I have a hunch that's not actually what's going on, the second "the" gets filtered out way earlier, before it even reaches the higher features of language processing: to compensate for the constant jittering and shifting of the eye's focus, our pattern recognition modules need to navigate by close-by features in order to build up a model of what we see. In reading mode, the second "the" never even makes it to language processing because it gets filtered out during error correction as an eye-alignment error.

I imagine the pseudo code backtrace would be something like this: "I saw a 'the' shape" > "move focus to next position on the right" > "still seeing the 'the' shape" > "move focus to next position on the right".

So unless you decide to consciously process the line, or your visual system explicitly learns to recognize a 'the the' pattern, the data simply never gets passed along.