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Can someone tell me how I will know when my applications are "received" and "complete" on lawschoolnumbers.com? I want to track my application process and just want to make sure I am doing it accurately.

The schools you are applying to will tell you when they receive your application, when you go complete, and when they render a decision. They have different models for doing that, but most just a status checker which lists your status and what day it last changed.

You then go onto your LSN profile and update the information as you learn it from the schools.