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It doesn’t matter what the source is (unless you have crt days flicker, bad tn panel polarization or what have you).

Irritation comes from the difference in brightness.

Reflection means it's got no difference (or a negligable technically) difference in brightness.

My mirror disagrees.

I think that reinforces their point if anything. With reflected light, you have a natural, inherent form of auto-brightness because the amount of light coming off the page depends on the amount of light in the room.

Books are not designed to reproduce colors though, and monitors are. If you have aggressive auto-brightness settings, that wouldn't actually make a monitor appear more like a book, it would just make it so the stuff that is actually supposed to look blisteringly white is merely mild. Which, sure, is an improvement for eye strain, but it's more of a workaround than a solution, and since it would muck up color reproduction a lot of users couldn't do this all the time anyways.

The retina can't tell the difference between reflected light and emitted light

Unless you're in the habit of reading your books with a flashlight, the context makes it very different.