Marauder's Map Folly?

[houyhnhnm102]

No: HPFGUIDX 150799

Rachel asks: 
 

> My question is - when Harry and Hermione use the 
> Time-Turner to save Buckbeak and Sirius, why does 
> Lupin not see two Harrys and two Hermiones?  Snape 
> as well does not notice this when he arrives to give 
> Lupin his potion and sees the map.  Is there any 
> explanation for this?

[Now Tammy says:]  
> > While I don't have my canon readily to hand, I do 
> > believe that Hagrid's hut doesn't show up on the 
> > Map, which is one good reason why Pettigrew, who 
> > helped write the Map, would have hidden out there.   
> > Also, if I remember rightly, at the time when Lupin 
> > and Snape would have each had a look at the Map, 
> > the Time-Turned Harry and Hermione were still over 
> > near Hagrid's place, hence off the Map.

houyhnhnm:

Lupin says:

"The important thing is, I was watching it carefully this evening,
because I had an idea that you, Ron, and Hermione might try and sneak
out of the castle to visit Hagrid before his hippogriff was executed ..."

And

"I watched you cross the grounds and enter Hagrid's hut.  Twenty
minutes later, you left Hagrid and set off back toward the castle. 
But you were now accompanied by somebody else."

Where were Time-Traveling Harry and Hermione when HRH1 were arriving
at and leaving Hagrid's hut?
*****************
They tore across the vegetable gardens to the greenhouses, paused for
a moment behind them, then set off again, fast as they could, skirting
around the Whomping Willow, tearing toward the shelter of the forest.
They made their way silently through the trees, keeping to the very
edge of the forest.  Then, as they glimpsed the front of Hagrid's
house, they heard a knock upon Hagrid's door ... And Harry heard his
own voice.
[...]
Harry watched the grass flatten in patches all around the cabin and
heard three pairs of feet retreating.  He, Ron, and Hermione had
gone...but the Harry and Hermione hidden in the trees could now hear
what was happening inside the cabin through the back door. 
*****************

It is possible they were off the map while on the edge of the
Forbidden Forest, and it does look, too, from Lupin's statement, that
the interior of Hagrid's hut does not show up on the map.

However, Lupin should have been able to see them running through the
vegetable gardens, past the greenhouses and around the Whomping Willow.

So it must be that Time Travelers, unlike people in invisibility
cloaks, do not show up on the Marauders' Map.  Or--consider the
difficulty of locating any single person or even two or three on a map
that shows the presence of hundreds.  Once Lupin had located the Trio,
it's unlikely he would have paid much attention to the multitude of
other figures moving on the map.  He was not expecting to see another
Harry or Hermione.

Snape would not have either, although under normal circumstances I
think his eyes would have been quick to spot any figure labeled "Harry
Potter".  He saw an unmedicated Lupin running through the passageway
towards the infamous Shreiking Shack and took off in pursuit, probably
with little thought of anything else.  At this time, Harry2 and
Hermione2 were still hiding on the edge of the forest, so they may 
have, indeed, been off the map.