Did he commit the murder, or is that too literal?

(spoilers)

The film is, from the very first shot, completely subjective. Nothing is literal, rather it is a combination of distorted memories and fantasies connected to try and recreate a narrative. So when, at the end of the film, he murders the little girl, do you think this is a memory (perhaps repressed) or is it a fantasy, the madness induced by following the Element of Crime method? People seem to be really sure that both Osborne and Fisher became murderers by following the method. The film we see is completely mad because it is about subjective memory and the psychology of a disturbed man, but it seems a bit extreme that both of the characters who follow this method become killers.

I've only seen the film once so there are many elements which I will have missed, but I wondered if many people thought Fisher committing the murder was more about the primal rage within man than a literal account of the events. Osbourne does say at the start of the film it is a mistake to look for crime (violence) in systems when it's true source lies in (universal) human nature. The idea that everyone has a murderer inside them is perhaps more powerful when we see that this is lying dormant than when it is actually emerging and our protagonist doing something so extreme (and yet, within this film, quite absurdly commonplace). Films like this are often more about understanding the mind than the actions, and although often the two can be inseparable, as in Lost Highway, it seems to be taken as a given that in this case he was actually a killer.

Thoughts?