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Okay, let me venture into the math underpinnings on this. If you can
mathematically demonstrate there *must* be a first turn advantage for
white, no matter how small, then I will predict in the 1-square game, the
"noise" will wash out the signal effectively totally. And the signal will
emerge from the noise very slowly, as the ranges of the pieces increase
toward the modern. That is apparently what I predicted anyway, since both
our explanations can fit the prediction above.

In Chief/Warlord, if there must be a first-turn advantage, then it *must*
be miniscule, because it doesn't become obvious if white gets 2 first
turns in a row [black passes turn 1.] The major part of the game is
[nearly]mathematically chaotic, as far as I can tell. If there were an ad,
which I doubt seriously, then I think it would be effectively washed away
by the many turns of deterministic but effectively/essentially/[actually?]
chaotic behavior. I don't see how a signal gets through that. Step through
my first Chieftain game with Carlos Cetina, and my last Warlord: a Clash of
Arms game with elkitch, for a look at the range of behaviors the series can
display. 

*To fix the knight move in 1-square, base the parity on the other knight
and the king, then the queen and the king, and finally on the moving knight
and king, with the color of the king's square being the determinant color.
The last condition will freeze the N as a wazir for the rest of the game,
but that's the default state I want. Does that sound better? 
Rule: If other piece and king are on opposite colors, move to king's
color. If they are on same color, move to opposite of king's color.