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First of all, we stipulate that the weak interaction is only significant on the scale of 10^-18m since it is seven orders less than that of the strong interaction, which is the binding force required to hold two quarks together. It so happens, that at this order of distance, it has the same field strength of the electromagnetic force. 

It is more than just stipulation however, since we have measured the unification of electromagnetic, to weak interaction from the energies required in accelerators, therefore this exact number is actually a matter of electroweak unification from strictly an experimental basis.

We know of the existence of the Z boson, a chargless particle and it's primary cousin, the W boson was observed, again from experimentation at certain energy levels required for this unification - in short, together they carry the electroweak force. 

Their mass limits the range of the weak force to about 10^-18m and it vanishes altogether beyond the radius of a single proton! This is how the measurements apply to the proton radius. To get exact calculations of this, requires a broader knowledge of the field equations, but can be summed up in some simple formulea that describe the decay processes that describe this interaction of the weak forces. Those calculations led scientists to speculate on the Higgs and spontaneous symmetry breaking, which is neither a trivial nor a simple set of things to discuss here, but to get some idea on how symmetry breaking is described mathematically, I will refer you to Susskinds online lectures on YouTube if you want to digest those complex ideas. Basically, you get this number of 10^-18m, not from any one simple calculation, it's based on several premises in physics which involves a series of complicated ideas which led to the current unification approach that we know of. While this has not answered everything, it's the best way I can articulate it for you.

Edited by Dubbelosix

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