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007,

we end up with different radial lengths partly because shockwave is using a larger hoop and wider radial angle,

the hoop adds half the distance between a radial and its neighburs to the length of the radials,

secondly as the radials are swept closer to the central monopole impedance drops and resonant frequency of both radial sleeve and monopole goes up,   

conventional maths for electrical length go out of the window as you sweep the radials up towards the central monopole,

for instance a gamma fed monopole with 90 degree radials that is electrically 3/4wave will be less than 3/4wave when you sweep the radials up,


another variable that im not sure about but have a gut feeling is that we may be manipulating radiation angle as seen in professional sleeve monopoles,

 

not sure about your beta match idea, i like the gamma, imho it would not be a sigma4 style if you did not use the gamma,

there is absolutely nothing wrong with gamma matching if its done correctly with apropriate tube diameters and spacing, the lossy gamma claim is nonesense,

the stock gamma is shitty but it does work ok with an ACOM 1000 full power fm without arcing or melting,

you can lower system q and loss with a fatter gamma rod but i never tried one to compare signal performance or bandwidth.