Since  α is dimensionless, the dimensions of this expression are Energy x time x (velocity)^4. However the significance of this term can only be understood from the complete expression, in which it's found. It's proportional to the (square of) electric charge, since α is. Indeed, in typical applications, precisely because Planck's constant and the speed of light are dimensionful, it's possible to choose units such that actions are measured in units of Planck's constant divided by 2π and speeds are  measured in units of the speed of light, so this combination is, simply, (16π^2) α in those units. 

Can you provide a complete equation within which this term is being used?

To give some context the complete expression is

e2  = (eo /4pi.c3).8pi.h.c4 .alpha

Where e2  is the electron charge. and e0 is the electric constant.

The full context is given in the attached paper    

Article Harmonic quintessence and the derivation of the charge and m...

This just the definition of the fine structure constant, written in terms of the electric charge, that's all. So its content is nothing more, or less, than that of the fine structure constant, namely the coupling constant of the electromagnetic interaction. 

No, it seems to say that the electric charge squared is the huge electric charge squared by the fine constant (which is small).  7 mod 3.  There exists this kind of numerology. 

Stam is right. When you simplify this equation, you end up with the standard definition of alpha: alpha= e2 /(2 eps_0 h c).

Note also that when you multiply this equation term for term by equation lambda nu = c, and isolating h nu, which gives the energy, you get an equation giving the energy of any photon with just its wavelength as a variable: E = (h nu)= e2 /(2 eps_0 alpha lambda).

@Stam

Yes you are correct, but formulated as

e2 = (eo /4pi.c3).8pi.h.c4 .alpha

You can see the principle equation becomes a three sphere with 1/c as the radius. This part of the equation accounts for most of the value for the charge of the electron

e2 = (eo / 3(4/3pi.c3)

The rest of the equation

8pi.h.c4.alpha   is a small correction  factor from which one can calculate the electron magnetic moment to Bohr magneton ratio - from first geometric principles.

From here the rest of particle physics drops put from first principles

Article Harmonic quintessence and the derivation of the charge and m...

your use of the 3-sphere idea is an interesting avenue, and the value obtained for the unit charge is clearly ok since the whole equation resolves to the set of fundamental constants that mutually define each other (e alpha h c eps_0 and pi which is embeded into eps_0).

I tried to locate where you derive the electron magnetic moment to Bohr magneton ratio from the correction factor 8pi.h.c4.alpha (value 0.981621217) in your paper, but could not locate this part.

This is of interest to me because from my analysis, an electron captive on a tight circular orbit with short gyroradius will obviously have a larger magnetic moment than one moving in a straight line which, from what I understand, the Bohr magneton happens to represent by definition.

The Bohr magneton to electron magnetic ratio, has not yet been published.

I would be more than pleased to collaborate.

Is your shorter radius radius based upon the classical electron radius by any chance?

Not really. It is based on the assumption that the carrying energy induced at the mean rest orbital of the isolated hydrogen atom is electromagnetic in nature.

It is well established that when ambient magnetic energy density is higher than electric energy density, an electron will be forced to follow a curved path while equal energy densities will cause it to move in straight line.

Based on an experiment caried out by Samuel Jackson Barnett in 1909, the idea came to me that if an electron is forced to move on a closed orbit (as it could be assumed to be in an isolated hydrogen atom) this would in turn force such a drift towards increased magnetic density of this carrying energy.

Calculations based on the above assumptions show that the Bohr magneton seems to be a property of this carrying energy and not of the electron proper.

Since from this perspective the traditional manner of calculating the Bohr magneton implies equal density of the related electric and magnetic fields, then this would mean that the Bohr magneton actually applies to an electron moving in straight line with this amount of energy.

More calculation shows that if this energy density (which could be seen as locally ambient) is drifted to the magnetic field aspect at the expense of the related electric field with the Bohr radius acting as a gyroradius, then a ratio in the proper range is obtained. I may lack some more data to get it to the exact value.

Difficult to go into more details here. You could see my complete analysis in an article I had published in an engineering journal a few years ago:

http://ijerd.com/paper/vol7-issue3/E0703021025.pdf

Looking forward to your derivation. The more info is integrated, the faster the whole issue can be finally resolved.