user-1: Having been inspired by an article on ChessBase.com, what is the longest current tablebase mate with a unique optimal line? 
By unique, I mean for each of the winning side's plies, there is only one move that gives a shortest forced mate. and for each of the losing side's plies, there is only one move that gives a longest forced mate.

user10380: Guy Haworth gives an answer to this in his Chess Endgame Records.
[fen "8/8/7p/7n/k7/8/2K5/3R4 w - - 0 1"]

1. Rd4+! {dtc/m/z = -26/-49/-17m} Ka3 2. Kd3! Kb3 3. Rd6! Kb4 4. Kd4! Nf4 5. Rb6+! Ka5 6. Rf6! Ne2+ 7. Kc4! Ng3 8. Rg6! Ne4 9. Kd4! Nd2 10. Rg2! Nf3+ 11. Kc5! Ka6 12. Rg6+! Kb7 13. Kd5! h5 14. Rh6! h4 15. Rf6! Ne1 16. Rf1! Nc2 17. Rb1+! Ka6 18. Rb2! Ne1 19. Ke4! h3 20. Rb1! Nc2 21. Kd3! Na3 22. Ra1! h2 23. Rxa3+! {KRKP: dtc/m/z = -3/-15/-3m} {Nalimov and Lomonosov DTM EGTs} *

Haworth states that this is the longest-known decisive AUMS, Absolutely Unique Move Sequence, known to the author.
This move-sequence arose from White and Black adopting the following strategies. White's strategy is: first, preserve the value, i.e. keep the position won for White; second, maximize DTM. (Why "maximize" and not "minimize"? I don't know, but in the above line White has only one value-preserving option at each move anyway, so it doesn't matter.) Black's strategy is to challenge White to find the unique value-preserving move.
Every move in the above line is optimal for the player's strategy. In the cases of Black's 20th and 22nd moves, there is not one unique option which is optimal for Black's strategy, but for all Black's other moves the move given above is Black's only option which implements Black's strategy of giving White only one option which keeps the position won for White.
Note that the identity and length of the longest decisive AUMS depends on what White's and Black's strategies are. Black's strategy of preferring "restrict White to a unique winning move" seems calculated to enable there to be long AUMSes. If, instead, Black adopts the strategy "maximize DTM", then the game proceeds differently. 18...Ne1 is indeed Black's only option which gives White only one winning move, but it loses in 20 moves. By contrast, 18...Ne3 loses in 32 (i.e. keeps Black alive for 12 moves more); 19. Kc6 then wins in 32, and 19. Kc5 in 34.
Some of the jargon explained:
decisive: ending in checkmate, not in stalemate
AUMS: Absolutely Unique Move Sequence
DTC: depth to conversion (i.e. depth to capture, promotion, or the end of the game)
DTM: depth to mate
DTZ: depth to changing which ply is the earliest at which a 50-move-rule draw may be claimed (i.e. depth to capture, pawn-move, or the end of the game)
value: whether the position is (with best play) won for White, won for Black, or drawn
Upper-case DTC/M/Z refer to white; lower-case dtc/m/z refer to Black.
Depth is measured in plies, where a ply is either a white move or a black move.
Haworth cites Conrady, 2003; van der Heijden #70232, (2010). The relevant entries in his bibliography are:
Conrady H. (2003). Computerschach und Spiele, Vol. 2, No. iv-v
van der Heijden, H. (2010). http://www.hhdbiv.nl/. ENDGAME STUDY DATABASE IV
The longest AUMS I can see in Haworth which ends with the mating move is a 13-ply KNkp mate. Using Nalimov I have extended it back for a further two plies but only by dint of captures.
[fen "8/8/8/1p6/b1R5/8/N7/k1K5 w - - 0 1"]

1.Rxa4! bxa4 2.Nb4! a3 3. Nc2+! Ka2 4. Nd4! Ka1 5. Kc2 Ka2 6. Ne2 Ka1 7. Nc1 a2 8. Nb3#! {Nalimov and Lomonosov DTM EGTs} *


-1: <p>Having been inspired by an article on <a href="http://en.chessbase.com/post/longest-dual-free-direct-mate-problem" rel="nofollow noreferrer">ChessBase.com</a>, what is the longest current tablebase mate with a unique optimal line? </p>

<p>By unique, I mean for each of the winning side's plies, there is only one move that gives a shortest forced mate. and for each of the losing side's plies, there is only one move that gives a longest forced mate.</p>
 10380: <p>Guy Haworth gives an answer to this in his Chess Endgame Records.</p>

<pre><code>[fen "8/8/7p/7n/k7/8/2K5/3R4 w - - 0 1"]

1. Rd4+! {dtc/m/z = -26/-49/-17m} Ka3 2. Kd3! Kb3 3. Rd6! Kb4 4. Kd4! Nf4 5. Rb6+! Ka5 6. Rf6! Ne2+ 7. Kc4! Ng3 8. Rg6! Ne4 9. Kd4! Nd2 10. Rg2! Nf3+ 11. Kc5! Ka6 12. Rg6+! Kb7 13. Kd5! h5 14. Rh6! h4 15. Rf6! Ne1 16. Rf1! Nc2 17. Rb1+! Ka6 18. Rb2! Ne1 19. Ke4! h3 20. Rb1! Nc2 21. Kd3! Na3 22. Ra1! h2 23. Rxa3+! {KRKP: dtc/m/z = -3/-15/-3m} {Nalimov and Lomonosov DTM EGTs} *
</code></pre>

<p>Haworth states that this is the longest-known decisive AUMS, Absolutely Unique Move Sequence, known to the author.</p>

<p>This move-sequence arose from White and Black adopting the following strategies. White's strategy is: first, preserve the value, i.e. keep the position won for White; second, maximize DTM. (Why "maximize" and not "minimize"? I don't know, but in the above line White has only one value-preserving option at each move anyway, so it doesn't matter.) Black's strategy is to challenge White to find the unique value-preserving move.</p>

<p>Every move in the above line is optimal for the player's strategy. In the cases of Black's 20th and 22nd moves, there is not one unique option which is optimal for Black's strategy, but for all Black's other moves the move given above is Black's only option which implements Black's strategy of giving White only one option which keeps the position won for White.</p>

<p>Note that the identity and length of the longest decisive AUMS depends on what White's and Black's strategies are. Black's strategy of preferring "restrict White to a unique winning move" seems calculated to enable there to be long AUMSes. If, instead, Black adopts the strategy "maximize DTM", then the game proceeds differently. 18...Ne1 is indeed Black's only option which gives White only one winning move, but it loses in 20 moves. By contrast, 18...Ne3 loses in 32 (i.e. keeps Black alive for 12 moves more); 19. Kc6 then wins in 32, and 19. Kc5 in 34.</p>

<p>Some of the jargon explained:</p>

<p>decisive: ending in checkmate, not in stalemate</p>

<p>AUMS: Absolutely Unique Move Sequence</p>

<p>DTC: depth to conversion (i.e. depth to capture, promotion, or the end of the game)</p>

<p>DTM: depth to mate</p>

<p>DTZ: depth to changing which ply is the earliest at which a 50-move-rule draw may be claimed (i.e. depth to capture, pawn-move, or the end of the game)</p>

<p>value: whether the position is (with best play) won for White, won for Black, or drawn</p>

<p>Upper-case DTC/M/Z refer to white; lower-case dtc/m/z refer to Black.</p>

<p>Depth is measured in plies, where a ply is either a white move or a black move.</p>

<p>Haworth cites Conrady, 2003; van der Heijden #70232, (2010). The relevant entries in his bibliography are:</p>

<p>Conrady H. (2003). Computerschach und Spiele, Vol. 2, No. iv-v</p>

<p>van der Heijden, H. (2010). <a href="http://www.hhdbiv.nl/" rel="nofollow">http://www.hhdbiv.nl/</a>. ENDGAME STUDY DATABASE IV</p>

<p>The longest AUMS I can see in Haworth which ends with the mating move is a 13-ply KNkp mate. Using Nalimov I have extended it back for a further two plies but only by dint of captures.</p>

<pre><code>[fen "8/8/8/1p6/b1R5/8/N7/k1K5 w - - 0 1"]

1.Rxa4! bxa4 2.Nb4! a3 3. Nc2+! Ka2 4. Nd4! Ka1 5. Kc2 Ka2 6. Ne2 Ka1 7. Nc1 a2 8. Nb3#! {Nalimov and Lomonosov DTM EGTs} *
</code></pre>
