Thank you for your in-depth explanation inkspirit; I've played a few of these "doubled pawns being a strength" positions before, but I didn't recognize this position as such. Additionally, the central control aspect and the flexibility really resonate with the way I'd like to think more about chess. 
Not entirely true, although I do agree with almost all of your remarks.
It's important to bear in mind that one of the main weaknesses of doubled Pawns is not their vulnerability, but the reduction in "collective mobility" that the doubling entails.
Each Pawn has a certain amount of mobility, ranging from very high (for an unblocked Pawn in the center that is part of an unopposed Pawn-duo) to practically zero (for a member of a Pawn chain that is locked against an opposing Pawn and is not a candidate for exchange).
An entire constellation of Pawns can be characterized by a certain level of "collective mobility", which is roughly the sum of the individual mobility of its constituent Pawns.
Typically, doubled Pawns tend to contribute very little to the collective mobility of the Pawn constellation of which they form a part. This is one of the factors that tends to devalue doubled Pawns.
On the plus side, doubled Pawns can also control important squares which would not be covered if the Pawns were "healthy". In the present example, d6 is just such a square (as you correctly pointed out).
Am I missing something obvious in my evaluations?
Yes.
Controlling the centre and developing your pieces are both fundamentals of chess that you are underevaluating.
cxb6 captures away from the CENTRE and axb6 DEVELOPS your rook without even moving it.
Yeah, it's (much) more for the sake of the d pawn than the rook. Honestly cxb just looks bad to me and it's hard to explain why other than that... but that's probably about the sum of it anyway.
"Black's pawn structure is too costly just to open up the a-file for the Rook, in my opinion."
So, what is the weak thing about Black's queenside pawns after ...axb6?
This does not give white any mid-term structural advantage, while significantly increasing the mobility of Ra8.
Idk much about in depth analysis of pawn structures but I use some shortcut ideas like "Double pawns in the centre is good " and "capturing towards centre is preferred"... "Opening files for rook is advantageous ".
I think ya'll are thinking too complicated and strategic.
I think cxb6 is discouraged specifically because of the move Qd6, although the strategic advice given in this thread is certainly sound.
I think ya'll are thinking too complicated and strategic.
I think cxb6 is discouraged specifically because of the move Qd6, although the strategic advice given in this thread is certainly sound.
After 9.Qd6 the engine says 9...Qf6 is an equal position.
And if you move the e4 pawn to d4 on move 8, the engine still prefers 8...axb
I anticipated Qf6, I think white is doing quite well here.
Okay, obviously this is a losing battle for me. Eventually I will make a mistake or miss something as long as people use their computers to evaluate my moves. I just wanted to point out two things:
1: It seems like the weaknesses of the d6 square is by far the most relevant consideration of the move cxb6, rather than the activity of the rook or the strategic ramifications of doubled pawns towards the center. Most concretely, the black pawn is needed in c7 to defend d6 and it is worse on b6. Okay, if the engine doesn't play Qd6 immediately, it will probably play something similar later or use the threat of Qd6 to its advantage.
2: Sometimes when analyzing positions, decisions are made due to concrete tactical considerations, and it seems like OP didn't consider this and only asked the strategic generalizations of the ramifications of open rook vs. doubled pawns. I think even though Qd6 isn't the best move, the first question black should ask himself when looking at cxb6 is "What am I going to do after Qd6?" not "Is an active rook better than double pawns on the c file?" and it seems like OP never asked this question when choosing cxb6 as his preferred move. You can ask any question in any position, but in some positions, some questions are far more important than others.
Yes. By playing axb6 you're capturing toward the center, which is almost always recommended if you don't have any better choice. White can't easily attack any of the doubled pawns, so black is OK in this situation.
I anticipated Qf6, I think white is doing quite well here.
Okay, obviously this is a losing battle for me. Eventually I will make a mistake or miss something as long as people use their computers to evaluate my moves. I just wanted to point out two things:
1: It seems like the weaknesses of the d6 square is by far the most relevant consideration of the move cxb6, rather than the activity of the rook or the strategic ramifications of doubled pawns towards the center. Most concretely, the black pawn is needed in c7 to defend d6 and it is worse on b6. Okay, if the engine doesn't play Qd6 immediately, it will probably play something similar later or use the threat of Qd6 to its advantage.
2: Sometimes when analyzing positions, decisions are made due to concrete tactical considerations, and it seems like OP didn't consider this and only asked the strategic generalizations of the ramifications of open rook vs. doubled pawns. I think even though Qd6 isn't the best move, the first question black should ask himself when looking at cxb6 is "What am I going to do after Qd6?" not "Is an active rook better than double pawns on the c file?" and it seems like OP never asked this question when choosing cxb6 as his preferred move. You can ask any question in any position, but in some positions, some questions are far more important than others.
- The d6 square is not something Black would worry about a bit, as white cannot maintain a queen or piece there after 9.Qd6 Qf6 or 9...Qe7.
- 9.Qd6 Qf6 10.e5?! is the start of trouble for white after 10...Qxd6 11.exd6 Nf6 or even 11...h5: the d6 pawn is a permanent weakness.
- The best way to prove d7 as a mid-term weakness is taking measueres against ...d5, say with 9.c4 followed by Nc3 and eventually piling up on the d-file.
Interesting analysis, if you don't mind answer, I have some questions. Why do you consider the d6 pawn weak after Qxd6 exd6? After Nf6, I planned to play Bd3 preventing Ne4, and how does black literally ever attack the pawn you consider to be weak? He doesn't have a dark squared bishop. Does he maneuver his knight to e8? Then white would look pretty strong with his rook on e1. Does he play c5 and maneuver his rook to c6? That looks incredibly cumbersome, and allows white to maneuver his white from a3 to b5 to c7. Is there something obvious I'm missing? I don't see how that pawn is weak whatsoever, it seems like an incredibly thorn in black's position.
EDIT: Ah I se, the plan is i fact castles and Ne8 and white is just one tempo too short. Still, I think that Ne8 maneuver is important. I don't think the pawn on d6 is positionally weak, it's tactically weak to the maneuver Nf6 O-O Ne8 because white is underdeveloped. If black didn't have that exact maneuver, he would never capture that pawn in a million years.
Pawns are weaker the more they move down the board. You need pieces to use the extra squares they give you, otherwise they're just a liability. Particularly in an early endgame, a quick pawn to the 6th rank just looks weak.
OP was using an endgame frame of mind in the early phases, while you're doing the opposite, using middlegame evals to judge something closer to an endgame.
I was skeptical about White's 10. e5 cause I thought Black could win the d-pawn, so I put it into Fritz and no matter what moves I tried Fritz could always draw the game (for both sides). Here's one of 5 variations I tried (only to end up in a drawn position).
1. e4 e5 2. Nf3 Nc6 3. d4 exd 4. Nxd4 Bb4+ 5. c3 Bc5 6. Be3 Bb6 7. Nxc6 bxc6 8. Bxb6 cxb6 9. Qd6 Qf6
10. e5 Qxd6 11. exd6 Nf6 12. Nd2 O-O 13. O-O-O (Note: at one point Fritz recommended 13. f3 Re8+
14. Kf2 which ended up equal), but White can try for more with the move 13. O-O-O line as I will show.
13... Bb7 14. Bc4!? Rfe8 15. Rhe1 Kf8 16. Nf3 Ne4 17. Bxf7 Kxf7 18. Rxe4 Rxe4 19. Ng5+ Kg6 20. Nxe4 c5
21. f3 and no matter what I played for the best ideas the position was always equal.
I wouldn't play this line for White.
Other posters already said all there is to it with regards to 8...axb6, so I'll just note something different which struck me out.
I really don't like 7. Nxc6 at all. White has secured this lovely d4 square for his Knight: it's centralized, it can jump to excellent attacking square f5 later in the game, it can exchange on c6 later in better circumstances, and Black exchanging it on d4 would give White a full pawn center. And White, instead of using latent power of his d4 Knight, rushes to exchange it; along with exchanging his dark-squared Bishop - thus dulling the game and quickly reaching even mid-game.
until that a-pawn got "promoted" to a b-Pawn the black rook was like a maserati sitting in the garage with the door closed. the garage door has now been opened. Gentlemen, start your engines!
How is this position equal? Are you really indifferent to either color here? Or are you saying it's equal because the engine says 0.00? There's a difference between 0.00 because black has a series of only-moves that lead to a close draw, or 0.00 because the position is equal and nothing white does will win. White has a clear advantage in this position. The straw d pawn will never be captured except by black's king if white's rook move. If black isn't careful, he will lose this position either by missing a tactic that allows white to play Re7 (which is devastating due to the d6 pawn) or missing a tactic that allows white to sacrifice the d6 pawn in exchange for extraordinary piece activity. I don't buy this position is equal, I want the white pieces.
How is this position equal? Are you really indifferent to either color here? Or are you saying it's equal because the engine says 0.00? There's a difference between 0.00 because black has a series of only-moves that lead to a close draw, or 0.00 because the position is equal and nothing white does will win. White has a clear advantage in this position. The straw d pawn will never be captured except by black's king if white's rook move. If black isn't careful, he will lose this position either by missing a tactic that allows white to play Re7 (which is devastating due to the d6 pawn) or missing a tactic that allows white to sacrifice the d6 pawn in exchange for extraordinary piece activity. I don't buy this position is equal, I want the white pieces.
Any good reason for Black to drop a pawn with 16...Ne4?
Lately my chess study has been concerned with the implications of pawn structure and when to break conventional rules (especially for middlegame compensation). I currently play 1.d4 openings more, but I found this 1.e4 position that puzzles me a bit (Scotch Game position). Black's response on move 8 is what confuses me in this line. Here 8...axb6 is the "best move." The computer claims that 8...cxb6?! is an inaccuracy, but I don't see Black getting enough compensation for the pawn structure. Am I missing something simple in my evaluations, or is the computer not to be trusted here?