Frank's classical radius of the electron shows that the fields of matter are pinned into the structure of matter at that classical radius of the electron like you pin fields in to a superconductor by adding discontinuities. With that model you get the phase velocity equals the velocity the phase velocity equals the velocity of light the group velocity equals the movement of the structure and all of special relativity emerging under that condition
Looking at the Compton frequency, the De broglie wave being the Doppler shifted reflected Compton frequency emerges naturally from this model out of that condition
More gibberish.
True or not?the speed of light in a BEC is adjustable down
In certain BECs you can select a narrow wavelength band---very, very narrow---and adjust the speed of that light and nothing else. There is no unusual distortion of the speed of other frequencies, or of other particles. So what? All transparent materials change the speed of propagation of certain wavelengths---glass, water, air, diamond, etc. BECs just do it to an unusual degree.
There is certainly no distortion of the fundamental constant known as the speed of light. For example, if you're measuring the time-dilation of muon decay, and you need to calculate 1/sqrt(1-v^2/c^2), the value of c is 3e8 m/s as always---it makes no difference whether the muon is in vacuum, in glass, or in a BEC.