You might be interested in a simple proof I found of why c/2 and c/3 are speed limits for orthogonal and diagonal spaceships respectively.
Definition: In a gameplay of life, an "infinite lifeline" is a sequence of pairs (c_i,n_i) such that each c_i is alive in generation n_i and either c_(i+1)=c_i or c_(i+1) is adjacent to c_i.
Lemma ("Two Forbidden Directions"): Let x,y be any two 'forbidden' directions from among N,S,E,W,NE,NW,SE,SW. In any gameplay of life that starts finite and doesn't die out, there is an infinite lifeline that never goes in either direction x or y.
The lemma's proof uses biology. Say that (c,n) is a "father" of (c',n+1) if c' is the cell adjacent to c in direction x or y. Otherwise, (c,d) is a "mother" of (c',n+1). By the rules of the game of life it's easy to show every living (c,n+1) has at least one living father and at least one living mother. It follows (modulo some more details) that since the gameplay doesn't die out, there must be an infinite lifeline where each cell is a mother of the next, i.e., an infinite lifeline that never goes direction x or y.
Proof of c/2 orthogonal speed limit: If a spaceship went faster than c/2, say, northward, by the lemma, it would have an infinite lifeline that never goes N or NE. The only way it could ever go northward would be to go NW. Every NW step would have to be balanced out by an eastward step (of which NE is forbidden) or the spaceship would drift west. So every northward step requires a non-northward step, QED.
Proof of c/3 speed limit for diagonal: A diagonal spaceship faster than c/3, say, northeastward, would have an infinite lifeline that never goes N or NE. The only way for it to go northward would be to go NW. Each NW step would need at least two eastward steps in order for the ship to go eastward, QED.
I'm no life expert, but this proof doesn't work for me. Just because there exists some configuration that doesn't generate lifelines in your chosen directions, doesn't mean that any particular configuration doesn't generate lifelines in those directions.
The lemma says that in any configuration which starts finite and never dies out, there DOES exist at least one lifeline which DOESN'T go in a forbidden direction. (There are almost certainly lots of other lifelines which do go in forbidden directions, that's fine.)
Okay, I read that and now I get it. Each live cell has at least three ancestors. Two or fewer of these ancestors come from a forbidden direction, so at least one comes from an allowed direction. Among all of the ancestors from an allowed direction, arbitrarily choose one to be the “mother”. All of the other ancestors are called “fathers”. There's no requirement that a father has to come from a forbidden direction.
You might be interested in a simple proof I found of why c/2 and c/3 are speed limits for orthogonal and diagonal spaceships respectively.
Definition: In a gameplay of life, an "infinite lifeline" is a sequence of pairs (c_i,n_i) such that each c_i is alive in generation n_i and either c_(i+1)=c_i or c_(i+1) is adjacent to c_i.
Lemma ("Two Forbidden Directions"): Let x,y be any two 'forbidden' directions from among N,S,E,W,NE,NW,SE,SW. In any gameplay of life that starts finite and doesn't die out, there is an infinite lifeline that never goes in either direction x or y.
The lemma's proof uses biology. Say that (c,n) is a "father" of (c',n+1) if c' is the cell adjacent to c in direction x or y. Otherwise, (c,d) is a "mother" of (c',n+1). By the rules of the game of life it's easy to show every living (c,n+1) has at least one living father and at least one living mother. It follows (modulo some more details) that since the gameplay doesn't die out, there must be an infinite lifeline where each cell is a mother of the next, i.e., an infinite lifeline that never goes direction x or y.
Proof of c/2 orthogonal speed limit: If a spaceship went faster than c/2, say, northward, by the lemma, it would have an infinite lifeline that never goes N or NE. The only way it could ever go northward would be to go NW. Every NW step would have to be balanced out by an eastward step (of which NE is forbidden) or the spaceship would drift west. So every northward step requires a non-northward step, QED.
Proof of c/3 speed limit for diagonal: A diagonal spaceship faster than c/3, say, northeastward, would have an infinite lifeline that never goes N or NE. The only way for it to go northward would be to go NW. Each NW step would need at least two eastward steps in order for the ship to go eastward, QED.