I never noticed this about quantum conductance

Wiki: Note that the conductance quantum does not mean that the conductance of any system must be an integer multiple of G₀. Instead, it describes the conductance of two quantum channels (one channel for spin-up and one channel for spin-down) if the probability for transmitting an electron that enters the channel is unity, i.e. if transport through the channel is ballistic.

That is just the hyperbolic condition, cosh^2-sinh^2 = 1.  The equation insures that the spread of the outer surface minus the spread of the inner leave one unit of conductance between them. The G0 will be the cotangent, I thnk, in this case. Then we have this:

Alpha is the fine structure which we know is the residual noise when the system is perfectly adapted and likely at Lagrange number two.  I only get it close with Lagrange one.
But, the spectral enegy needed to make exchanges will be twice the noise energy, no? After all, maximum entropy conditions have to be met, we have an adapted probability system.. So the ratio: G0/alpha  is simply the center frequency to  power spectra. I may have bandwidth and power spectrum mixed here, bear with me.  But that ratio is just the bandwidth of space, including the coupling constant, so 1/Z0 is simply a measure of spectral capacity. The formula has all the usual time and distance fake units in it, so there is some reduction that needs to be done. But I know where that center frequcne is, it is hyperbolic angle 3/2 * ln(Phi()). This system is just making pi out of the 1.5 Quants supported by free space.