I agree... like I said, the calculation indicates that you'd actually see a 1/3 tick rate from a clock actually ticking at at rate of 0.6 (ie time dilated from a rate of 1.0) and moving away at 0.8c.
So how do you figure that "what you actually see would only be 1/5 or 20%"?
Here is where you and I likely part. You are content to apply this logic to both observers. I contend that only the accelerated clcok is dilated to 0.6 and that the denomintor therefore becomes "1" because "v" is the actual "v" induced by acceleration and not merely relative velocity. "v" for the station = "0" hence f' = f * (1 - v/c) / (1 - v^2/c^2)^0.5 = f * (1 - 0.8) / (1 - (0.0)^2)^0.5 = 0.2 / 1.0 = 0.2.
That is there has been no cause to change physics (tick rate) of the station clock. That is indeed what empirical data shows.
However, one can then argue that since the shuttle clock IS indeed ticking physically slower, it appears to the shuttle observer that the station clock is ticking at(0.2/the shuttle dilated rate) or .2 / .6 = 0.3333.
However, IF you also wish to insist that the "v" IS relative then you would get 0.333 / .6 = 0.5555.
My point is that there is no data, has never been a test nor observation made from the accelerated frame to verify what is "Seen" from that frame. It is wrongfully assumed that both frames are equal and the formulas apply and describe physical reality. That isn't supported by data. The resting clock nevers loses time. If it did then both clocks would dilate equally and no differentail could ever have been measured.
And you didn't answer the first question: What do you mean by "See", when you put it in quotes?
I think you mean "Measure after accounting for any signal delays." Is that right?
If you are measuring (Seeing) frequency, once the signal has arrived delay has no bearing. So not sure what you are getting at. "See" means "See" and assumes some super new technology to peer at the other clock and "See" it's tick rate (receipt of flashes of light per tick) while moving at relavistic speeds.
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