Bizarre "Elementary, Dear Data" plot hole

Moriarty specified the computer replicate a "real" sheet of paper because he knew Data would take it to Picard.

Where's my no-prize?

The people are holographic projections. All the items are replicated.

From 'Encounter at Farpoint' when Riker and Data first meet in the holodeck:

Data: This woodland pattern is quite popular, sir. Perhaps because it duplicates Earth so well. Coming here almost makes me feel human myself.
Riker: I didn't believe these simulations could be this real.
Data: Much of it is real, sir. If the transporters can convert our bodies to an energy beam, then back to the original pattern again--
Riker: Yes, of course. And these rocks and vegetation have much simpler patterns.

Much of the matter on the holodeck are replicated simulations, but the holodeck, and the replicators, can't actually creating living things.

I read that this was intended to disintegrate later but they never filmed that or it was cut, and that Picard was originally going to trick Moriarty into leaving the holodeck knowing this, intentionally allowing him to vanish. Roddenberry thought this was too cruel of Picard.

The piece of paper could have been replicated off screen.

It’s possible the paper was physical/replicated.

In universe it boils down to whether of not the paper in question is actually replicated or an illusion via force fields, but it's always been at the service of the plot more than anything..

The only constant in this scenario is that it can't have it's elaborate things, like Moriarty, exit.

Not only that, Geordi holds the picture upside down. :)

The holodeck mixes holograms with replicated items. Anything that generally doesn't move is a hologram. The book that Picard threw out the door was really just decoration, and would likely never be touched, so it was a hologram. But anything that will likely get moved, consumed, burnt, wet, etc gets replicated. A piece of paper could easily be used and carried around, or burnt, or get wet, so it was replicated. It's more efficient for the computer than keeping tabs on billions of molecules that can move and interact with other things.

Everyone else is answering the question with presumptions and guesses. According to the original script, after convincing Moriarty to surrender and ending the program, Data and Picard went on to explain that Moriarty could have actually left at any time without disappearing, and uses the existence of the paper out of the holodeck as proof.
Link to the final draft, Ctrl-f: Act 5

My guess is that it's a programming parameter. Whoever writes the program can decide if an object is replicated or projected light, allowing use of the holosuites for food/drinks, classrooms, emergency quarters with functional pluming. Whatever, really.

Or if unspecified, the computer may do whatever is most efficient. This may be how turning off safety parameters works. Imagine a firearm; the bullets move too fast to see so the computer may only replicate their effect on their point of impact. With the safeties off, the most efficient thing to do would just be to replicate a real firearm and bullets.

well perhaps because its far far far less complicated to replicate a holographic paper then a complex holographic being.

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is my guess.

but yes that is a bit of a glaring plot hole they should have explained.

didnt think about that till now

lol.

Holodeck stories are full of plotholes