Before I begin, let me say that I personally feel Serial Experiments Lain was created to be contradictory and I still do not fully comprehend SEL even after watching it 3 times and reading multiple theories.
Lain is software developed by Tachibana Labs and the Knights, Eiri tried to add protocol 7 to Lain unauthorized and was killed but knew the risks beforehand and programmed himself into the wired. Lain ran off human brain power and the version version existed in the wired to socialize with others.
Since the Knights were scatted across the world and improving Lain would be difficult, she had to come to the Wired. Eiri intentionally made Chisa kill herself and send the email so Lain would visit the Wired.
Afterwards, the Knights decided to set up a LAN in Cyberia so they could project Lain into the minds of others in reality. Whenever someone took Accela they would have the hallucination of Lain. Later on though the drug wasn't needed for people to see Lain as indicted by the 3 girls who could see Lain in Episode 2.
The next step in the software was for Lain to actively exist in the physical world. In episode 1, Lain wasn't as responsive as later on because she was still in her beginning stage. The Wired Lain could still take over her physical self if she was in danger which you can see when she was in danger at Cyberia and her personality took a 180 degree turn.
Lain was up and running, what else is needed? Speed and more functionality! After they installed the psyche processor Lain was able to improve herself through programming and was more positive. The Knights were constantly improving Lain though and they added in another personality and body, aka Evil Lain.
Lain later on started hacking and surpassed the Knights and achieved her original goal to get revenge on Tachibana Lab so the Knights were killed by the men in black, though the secret was exposed and they were killed also. After all of this, only Lain and Eiri knew the secret with Eiri having full control over Tachibana Lab and the Knights & MIB killed.
Soon after, Eiri cut off reality to Lain and tried to force Lain to love him. Lain refused and connected back to reality and tried to share her true self with her best friend, Alice. In episode 12, she had full control over all her personalities and was happy though Alice was sad.
When Alice visited Lain, Eiri found out that Alice was starting to understand Lain and asked Lain to control Alice and of course Lain denied the request. After the fight with Eiri, Lain changed reality by making everyone forget her and changing Alice's age to make her relationship with her teacher possible.
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