Can it be proved that knight or wizard(a piece in a variant) can not lose tempo?

If we talk just about knight, the equation/statement would relate to fact that it travels only to opposite colour. But in a variant, there is a piece called wizard.

http://www.chessvariants.org/piececlopedia.dir/wizard.html Pieceopedia Link to Wizard

Can it be proved by a mathematical equation or statement that it can not lose tempo or by experiment that it can?

Equation for Knight(as a Sample):-

Let knight move to square X from Y in z moves.

If colX=colY, z=2m(even)  [col=colour]

If colX is not equal to colY, z=2m+1(odd)

Since colX=colX, if it comes back from any route in b moves, b=2n=even

Therefore, it is proved that knight can not lose tempo.

So, can it be found that wizard may or may not lose tempo? BTW, it's a piece in Omega Chess.

Theories that you can use:-

Any, if it is true, though I'll myself confirm that if it's true or false. Should there be theories in chess? There can be many, but should there be? I mean mathematical theories which can be reasoned.