What do you mean with the initial box? The radar prototype has two static ranges: one part which always uncovers a certain range (also shown in blue in the minimap) and one range which scans for new chunks every so often but the disappear in the fog of war after a while. The downside with the built in radar functionality is that those ranges and time between scans are hardcoded (during the data stage) and can't be changed when the game is running (during runtime stage). Unfortunately this also means that the information in the tooltip can't be updated for the infiniradar based on the research progress.
What do you mean by keeping the radar on if the radar is not scanning? Is it only the animation? In that case the radar shows an additional warning icon and further research is required. Do you expect other functionality?