When Neo and Morpheus are in the Construct and Morpheus is showing Neo the 'Real World', he says the line "Fate, it seems, is not without a sense of irony", and he looks up to the stormy sky, and you can see the clouds' reflection in his glasses. If you look close, you can see a reflection of the boom mike in the left lense.
You are absolutely right. As a filmmaker, I can tell you right now that that is positively the boom mic. Also, just before this scene, Morpheus first meets up and walks towards Neo in the white room, his glasses are black, and then in the next shot, they are reflecting the white room!
As a "film maker", you should also know a little more about light reflections. It's highly possible that from one angle the lenses appear black whilst from another angle it reflects it's surroundings.
Maybe this is not a slip-up but an easter egg... what if the reflection on the lenses was added digitally and they intentionally put a boom mike in the scene? Anything can happen in the science fiction universe.
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