When I exit the BOINC manager and stop running applications(File -> exit -> stop running applications), and then reopen the manager, I have lost progress on my running tasks. The percent complete has dropped down, but the elapsed time remains the same. However, it seams to regain progress much quicker than normal. Additionally, the estimated time remaining drops rapidly (by 15-30 second increments).
Is BOINC not saving progress properly, or is it required to verify things before continuing?

Not BOINC, but the science applications write checkpoints to disk. Whether or not a science application checkpoints and how frequently it does so, is all up to the project. So yes, it is quite possible that when you close BOINC and restart it, that your work continues from an earlier checkpoint.

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