As a California driver who has never received a ticket in Arizona, this case seems rather confusing with the available information. Isn't the AZ officer supposed to write a paper ticket and give you a copy with the code violation listed, just like in California? If a ticket was written, was it for speeding or being a moped in a bike lane? If speeding was a violation, was the code violation and observed speed written on the ticket? If the speeding code violation and clocked speed was not written on the ticket, then I think it is a pretty weak argument to add it after-the-fact.
In any case, assuming there is a valid speeding violation, here are some things to consider as part of a defense -
1) Why is your claimed speed and the officer's radar speed so different? Were you clocked by radar while riding in busy traffic? Radar works by bouncing a reflected wave off of a target. A bicyclist is a rather narrow and SOFT target. If you were in busy traffic, there is a possibility that the radar picked up reflections from faster moving high profile hard objects like adjacent cars or trucks near you. To bolster your argument, you could take pictures/measure traffic patterns on same weekday/same time as your violation for evidence. This could provide a graceful face-saving explanation of why the large difference in speeds claimed by you and the officer, and enable the judge to throw out the case without having the officer admit being wrong.
2) Was the place you were clocked speeding with radar a downhill grade or just after a downhill grade? See this link for the definition of a moped
http://www.azleg.state.az.us/ars/28/00101.htm (item 30). Essentially, a MOPED is maximum 25mph on a flat surface with less than a one per cent grade. See this link for the definition of a MOTORIZED BIKE
http://www.azleg.state.az.us/FormatDocument.asp?inDoc=/ars/28/02516.htm&Title=28&DocType=ARS (item 7c). Essentially a motorized bike is defined as maximum 20mph. It is common knowledge that a moped is more powerful than a motorized bike. So, one can conclude the legislation sloppily left out the rest of the motorized bike definition. It should have been written that a motorized bike is maximum 20mph ***as measured on a flat surface with less than a one per cent grade***. In fact, that is how California legislation is written
http://www.dmv.ca.gov/pubs/dl655/dl665mcycle.pdf (pg 3). In California, a motorized bicycle is ***...no more than 30mph on level ground, and ....***. So, by establishing a motorized bike is 20mph and should be measured on a flat surface, one can make an argument that you can coast downhill faster than 20mph, as long as you return to 20mph when you hit the level ground areas again. If you were on or after a downhill, use this to your advantage.
But, in the end, how to fight this really depends on exactly what violation was written up and how it was recorded. I don't think a bunch of Emails will sway the judge or local government either way.
Good luck!