1: Fine Structure Constant Alpha and sqrt of Planck Momentum
The fundamental physical constants:
G (5 digit precision)
h (8 digit precision)
e (8 digit precision)
m_e (7 digit precision)

can be defined (with 10-digit precision) in terms of:
c (exact value)
µ₀ (exact value) ... permeability of vacuum
R_∞ (12 digit precision) ... Rydberg constant
α (10 digit precision) ... inverse fine structure constant alpha

we first define G, h, e, m_e in terms of Q, c, l_p, α. We can then define:
Q in terms of R_∞ where Q = sqrt of Planck momentum
l_p in terms of µ₀ where l_p = Planck length

The online calculator demonstrates this relationship.
[ physical constants calculator]

For a list of formulas and derivations refer to:

2: Electron as standing wave magnetic monopole
Proposes electron as a mathematical formula for a spherical standing wave formed from 2 magnetic monopoles. Suggests both a Planck-unit theory and a Mathematical-Universe Hypothesis.
[ electron formula]

3: Gravitational waves as atomic orbitals
Proposes both quantum gravitational waves and atomic orbitals as physical standing waves. Graviton as pulse of Planck momentum. The online calculator calculates gravitational wave frequencies for 1kg object lifted to height h.
[ gravitational wave calculator]

For a list of formulas and derivations, refer to:

4: Expanding black-hole universe with contracting white-hole universe twin
Proposes an expanding [ black-hole universe] with a contracting white-hole universe twin. Calculates universe mass density.

Note: The above is condensed from the book Plato's Cave: a model of the universe in terms of the geometry of momentum.