Doesn't Snape see Peter Pettigrew on the map in Prisoner of Azkaban?

Question

So, I see lots of posts on why Fred and George didn't notice Peter Pettigrew on the Marauder's Map, but why does Snape not see Peter Pettigrew on the left of the map when seeing where Lupin went, while trying to bring him his potion. Snape would definitely recognize his name. Snape would know the wizard who Sirius killed having been so invested in the event.

I get that he truly hates Sirius and wants him dead. I believe he is truly prejudiced against Lupin. But in his effort to be the best undercover agent ever, he just makes no acknowledgment of Peter Pettigrew in the room and what he's doing there or how he might be validating Lupin and Sirius' innocence.
I'd just like to hear other people's thoughts on it.

Answer 1

No, because the Shrieking Shack is located in Hogsmeade, not Hogwarts.

Snape was slightly breathless, but his face was full of suppressed triumph. 'You're wondering, perhaps, how I knew you were here?. he said, his eyes glittering. 'I've just been to your office, Lupin. You forgot to take your Potion tonight, so I took a gobletful along. And very lucky I did ... lucky for me, I mean. Lying on your desk was a certain map. One glance at it told me all I needed to know. I saw you running along this passeway and out of sight.'

"Out of sight": The map ended there. It only showed the compounds of Hogwarts and not beyond that. The Shrieking Shack is an abandoned house in Hogsmeade.

Thus Snape never saw Harry, Ron, Hermione, Sirius, or Pettigrew on the map at that time, because they weren't technically in Hogwarts territory (which the map covered).