what we would like to explain. Still, I wonder how mathematically difficult it is to obtain this within a QFT.
Feigenbaum's constant describes the approach to chaos - the rate at which a point attractor bifurcates, as a control parameter varies. It does show up in the theory of phase transitions, which sounds like QFT, but so far I only see it appearing in an indirect way, as part of a formula for some Lyapunov exponents.
It's unclear how one would go from that, to the constant appearing with such simplicity, in a formula for a coupling. Also, I have not found any work on infrared fixed points in which a weak U(1) coupling is part of the attractor.
But I admit that my survey of the possibilities so far is preliminary and superficial. So, maybe it has a chance of being true.