The Schwarzchild radius, for example

The radius at which liquid mass had strong enough gravity that light cannot escape. This is all based on an assumption about light matching mass (energy equation) and the Compton frequency. Instead treat light as a finite spectra signal, and the sphere captures more spectra toward the center, as that is the Hamiltonian assumed. So Schwarzchild simply integrates the spectra from the center out until there is no more. The built in assumption is that light has a bounded cubic spectrum.

Do it different. This is problem in subdivision, what is the bounds when the subdivisions increase? But the subdivisions are recursive of order n, which is hyperbolic, spherical. and composed of linear Bournilli numbers. The Compton radius goes to zero faster than light has spectra. Tanh(n) gives you the bounds on the slope of n when n is an exponent of e.  It is just the general form, the relationship between n and the slope of the function is created. Tanh(n) is the mean over the difference in at n: Of the functional on tanh, limit ourselves to spherical and small n.  n [1,2,..N] has to be symmetric with respect to a unit sphere. The maximum rate of action is along the hyperbolic arms, and has sign.

Time dialates as R increases?    So that makes Tanh*(n) the Compton wave at n? Bandstop is lower, frequency is lower, mass is lower, n is lower.  If the unit circle moves as 1/F, such that the sum 1/F, to some integer power, f=1,2,3,.. is the Zeta function, and likely finite. So the unit sphere emits and absorbs  flow in units of 1/F, F is short order recursive. to make the Zeta = 1. Like the sum of 1,2,3 all taken to the mth power, m <=3. Doing so marks its bandwidch in sinh^2 - cosh^2 = 1.

So, any unit sphere in the spherical environment can emit and absorb the 1/f, in some limited order. The 1/f are Higgs Nulls, and act as band stops in the spherical system. Unit spheres have three groups of recursive 1/f, I think.at most, recursive in quant number.

Great model, but says nothing about physics, it talks to the grid size in spherical systems matching the equations of motion. Shift in N such at:
a*Sinh(n)^2 - b*cosh((n)^2 = 1 are considered.