How did Neo defeat the architect at the end of The Matrix: Reloaded? Was it solely due to his belief in himself or were there other factors at play?

They didn’t have to respect the laws of physics or any coherent system of rules, for that matter, and most of the events in the movie are conceptually impossible or very unlikely. You may ask them whether there is a “scientific explanation” why this victory was possible, they may answer by silence or by a would-be explanation but you may be pretty sure that the explanation is BS according to consistent frameworks to judge ideas.

The Matrix is a fiction. There are several levels of fiction when it comes to their realism and the Matrix simply isn’t a particularly realistic piece of fiction. In particular, when someone creates something or someone - especially by a method like coding - this agent (creator) has in principle a control over what happens and the basic aspects of the creation’s fate may be said to be the creator’s decisions. This common sense rule was violated in the Matrix movie - but it wasn’t violated in the mechanisms that created the movie. Both Neo and his creators are actually creations of the writers of the Matrix so they just decided who would win!

Neo defeated his architect because the authors of The Matrix - whether it was the Wachowski brothers-turned-sisters or Sophia Stewart - figured out that the profit from the movie would be higher if it had a sexy story showing the success of a creation against his creator. In essence, the viewers’ preference for this kind of a story is nothing else than the usual left-wing preference of rooting for the underdog.

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Now, someone may complain that I am ruining the story, it is right for people to be immersed into a work of fiction so that they don’t distinguish it from reality. Great, I happen to disagree. When they leave the movie theaters, people should regain the status of at least partially rational entities.