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Na--,Nd--: I only =appear= to control the square
Compiled by notyetagm

OBLIGATION RESTRICTS MOBILITY so that a defender against a threat cannot leave the defensive complex of squares from which it meets that threat. Due to the threat restricting the mobility of the defender, the defender only =APPEARS= to control squares outside of its defensive complex. Hence when computing the true value of Na and Nd for a square outside of the defensive complex, the defending piece =cannot= be counted there (Na--, Nd--).

Consider the two Topalov White games in the collection.

In the first game against Kasimdzhanov, after Black's brilliant 41 ... ♘e2-f4! the White d6-pawn must meet the threat of ... ♖d8x♖d1# by <BLOCKING> the d-file. Since <BLOCKING> a line is a full-time job, the White d6-pawn cannot also <CONTROL/ATTACK> the e7-square/Black e7-queen. So the White d6-pawn only <APPEARS> to <CONTROL/ATTACK> the Black e7-square/e7-queen. In fact, the White d6-pawn takes the Black queen only under penalty of mate and so it does not <ACTUALLY> threaten the Black queen (Na--).

In the second game against Nisipeanu, after 43 ♕d3? both the White e3-bishop =and= White e2-knight must <DEFEND> the f4-forking square to meet the threat of a double <KNIGHT FORK> on that square. So since =both= of these pieces are tied to the defense of the f4-square, neither of them actually <DEFENDS> the d4-square! Therefore the d4-square is not one attacker (Black c3-bishop) versus two defenders (White e3-bishop and e2-knight), it is actually one attacker versus <zero(!)> defenders (Nd--, Nd-- twice!) since both of the alleged "defenders of the d4-square" are occupied keeping the light-squared Black knights out of f4. Hence the d4-square is not <ACTUALLY> defended at all; the White e3-bishop and e2-knight only <APPEAR> to <DEFEND> the d4-square. This <OVERWORKED> nature of the White e3-bishop and e2-knight allows the stunning winning shot 43 ... ♗c3-d4!!.

41 - Ne2-f4! Black e7-queen taboo, White d6-pawn BLOCKS d-file
Topalov vs Kasimdzhanov, 2007 
(E06) Catalan, Closed, 5.Nf3, 70 moves, 1-0

43 - Bd4!! White e3-bishop, e2-knight =both= tied to f4-square
Topalov vs Nisipeanu, 2007 
(B01) Scandinavian, 47 moves, 0-1

Black g6-bishop does not attack White d3-knight, BLOCKS g-file
I Nei vs Petrosian, 1960 
(A56) Benoni Defense, 33 moves, 1-0

32 ... g7xf6 Black f5-rook does =not= DEFEND f6, BLOCKS b1-h7
Van Wely vs Judit Polgar, 2007 
(E17) Queen's Indian, 40 moves, 0-1

4 games

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