WRITTEN BY BUTTERNUT / SHRIMP SYSTEM
Integration is the process of amnesic barriers lowering due to understanding more about your system. The more you understand, the less need there is for amnesia, since the amnesia is a strategy to keep away negative, traumatic, unwanted memories. When integration occurs, memory between alters also gets better. Integration is often used to refer to fusion as well, which, they just are not the same at all.
Fusion is what can come after integration. After amnesic barriers have lowered between alters, it depends from system to system, but sometimes alters can kind of choose to fuse, other times, it just happens, etc. Regardless of if you want to fuse or not, it most likely will just happen naturally with the healing process.
Final Fusion is when the system collectively decides their end goal is to all fuse and become one alter. This isn't the case for everyone, but it is a valid decision nonetheless.
Functional Multiplicity is when you choose to not to take the path of Final Fusion, but instead, choose to just Integrate as a collective. Fusion can and will still happen, even if it isn't the goal. The reason Functional Multiplicity is possible is because of the fact that with integration, memory becomes better.
Integration is easier to pinpoint over fusion. You can usually tell if alters have integrated if they have very little to no amnesia between themselves. With fusion, however, there is no one specific thing to look out for.