Why is the Crowned Knight worth so much?

@HGMuller okay, I understand why the Centaur is worth that much, but I still have one question:

why the non-royal King is worth nearly the same as a knight when it can’t even fork a the King and other piece without being protected, and can’t even attack a Queen without being protected and most of the things it can are defensive?

the Knight is a really good offensive piece:

(the King on d4 is not royal)

Well, like the Knight it attacks 8 squares. The weaknesses you mention are dependent on what material the opponent has. Although this has some effect, it is usually pretty minor. (Except for the mentioned devaluation of strong pieces facing weaker ones.) If the opponent had an Archbishop and a Chancellor, a Knight could not fork those without being protected, but a Commoner could.

The main value of the moves is what they can individually do. For things like forking, which depend on the relative positioning of the moves, you need a special pattern of opponent pieces, which will occur only very rarely. An exception could be the ability to attack two adjacent squares on the same file: this allows you to attack a Pawn and the square it could move to. You could also consider that a fork, but the pattern it attacks will be very common, because it just depends on the presence of the Pawn, as the square it can flee to will automatically accompany it, and Pawns are abundantly present. This is why I think there is significantly extra value in attacking orthogonally adjacent squares.

Things a Commoner can do that a Knight cannot, are checkmating a bare King, and destroying an isolated passer if it can reach it. The Knight has better chance to stop the passer, because it is faster, but it will then be bound to stop it. So if on one wing the Kings are bound to the Pawns there, a passer on the other wing will draw against the Knight. If you had a Commoner instead of the Knight it would gobble up that passer, and then come to the other wing to break the stand-off there.

So this is roughly balanced. a1 = centaur.

GM Yasser Sierawan has said it is worth about 4.5 pawns.

It is strong enough to stalemate a king, can checkmate with any other piece

King can fork rooks and bishop and knights and take opposition

Think about forks

The Crowned Knight is also known as Centaur.

https://www.pokemon.com/us/pokedex/horsea

Chess Terminology: Castling

Its strong enough to stalemate a king by itself

Wait isnt knight worth 3.25?

Pieces have debuffs based on their weaknesses. The most common example is bishop's worth of 3 or 3.25 after the debuff of being stuck to 1 color, most players know of the bishop pair being worth 0.5 pawns by itself, which is an intuitive way of quantifying it. A basic synergy of complementary moves is usually worth about a pawn. The queen is worth between 9 and 9.75, depending on position and stage of the game, more than the rook and the bishop, which can heuristically be represented as 5+3.25+1+0.5=9.75, as it has the bishop pair built in.

Both knight and man (nonroyal king), have a speed debuff, the man moreso, and both have an ideal square they can reach by investing tempo. Tempo is generally worth a 1/5th to 1/3rd of a pawn, so they have a variable debuff based on how much extra tempo is required to reach their ideal squares, compared to ranged pieces. However, they are both defensive powerhouses, which can make them more valuable than ranged pieces based on position being closed.

Taking a shot at approximating synergy value gain without running complex mathematical calculations or extensive simulations: 3.25+3.25+1=7.5. As you notice they still have the speed debuff as they're both short range pieces. Chancellor(R+N): 5+3.25+1+0.25=9.5 removing knight's less severe speed debuff. If a king's worse speed debuff is similarly removed with a completely complementary nightrider for example, it would be worth 5+3.25+1+0.5 = 9.75. Archbishop(B+N): 3.25+3.25+1+0.5+0.25 = 8.25 which accounts for removing both the bishop's color debuff, the knight's speed debuff, and synergy, but I guess as someone said in practice it's stronger due to orthogonal concentration of moves, and only about 0.75 or 1 pawn weaker than a queen. I'm guessing the centaur has a similar effect, but I doubt it can be more valuable than 8 unless you're in a closed position.

I also wonder if there's a significant debuff (0.25 pawns or more) for being able to be attacked by unprotected pawns and it impeding centralization and costing tempo. Queen, Archbishop, bishop, and centaur would not have this debuff, but chancellor, rook, and knight do. Also being able to be attacked by the king without attacking him would debuff every piece besides queen and centaur.

Centaur 7 - 7.5 range. While Archbishop 7.75-8. There is a clear difference.

How does archbishop+pawn beat a queen then, according to HGMuller's simulations?

To be honest. I don't know. I haven't conducted tests with Arch vs Queen. I did however try removing archbishop for two bishops in Seirawan and Capablanca chess and it's not certain the Arch prevail each time. A few games ended in draw.

I extensively tested Archbishop versus Queen, Chancellor, Rook + Knight, possibly with Pawn odds on one side to revert the advantage, as replacement in an opening position. The result was that A+P vs Q on the average beats Q. (Score ~54%, where pure Pawn odds gave 65%.) And that C+P beats Q by about 58%, while Q beats C by about 58% with equal Pawns. A also beat R+N (I forgot by how much), both in an opening position as in end-games with just those pieces and (4-6) equal Pawns. C vs A as Q replacements in the FIDE opening position was a slight advantage for C (also ~54%).

This led to the conclusion that Q-C = 0.5 Pawn, and C-A = 0.25 Pawn.

Have you thought about retaking the test with Fairy Stockfish to see if your conclusions are the same ?