Can White engineer a winning breakthrough? I have managed to create a temporary "fortress" after losing the exchange. Analyzing the position after the game ended, I was able to find a plan for White to breakthrough. After further analysis I was able to find the defense for Black. 

My "gut feeling" tells me that White should win anyway but I need help in finding the solution. 

Below is the position:

[Title "White to move"]
[fen "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]


EDITED on July, 21th 2014 :

Since nobody ( not even engines! ) considered option of penetrating on the kingside with the King via light squares, I am submitting lines that can guide others when constructing their answers.

[Title "White to move"]
[fen "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.b4 c5!? 2.dxc6+ Kc7 3.bxa5 bxa5 4.Rb1! Ra7 $8 5.Kf2! Ne6 6.Kg2! Ng5 ( 6...Nc5 7.Rxc5! dxc5 8.g5!+- Kxc6 ( 8...hxg5? 9.Rb7+ Rxb7 10.cxb7+- ) ( 8...Ra8 9.Rb7+ Kxc6 10.Rxf7 hxg5 11.h6 Kd6 12.h7 Ke6 13.Ra7! Rd8 14.Kh3 Kf6 15.Kg4 Kg6 16.Rxa5+- ) ( 8...f6 9.g6!?+- )  9.Rb8!+-  ) 7.Rb5! f6 ( 7...Ne6 8.g5!+- ) 8.Rb1! $22 Ne6 9.g5!! Nxg5 ( 9...fxg5 10.Rb7+! $8 Rxb7 11.cxb7 Kxb7 12.Kh3+- ) ( 9...hxg5 10.Rb7+ Rxb7 11.cxb7+ Kxb7 12.h6+- ) 10.Rb5! Ne6 11.Kh3 $22 Nc5 ( 11...Ng5+ 12.Kg4+- ) ( 11...Ng7 12.Kg4+- ) 12.Rcxc5!+- dxc5 13.Rb7+ Rxb7 14.cxb7 Kxb7 15.Kg4+-


Based on the above lines, I have tried to "sit in place", and came to the critical position where Black seems to hold:

[Title "White to move"]
[fen "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.b4 f6 2.bxa5 Rxa5 $8 ( 2...bxa5? 3.Rb1+ Kc8 4.Rbb6! Kd7 5.Rb7+- ) 3.Rcb4 Nf7 4.c4 Nd8 5.Rb5! Rxb5 ( 5...Ra8 6.a5! ) 6.cxb5! $8 Nf7 7.Ra3! Ng5 8.Kd2 Kb8 ( 8...Nf7!? ) 9.Kc2 Kb7 10.Rc3! Nh7 11.Kb3 Nf8 12.Kb4 Nd7 13.a5 bxa5+ ( 13...Nc5!? 14.a6+! $8 Kb8 15.Kc4! Nd7 16.Kd3 Nc5+ $8 ( 16...Nf8 17.a7+!+- ) 17.Ke2 Ka7 $8 ( 17...Na4? 18.a7+!+- ) 18.Kf2 Kb8 $8 19.Kg2 Ka7 20.Kh3 Kb8 21.g5!+- fxg5 $8 22.Kg4 Ka7 23.Kf5 Kb8 24.Rc2!! Ka7 25.Rc1! Kb8 26.Rg1! Nb3 27.Rxg5!! Nd4+ 28.Kg4! Nxb5 29.Rg6 Ka7 ( 29...Nd4 30.Rxh6 b5 31.Rh7! b4 32.a7+! Kxa7 ( 32...Kb7 33.Rxc7+!+- ) 33.Rxc7+ Kb6 34.Rc1+- ) 30.Rxh6 Kxa6 31.Rh7 Kb7 ( 31...Nd4 32.Rxc7+- ) 32.h6 Nd4 33.Rg7+- ) 14.Kxa5


It seems that Black can indeed create an impregnable fortress. Sacrifice with b6 does not work, and knight will always be able to reach g5 in time to stop White king from penetrating via g5 sacrifice. If White sacrifices on g5 too early, Black will always have Nf6 and will move around his king, maintaining blockade. Lines where White sacrifices rook on g5 end bad since Black sacrifices the knight and has won pawn endgame.

I will investigate few more ideas ( like c5 sacrifice ), before I officially accepting an answer...

END OF EDIT
After 10 minutes, Stockfish says:



[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.Rb1 c5 2.dxc6+ Kc7 3.b4 Ra7 4.Rb3 Ne6 5.Rb2 Ng5 6.Rb1 Ne6 7.b5 Ng5 8.Kf2 Ra8 9.Rd1 Ne6 10.Rd2 Nc5 11.Ke2 Ne6 12.Rd1 Rg8 13.Rc1 Ng5 14.Rh1 Ra8



...with the machine clearly not finding a plan. But what happens if we play a 'do nothing' move for Black's first move?



[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.Rb1 f6 2.b4 Ra6 3.bxa5 Rxa5 4.Rb3 Ra8 5.Rcb4 Nf7 6.Ra3 Ra5 7.Rb5 Ka6 8.Ra1 Nd8 9.Kd3 Nf7 10.Rxa5+ Kxa5 11.Rb1 Ng5 12.Ke2 Ka6 13.Rb4 Ka7 14.Kf2 Nh7 15.Rb5 Ka6 16.Ke2 Ng5  (663.81)



White removes the Black Rook but then it gets pretty slow. Black's superb Knight and that weak f pawn are keeping Black in the game.

So what about the obvious 1.b4?



[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.b4 f6 2.bxa5 Rxa5 3.Rb4 Rc5 4.Ra3 Ra5 5.Rb5 Ka6 6.Ra1 Nf7 7.c4 Nd8 8.Ra3 Nf7 9.Kd3 Ng5 10.Kd2 Nh3 11.Ra1 Ng5 12.Ke2 Nf7 13.Kf2 Nd8 14.Ke1 Nf7 15.Ra2 Ng5 16.Kf2 Nf7 17.Ra3



No dice.  Black looks solid.

From what I can see, Black holds.

EDIT

ALNS asked about this variation: 1.Rc6 Ne6! 2.g5 Nd8!? so I fired Stockfish back up.

I couldn't get past that hanging g-pawn, so the first variation is 1.Rc6 Ne6 2.g5. Stockfish used 4800 seconds on this (80 minutes).


[Title "1.Rc6 Ne6 2.g5"]
[fen "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.Rc6 Ne6 2.g5 Nxg5 3.b4 Ne6 4.Rc4 Ng5 5.bxa5 Rxa5 6.Rb4 Nh7 7.Rbb1 Nf6 8.Kd3 Nxh5 9.Rh1 Nf6 10.Rxh6 Nd7 11.Rh8 Nc5+ 12.Kc4 Na6 13.Rf8 b5+ 14.Kb3 Nc5+ 15.Kb4 Rxa4+ 16.Rxa4 bxa4 17.Rh8 Kb6 18.Rh6 Kb7



So it's kinda sharp, but again it is hard to see how any progress will be made.  The position is rated .59 White's favor - the least favorable score for White for any variation we've discussed.

Now, on to the main variation 1.Rc6 Ne6 2.g5 Nd8


[Title "1.Rc6 Ne6 2.g5 Nd8"]
[fen "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.Rc6 Ne6 2.g5 Nd8 3.Rc4 hxg5 4.h6 c5 5.h7 Nc6 6.dxc6+ Kxc6 7.Rh1 Rh8 8.Kf2 Kd7 9.Rh6 Ke7 10.Kg2 f6 11.Kf2 Ke6 12.Rh1 Ke7 13.Rh2 Ke6 14.Kg2 Kf7 15.Kf1 Ke7 16.Rh6 Kf7 17.Kf2 



That's after almost 2 hours.  White is rated +3 and seems to be able to engineer a break on the Queenside. This is enough for a win. The Knight seems misplaced on d8.  I'd grab the pawn on g5 instead.

EDIT - ALNS has 2 more variations to look at


[Title "5... Ne6"]
[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1. Rc6 Ne6 2. g5 Nd8 3. Rc4 hxg5 4. h6 c5 5. h7 Ne6 6.dxe6 fxe6 7.Rh1 Rh8 8.Kd3 Kc7 9.b4 Kd7 10.bxa5 bxa5 11.Ke2 Kc6 12.Kd2 Kd7 13.Kd3 Ke8 14.Ke2 Kf7 15.Rd1 Rxh7 16.Rxd6 Rh2+ 17.Kf1 g4 18.fxg4 Ke7 19.Rc6 Kf6 20.R4xc5 Kg5 21.Rxe5+ Kxg4 22.Rcxe6 Kf3 23.Kg1 Rg2+ 24.Kh1 Rc2 25.Rxa5 Rxc3 26.Rf5 Rc1+ 27.Kh2



With, after several hours of processing, a healthy score of +7.5 in White's favor. Black is dead lost. I like to add complete variations to these threads, but all these moves are suspect after moves 4 or 5. In this variation, Black captures the pawn on h7 but in doing so allows White to run amok.  I don't care much for the Knight sacrifice; holding when down a full Rook is tough.

edit - 8.Kd3 is a killer move. Black can get the Rook with 8... d5 but after 9.Re4 dxe4 10. Kxe4 Black's king-side pawns all drop.



[Title "5. dxc6"]
[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.Rc6 Ne6 2.g5 Nd8 3.Rc4 hxg5 4.h6 c5 5.dxc6 Nxc6 6.Rd1 Kc7 7.b4 Kd7 8.bxa5 Rxa5 9.h7 Ra8 10.Rb1 Kc7 11.Rh1 Rh8 12.Rh5 Kd7 13.Kd3 f6 14.Rh6 Kc7 15.Ke2 Kd7 16.Rh5 Kc7 17.Rh2 Kd7 18.Kd3 Kc7 19.Rh1 Kd7  (10405.71)



This comes up as +2.2 for White.  I don't like this because it allows the misplaced Knight on d8 to reposition with tempo. The simpler 5. h7 puts the pressure on Black and wins the Knight and then the game per the previous diagram.

EDIT
This is in regards to the "White King Infiltrates" variation mentioned in the OP's post.



[title "Infiltration"]
[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.b4 c5 2.dxc6+ Kc7 3. bxa5?!+= (3.Rc1 Ne6 4.b5 Rg8 5.Rb1 Re8 6.Ra1 Ra8 7.Rd1 Rd8 8.Kf1 Ra8 9.Rd2 Nc5 10.Kf2 Ne6 11.Ke2 Rd8 12.Kf1 Ra8 13.Kf2 Nc5 14.Ke2 Ne6 15.Rb2 Rh8 16.Kf1 Rh7 17.Kf2 Nc5 18.Rd2 Rh8+/-) 3... bxa5 4. Rb1 Ra7 5.Kf2 Ne6 6. Kg2 Ng5 7.Rb5 f6 8. Rb1 Ne6?! ( 8...Nh7 9.Kf1 Ng5 10.Ke2 Ne6 11.Rb5 Ng5 12.Rb2 Ne6 13.Rd2 Ra8 14.Rd5 Ng5 15.Kf2 Nh3+ 16.Ke1 Ng1 17.Rd3 Nh3 18.Rd1 Ng5 19.Rb1 Ra7 20.Ke2+/-) 9. g5 Nxg5?! (9...fxg5 10.Kh3 Nc5 11.Kg4+/-) 10.Rb5 Ne6 11. Kh3 Nc5?? (11...Ng5+ 12.Kg4 Nh7 13.Kf5 Ng5 14.Kxf6 Nxf3 15.Rb2 Ng5 16.Kg7 f3 17.Kxh6 Ne6 18.Rb1 Nf4 19.Kg5+/-)



The variations show a significant deviation from what Stockfish thought was best. All variations except for the last indicate a move that loses perhaps a pawn. Black's 11th move is fatal, however.  The associated variation shows White with a moderate advantage.  Black can make White work for it.

FINAL EDIT

Here's Stockfish playing itself at 15 minutes per move.



[title "Granite is tasty"]
[FEN "r7/1kp2p2/1p1p3p/p2Pp1nP/P1R1PpP1/1PP2P2/4K3/R7 w - - 0 1"]

1.Rc6 Ne6 2.Kf1 f6 3.b4 Nd8 4.Rc4 Kc8 5.Kf2 Kd7 6.Rc1 Nf7 7.Ke2 Ra7 8.Rc2 
Ra8 9.Ra2 Ra7 10.Ra1 Ra8 11.Kf1 Nd8 12.Ra2 Nf7 13.Kf2 Ra7 14.Ra1 Ra8 15.
Ke2 Nd8 16.Kd3 Nf7 17.Ra2 Ng5 18.Ke2 Nf7 19.Ra3 Nd8 20.Kf2 Nb7 21.Ra2 Nd8 
22.Rc2 Nf7 23.Rc1 Ng5 24.Rc6 Nf7 25.Ke2 Nd8 26.Rc4 Nf7 27.Rc2 Kc8 28.Ke1 *



...with White clearly not finding a break-through. Black holds, thanks to that magnificent Knight... and White's Rook being bottled up.