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Hey, HG, can I ask a favor if/when you edit your earlier remark? I'd
appreciate it if you add the edit either after or before the comment you
are correcting, and leave both. Don't just change the original, so we have
context for the following remarks when people go through this topic in the
future. [I'd like to see that as a general policy.]

Now, as for centralization and mobility, I see what you are saying - there
are differences between the two, although in chess, they correlate
strongly. Are there situations where they don't? Let's build up to this
slowly. I'm interested in establishing some [very] general principles.

First, while a wazir on e4 is much better positioned than one on b2, look
at the wazir on b4. Is it better-positioned than the one on e2? Both
require 2 turns to get to e4, which we will agree, for now, is the 'best'
square. However, the piece on b4 is already 2 steps closer to the enemy,
and thus much better positioned, no? 

What's the mean free path of a bishop? Stipulate that it changes during
the game, starting at zero in the setup and approaching the length of the
side/shortest dimension of the board at game's end. During 'mid-game',
the path length would be roughly 3-4 squares, no? On a small board, ie:
8x8, that still means that the bishop, for greatest effect, must be in the
center. Suppose the board is bigger? On our 12x16, there's a 6x10 area
where, on an empty board, a bishop on any square can travel at least 3
squares in any of its 4 allowed directions. 

Next, suppose no piece in the game moves more than 3 squares. Now what
have you got? A very large game, lots of pieces, all with short range moves
- how does this affect the equation? How about having more than 1 goal
target? Suppose there are 2 kings per side, not all that near each other.
Or more kings. Make it multi-move. 

On an infinite chessboard, there is no center. - Unless you define it by
goal positions, which change as kings move. I'm sure I'm looking at this
far too broadly for most people's taste, but who gets anywhere new by
following the crowd?