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Am I understanding correctly that the score of 0 at the top means that the chess engine thinks their positions are exactly equally good?

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When it's close to 0, it means the engine thinks the position is equal. But usually when it's exactly 0, the machine thinks the best move to play next must lead to a draw. As, in the next best move will lead to a worse position for the player making the move.

Huh, never thought of it that way but that makes sense

Exactly 0.00 can also mean that the engine has no idea what's going on.

Err...sorry no. At least not for Stockfish (maybe for AlphaZero or LeelaZero).

0 means that the position is exactly equal (material + positional)

The engine always has an idea of what's going on through simple material calculation + positional heuristics (although it maybe be flawed)

Yes

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