Well, I don't know about Carlsen's computer but my computers certainly "think" that the rook exchange sac is OK, at least for a draw. At d=50 after 19...Rxa4 20.bxa4 Stockfish 12 evaluates its top 2 lines at [0.00]:

1. [0.00]: 20...Bf5 21.Rde1 Ng4 22.Rhf1 Be5 23.h3 Rb2 24.Qd3 Nh2 25.Bc3 Qb8 26.Bxe5 Qxe5 27.Rh1 Rb4 28.Qc3 Rb1+ 29.Kd2 Rb2+ 30.Kc1 and a draw by repetition.

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2. [0.00]: 20...Qb6 21.Rde1 Nxc4 22.Nxf6+ gxf6 23.Re8+ Bf8 24.Rxc8 Rxc8 25.Qxc4 Bd6 26.Bc3 Rb8 27.Kd2 Be5 28.Bxe5 fxe5 29.Rc1 Qh6+ 30.Ke1 Rb4 31.Qxc5 Qf4 32.h3 Re4+ 33.Kf1 Qh2 34.f3 Rxa4 35.Qc2 Rd4 36.Kf2 Rxd5 37.Rd1 Rxd1 38.Qxd1 Qf4 39.Qd6 Qc4 40.a3 Qf4 41.g3 Qc4 42.Qb8+ Kh7 43.Qxe5 Qa2+ 44.Ke1 Qb1+ 45..Ke2 Qa2+ and a likely draw, by repetition if White is so inclined. Which, when you are having a bad tournament and your opponent is the world champion, you are likely to be.

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Komodo 12.3 had similar lines and evaluations at d=36:

1. [0.00]: 20...Bf5 21.Rde1 Ng4 22.Rhf1 Be5 23.h3 Rb2 24.Qd3 Nh2 25.Bc3 Qb8 26.Bxe5 Qxe5 27.Rh1 Rb4 28.Qc3 Qf4+ (only here does Komodo deviate from Stockfish) 29.Qe3 Rxc4+ 30.Kd2 Rd4+ 31.Kc3 Qe5 32.Qg3 Rd3+ 33.Kxd3 Qxd5+ 34.Kc3 Qd4+ 35.Kc2 Bxe4+ 36.Rxe4 Qxe4+ 37.Kd2 Qd4+ 38.Qd3 Qxf2+ 39.Kc3 Qxa2 40.Rb1 g6 41.Rb8+ Kg7 42.Qd8 Qa3+ 43.Kc2 Qxa4+ 44.Kd3 Qa3+ 45.Kc4 Qa2+ 46.Kxc5 Qc2+ 47.Kb6 Qf2+ 48.Kxa6 Qa2+ 49.Kb6 Qb3+ 50.Kc7 Qc4+ 51.Kd6 Qf4+ and I doubt that White can prevent a draw by repetition, although in the open board and possible time trouble it would take a great memory to claim it.

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2. [0.00]: 20...Qb6 21.Rde1 Nxc4 22.Qb3 (this time Komodo deviates from Stockfish fairly quickly) 22...Bf5 (and Komodo allows the exchange of queens, probably evaluating that the greater activity of its pieces and its 2 bishops compensate for Black's material deficit, even without queens on the board) 23.Qxb6 Rxb6 24.Bc3 Bf4+ 25.Kc2 Na3+ 26.Kd3 Rd6 (here 26...Nc4 might have saved a tempo) 27.f3 Rxd5+ 28.Ke2 Rd8 29.Kf2 Rd3 30.Re2 Nc4 31.Bxf6 Bxe4 32.Rxe4 Rd2+ 33.Ke1 Rc2 34.Rxf4 gxf6 35.Re4 Nd6 (35...Rc1+ was apparently also possible, Komodo evaluated both 36.Kf2 and 36.Ke2 at [0.00]) 36.Kd1 Rxa2 37.Ree1 Kf8 38.g4 hxg4 39.fxg4 Ne4 40.Rhf1 Nc3+ 41.Kc1 Ne2+ 42.Kb1 Nc3+ 43.Kc1 and a likely draw by repetition, although I think it would have been impossible for any human to see all this.

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Long lines to be sure, they don't prove anything since the likelihood that the game would have progressed along these lines is slim to none. But they do illustrate the virulence of Black's potential attacking chances, and seem to show that Black has enough compensation for his material deficit.

If by "tangible" you mean only material then of course there isn't enough. White is up a pawn and the exchange. But White's q-side is a mess, and as the four lines above show, particularly after ...Bf5, Black's control of the open b-file and the pinned Ne4 greatly limit the activity of White's pieces, and Black's pieces are certainly more active. And in the final position of Stockfish's first line not only does White is still up a pawn and the exchange but Black's Nh2 is trapped. Yet the activity of Black's pieces allow him to get a draw by repetition. So "tangible compensation" is sometimes based on more than optimism, but of course that depends on your experience and your talent.

If you are going to quote my "Ouch! :-)" in Norway Chess (2020) (kibitz #121) than you should at least do in its proper context. As you well know my "Ouch!" was referring to his game against Duda the following day, when I first said that "Maybe he partied too much during yesterday's day off. Or maybe the pressure of this 125-game unbeaten streak got the better of him." and then "If the latter was the case then he will be relieved and relaxed tomorrow and be out for blood. I feel sorry for his opponent tomorrow. Oh, wait! He's playing Duda again tomorrow! And he has the White pieces. Ouch!" And as we all know by the results of that not-yet-played game on the following day (and not yet uploaded), I was quite correct. A blow-out by Carlsen in 26 moves.

16...Ne5 allows 17.Bd2, with strong advantage for white:

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Stockfish_20100519_x64_modern:

39/57 06:01 +2.18 17.Bd2 Ng6 18.Rhe1 Re5 19.Rxe5 Bxe5 20.Bc3 Qc7 21.b3 h4 22.Ne4 Bf5 23.d6 Qb7 24.d7 Bxe4 25.d8R+ Kh7 26.Qe2 Bf4+

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Stockfish_20100519_x64_modern:
NNUE evaluation using nn-baeb9ef2d183.nnue enabled

This kind of proves my point. No one but computers would, and probably should, go a rook and pawn down for a bishop because they forsee a possible or even probable draw somewhere in 10 to 30 moves. That might be stretching the definition of adequate compensation.

I get the feeling that Carlsen didn't spend a whole lot of time analyzing or memorizing much after 19...Rxa4. Probably saw the assessment at near equal and saw the position. A position that admittedly looks like there should be enough tangible compensation. At least to come.

The problem might be that it's computer assessed at near equal only if you go for the draw. The computer probably let Magnus down by not warning loudly after 19...Rxa4 "Must use drawing lines to continue. All others clearly lost!" Otherwise with an apparent visual positional advantage and computer backing of near equality, only seems logical to play optimistically and actively. Which unfortunately looks to only get black in trouble.

One must <ALWAYS> question the silicon monsters. In the "classic" chess engines where the best move is based on the minimax algorithm it reflects the evaluation of the leaf node (position) of the branch of the search tree that gives the best results in the minimax sense. And, as a result its iterative deepening technique, in any line, no matter the search depth, that result is based on the pairwise comparison of the tree of node evaluations. And the last few nodes of the search tree do not have the benefit of a reasonably deep search and are therefore susceptible to errors.

In the extreme case the evaluation of the nodes at the leaf nodes of the search tree do not have the benefit of any lookahead. The chess engine is literally blind. It may indicate a winning advantage for White when Black has a mate in one on its next move.

I did one analysis in which this happened but unfortunately I didn't save it. But I did save this one: Stockfish (Computer) (kibitz #115). In this position;

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Komodo 9.2 (the strongest chess engine at that time) evaluated its 3rd best line at [+0.16], indicated it at having equal chances for both players. But Black has a forced mate in 5 after 1...Ra2 2.Rb8+ Kg7 3.Rb7+ Kh6 4.Rh7+ (a version of the horizon effect, an engines sees that it is lost and tries to delay the inevitable as long as possible) 4...Nxh7 5.Bxh7 Rxh2#

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So it's very important to review the silicon monster's lines for reasonableness and see if there are any alternate moves during its analysis lines that might result in a different evaluation of the initial position. This can be done either by human or chess engine analysis. In the latter case it's called "forward sliding" and involves restarting the analysis at a position further down from the analysis' starting position in order to push the horizon effect further into the future. At the extreme you can restart the engine analysis at the last position listed in the engine's analysis (which may not be the last position in its search tree) to see if the engine's initial evaluation was reasonable. I call this "forward leaping" in a pun to "forward sliding" and, unlike forward sliding, it doesn't have as its goal to find better alternative moves than those shown in the engine's principal variation (the line representing best play by both sides), but just to check if the engine's original evaluation was reasonable.

By the way, when I present analysis this way, as a long computer run, it might best be called "unverified," much as how artificially intelligent software interprets a Electro Cardiogram or ECG, before review by a doctor. I do not seek perfection in my analysis, merely a reasonably good first guess, where I invite others to look further and improve where possible.

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Stockfish_20100519_x64_modern:
NNUE evaluation using nn-baeb9ef2d183.nnue enabled

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Stockfish_20100519_x64_modern:
NNUE evaluation using nn-baeb9ef2d183.nnue enabled

Glad you looked. Wondered about it but didn't check it out.

Think it might of worked out pretty well if in your line Duda played 28. Kd1. He might've as he stated he was worried about blacks N check possibilities on e2 and did play it in the game.

Then 28. Kd1 Be5 29. Rb3 Rxb3 30. axb3 Nxh3 31. Rg2 Qb8 32. Qd3 Bxe4 33. Qxe4 Qxb3+. Where Carlsen had a N and p for the rook, a monster B and an active Q. Probably good compensation for 19...Rxa4.

But maybe not keen on the simpler 26.g4 Bxe4 27. Qxe4 Ng3 28. Qg2 Nxh1 29. Qxh1 being a pawn down but likely draws. Could be looking for more. Thinking an eventual passed pawn will be the payoff for the exchange sac and the advanced h pawn might be the one.

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Stockfish_20100519_x64_modern:
NNUE evaluation using nn-baeb9ef2d183.nnue enabled

I'm not sure which of your points it proves. Your original post (J K Duda vs Carlsen, 2020 (kibitz #12)) dealt with Carlsen's computer steering him wrong in believing the exchange sac 19...Rxa4 was OK, and that you had to believe that it was part of his prep. I'm not sure. The <Chess24.com> side indicates that Carlsen didn't spend more than 1 min/move until 11...b5 when he spent about 2.5 mins. He then spent about 8.5 mins on 15...Re7, likely preparing to move this rook to b7, with or without sacrificing the exchange with 19...Rxa4. He then spent a whooping almost 34 mins on 17...Reb7 so I think that this is where he first contemplated 18...Rb4 and 19...Rxa4. And he only spent a little bit more than a minute on 19...Rxa4, as though he had already made up his mind to do earlier.

So I don't think that 19...Rxa4 was home prep, otherwise he would not have spent almost 34 mins on 17...Reb7. And if this was home prep, I would have thought that he would have looked deeper to see what the most likely result of this exchange sac. After all, I was able to do that and I suspect that with a reported net worth in the range of $ 8M - $ 10M can probably afford a small staff with fairly powerful computer(s) to run his analyses I would think that Carlsen would also be able to do that.

And if this was home prep then I don't think that he would have gone for this line if the analyses had shown that it would have likely resulted in a loss, and with a significant rating differential over Duda and Duda's poor play in the tournament, I doubt that he would have gone for this line if the analyses had shown that it would have likely result in a draw; he probably would have expected better than that.

You then said that your rule of thumb was that if you didn't see tangible compensation by six or so moves out, then it's a sac based mostly on optimism rather than principle, but that you could maybe stretch it to ten or so moves for these guys. OK, but I suspect that different players have different thumbs, and what might be "optimism" for you and I might represent a calculated risk for these guys, particularly Carlsen. So, I think that he came up with this exchange sac OTB and it was not part of his home prep.

You finally said that Carlsen's computer should be punished for optimistically suggesting that 19...Rxa4 was a good way to go. But I doubt that his computer(s) suggested this and, if it had, Carlsen probably would not have thought that a draw against a considerately weaker player that was playing poorly in the tournament would have been a good way to go. Even with the Black pieces.

And, of course, this is all pure conjecture on my part. I don't have any idea what Carlsen's thought process really was when deciding whether to go into the 19...Rxa4 exchange sac nor when it first occurred to him.

I do agree with you that Carlsen didn't see a probable draw somewhere in 10 to 30 moves or more ahead. It was simply too long given that there were not many forced moves in the drawing lines. And his computer(s) would have warned loudly that the best he could do with this approach was a draw; at least mine did and his computers are certainly better than mine. Mine clearly indicated that if he tried to go for more than a draw that a loss was the most likely outcome. But I suspect that a motivated software-savvy chap could interface a voice synthesizer to whatever chess GUI Carlsen or one of the member of his staff was using and then, as you suggested, if the real-time evaluation of the GUI exceeded [2.00] it would be made to exclaim "Danger Magnus Carlsen, Danger! Along with flashing all the room's lights and/or a red light to make it even clearer.

But you didn't even have the courtesy and decency to quote me correctly and in the proper context. I suppose that's what happens when you are a disrespectful, rude, obnoxious, and childish kibitzer who has to resort to misrepresenting what others say in an attempt to justify his silly vendetta against Carlsen. #NotMyFavoriteTroll.>

My dear friend,

1. There is no need for any kind of mental breakdown. Let meltdowns remain the privilege of our great friend, Perfidious.

2. You should inspect things a little bit deeper: in the past several months I already commented 'Ouch! :-)' numerous times when I encountered Justin losses. There was no attempt to (mis)quote you, but if it appeared to be a misquotation, indeed, then I am very sorry. It is a very unfortunate coincidence.

Video analysis of the game:
https://youtu.be/Dp6dMSgBsN4

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Stockfish_20100519_x64_modern:

67/88 128:07:42 +2.51 17. ... Ng4 18.Rhe1 Nxf2 19.Rxe7 Bxe7 20.Re1 Ng4 21.Nxh5 Ne5 22.Kb1 Bd6 23.Ng3 Bd7 24.Bxd7 Nxd7 25.a3 Be5 26.Bc3 g6

67/91 128:07:42 +2.99 17. ... h4 18.Ne4 h3 19.gxh3 Reb7 20.b3 Bxh3 21.Rde1 Bf5 22.Re3 Kf8 23.Rhe1 Re7 24.h4 Ng4 25.R3e2 Ne5 26.Qc3 Bxe4

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Stockfish_20100519_x64_modern:
NNUE evaluation using nn-baeb9ef2d183.nnue enabled