K. White's Win against 62...Kc2.
68. ... Qc8!?
It is a paradox of this ending that Black willingly hangs his pawn
with check: 69. Qxd5+? is a computer-proven draw. Now White's King is
contained for the moment: 69. Kh7 Qc2! seems so good as to be an easy
draw for Black, 69. Kf7 simply meets ...Qc7+ if not ...Qd7+, and 69.
Kh6 Qh8+ 70. Kg5 Qe5+ 71. Qf5 Qe7+ is not great progress for White.
Thus it is time for White to re-position his Queen with checks. Three
target squares seem particularly inviting: e2, b4, and b6; the last
of these seems the one to fear.
It is worth inserting Black's defenses to the first two, since they also show rich themes: 69. Qe2+ Ka1/b1 70. Qd1+ Ka2/b2 71. Qd2+ Kb1 72. Qb4+ is met by 72...Ka2! and on 73. Kf6 Qc6+ 74. Kf7 Qd7+ 75. Qe7 Qf5+ we transpose into positions White could have had long ago, in which Black is able to block White's King from going to the Q-side. The other line leads to pretty play if Black moves our King to b1: 69. Qe2+ Kb1!? 70. Kf6(!)(?) Qf8+ 71. Kg5 Qd8+ 72. Kf5! Qf8+ 73. Kg4 Qc8+ 74. Kg3 Qc7+ 75. Kf3! Here 75. Kg2 allows ...d4! since Black has a forking check at c6, but now that check will be met by the cross-check Qe4+ trading Queens. Of all Black's tries, the best seems to be the immediate 75...Qg7!, when 76. Qd3+ Kb2 77. Qf5 d4(!) 78. Qf7 Qh6! 79. g7 Qh3+ is a pattern the World Team found to be good in other lines with White's Queen at f7---Black can deny White a useful interposition, and 80. Ke4 Qe3+ 81. Kf5 Qf3+ 82. Kg6 Qg3+ 83. Kh6 Qe3+! 84. Kh7 allows Black both the defense 84...Qe4+ 85. Qg6 Qh4+ 86. Qh6 Qe4+ 87. Kh8 Qe5 that the misguided 58...Qe4? originally sought to achieve, and the probably better 84...Qh3+! 85. Kg8 d3! and Black seems to have caught up in the pawn race. Nor does 79. Kg4? d3! seem to work as Black is queening with check. Moreover, 69. Qe2+ Ka1 looks possible on account of 70. Qe5+ d4!, and this is a case where Black draws even if the pawn falls on d4 with check, and Black also appears able to avoid the dreaded "Pin Battery" idea (White running with his King so as to play Kxd4 and a discovered check) because the Queen on e5 is not guarding g6. But enough of second-best moves for White...
69. Qf2+! Ka1 70. Qa7+! Kb1 71. Qb6+! Ka2
Now the ONLY difference from White's option of playing 59. Qb6+ right
back at the beginning is that Black's Queen is on c8 not f5. Also
note that 70. Qd4+ Kb1 71. Kh6 Qe6! makes White have to start all
over again. But here White's King can emerge effectively. As in the
first alternate line above, Black avoided 71...Ka1 in order not to
get crosschecked on a1 or drop d4 with check, but finds the King on
a2 sensitive again. Moving to the c-file seems to lose quickly as in
our present game---see below.
72. Kf7! Qf5+!? 73. Ke7 _ Qe5+
Now 74. Qe6!? Qc7+ leads to a desperate battle of "Tablebase Chess"
against the Pin Battery that I think Black survives: 75. Ke8 Qb8+ 76.
Kd7 Qb7+ 77. Kd6 Qb8+! 78. Kc5! Qf8+! 79. Kc6! Qa8+! 80. Kb5 Qb8+! In
all cases Black has played only-moves to avoid EGTB losses---e.g. it
is curious that 79...Qg7 80. Kxd5! wins while it didn't win the
previous move with Black's Queen on f8! (Someone please double-check
all this...) Now 81. Qb6 seems not so clear: 81...Qe8+ 82. Ka5 Qe1+
83. Qb4 Qe5! looks like nothing special for White, and ditto 83. Ka6
Qe2+ 84. Ka7 Qe7+ 85. Qb7 Qe3+! 86. Ka8 Qe8+ 87. Qb8 Qc6+ etc. White
may have a way of winning in this line, so it certainly needs
attention too, but the following seems a more precise and incisive
means of reaching the desired goal, if the final position in my
mainline is winning for White. Note that 74. Kd7?! gives Black an
extra option by 74...Qg7+! 75. Kc8 d4!? 76. Qe6+ Kb2 (White can
attempt to force this anyway on 76...Ka1 77. Qe1+ Kb2 78. Qf2+ Ka1
79. Qf1+ and then 80. Qf7; for 76...Ka3!? here see more below) 77.
Qf7 Qh8+! 78. Kb7 Qh1+ and it doesn't look clear to me: 79. Kb6 Qh6!?
or 80. Ka7 Qg1!? or ...d3!? 81. Qf2+ Kb3! are all obscure. (My
MacChess 5.0.1 has examined 41 million nodes after 78...Qh1+ and is
currently allowing Black to queen with check in a 4Q position, so
don't ask me for any computer evals.:-) White "must" be winning
somewhere here, but if not, then White is probably not winning in
lines C or D either.
Since 74. Qe6!? Qc7+ 75. Kf8?! Qb8+ 76. Kf7 Qc7+ 76. Qe7 Qf4+ 77. Kg8
Qb8+ 78. Qf8 Qe5 enables White to force the above-marked critical
position, Black cannot be better off here with 68...Qc8 than with
68...Qe5+(!), but the resulting positions are still relevant to
White's general strategy of running Q-side with his King.
[Addendum 11/8: After posting a query about 74. Qe6 on both BBS-es today I have found a new idea for White: not 81. Qb6 but 81. Ka5!? with the funny idea of Qa6 coaxing Black into ...Qg7 and then a discovered-check combination to force a trade of Queens! The main ideas are shown in these two lines so far: 81. Ka5 Qa7+ 82. Qa6 Qg7 83. Kb4+! (+/-) Kb2 84. Qe2+ Kc1 85. Qe3+ Kc2 86. Qb3+ Kd2 (all forced to avoid Q-trade), but 87. Qxd5+ Kc1 88. Qd6!! is EGTB+-. (A general theme that I first noticed in my "Chess Poetry" post on 52...Kc1 is that White Qd6 can be a death-grip if White's King is active and Black's Queen is not, even though White's pawn is still on g6, if Black's King is not tucked all the way into the corner.) Or 81. Ka5 Qa7+ 82. Qa6 Qc5+ 83. Ka4 (hey, Black almost gave mate:-) Qd4+ 84. Kb5+ Kb1/b2 85. Qb6! looks close to winning. A White win here is not bad news for Black, as Black can force "Position G" earlier, but it is good news for the artistic tension of ideas in this endgame---it would be horrible if Black can hold like this!]
74. Kd8(!) Qh8+ 75. Kc7 Qg7+ 76. Kb8 Qh8+ 77. Ka7 Qg7+ 78. Ka6 d4!?
Notice alas that 76...d4 was impossible yet again owing to Black's
King being on a2 on account of 77. Qa7+! Black has other checking
options in these moves as well, ones I haven't had time to explore
yet---hopefully they could produce an e-file hold like in the 74.
Qe6!? variation given above. But the position now on the board seems
highly thematic, and my final question is whether White's master plan
of 79. Qe6+ and an eventual Qf7 is winning here. I see:
(a) 79. Qe6+ Kb2 (or ...Ka1 80. Qe1+ Ka2/b2 81. Qe2+ Ka1/b1 82. Qf1+ Kb2; other King placements may be worth mention here) 80. Qf7 Qh6!? 81. Qh7!? Qg5(!)---is this winning after 82. g7?! or 82. Qb7+!? ---?
(b) 79. Qe6+ Ka3!? 80. Qf7 Qh6 81. Qh7?! Qe3! and maybe better 81...Qg5! 82. Qh3+ Kb2 both deny White useful checks.
(c) In both (a) and (b), White can make a King
move after 80...Qh6, and things get hairy. For instance, 79. Qe6+ Kb2
80. Qf7 Qh6 81. Ka7 d3 is desperate stuff, staring 4Q losses in the
eye, but 82. g7?! Qe3+! 83. Ka6 d2 84. g8=Q Qd3+! seems to be one of
those weird 1Q-vs.-2Q perpetual check or trade-of-Queens draws. And
Black may be even better off in the analogous 79. Qe6+ Ka3 line with
80. Qf7 Qh6 81. Ka7 Qe3! and suddenly White does not have so many
useful checks. Anyone see a quick kill for
White hereabouts---or simply a winning line?