Rook vs 4 pawns

http://www.chess.com/livechess/game?id=551965694

Watch this game. i played black

Very nice! Do you know if 4 pawns is a theoretical win vs a rook? I actually think 3 pawns is even a theoretical win vs a lone rook, but I could be wrong. I had a similar situation occur in one of my OTB games - I could have sacrificed one of my rooks to reach a 3 connected passed pawns vs rook situation, likely guaranteeing at least a draw, but I didn't see it 

Instructive to see how to stop them.  The same principals apply when trying to stop 3 or 2 also (usually rook goes behind and king stays in the area of action as much as possible).  In the game 52.Kd4 and 53.Ra8 were very bad, and at that point black should be winning.

52.Kf6 wins a pawn.

But sometimes 3 backward pawns can't win against rook when king is placed well

Two (connected passed) pawns on the sixth (rank) beat everything up to a royal flush GM Ian Rogers (FACT)

Three connected passed pawns on the 5th-rank beat a lone rook (FACT)

I can't recall actually reading anywhere that 4 connected passed pawns on the 4th-rank beat a lone rook, since that appears to be the next logical conclusion here. Using search engines has only revealed that search engines suck ass when it comes to looking for exactly what I tell them to look for, or perhaps more likely that I suck ass when it comes to using search engines properly.

"As is to be expected, 4 pawns will normally win vs the rook. However, when they are not far advanced, the rook will be able to stem the tide" BCE, p. 286. (Too bad GM Fine doesn't explain what "far advanced" means) "More than 4 pawns should win without any trouble" (IBID)

All the above presumes that the rook's King is unable to help stop the pawns from advancing (he's too far away or out to lunch, whatever dude)

What is the score of the position after a8=Q Rxa8 Rxa8 when engine analysed. Is it + or - ?

But Kf6 makes it easy for the pawns to promote quicker because the kings not around to prevent promotion

You are right. Kf6 draws the game. I checked it with Scid

I just played a game and won with 4 pawns against a rook, some of the examples here are great but when helped me was to properly shoulder the king from getting close and then supporting the pawns with other pawns and my king

#5
4 connected passed pawns on the 4th rank do not beat a rook.
This is a draw

3 connected passed pawns on the 5th rank do not beat a rook
this is a draw

Often true, but this is a win for the pawns (or with the position shifted right). With the black king on c8 - draw.

or this (but the pawns haven't all reached the 5th. rank). With the black king on e8 Black wins.

Edit: Just understood what you're saying. You mean 3 connected passed pawns on the 5th rank do not always beat a rook.

#13
Indeed I provided some counterexamples to the 'rule'
Here is another counterexample that shows 2 connected passed pawns on the 6th rank do not beat a rook: the rook wins