Hal Jordan returns from a visit from Oa and arrives to see the earth decimated. the machines attack him as soon as he enters the earth atmosphere catching him off guard. GL goes into the ruin city to escape the machines. He uses his ring in an attempt to analyze one of the machine scraps. his ring interacts with it and uploads him into the Matrix.
Neo sees him arrive and does not recognize his computer code and thinks he is a new agent specifically tailored to defeat him.
If they are actually in the Matrix Neo wipes the floor with him. By the end of the final film he has ridiculous power in the real world and pretty much absolute control within the Matrix.
Matrix master crushes Matrix newbie...at first. But once the Ring (which supposedly is a far, far more advanced tech than the Matrix) analyzes the Matrix's workings, GL's wins start mounting. The Ring is, after all, "the most powerful weapon in the universe," hardly a claim ever made for the Matrix.
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Shinier than a speeding bullet.