[seebs]

There is a huge distinction between identity and other kinds of equality.

Michali, your basic problem is that everything you're doing is devoted to trying to recover your "end states" notion. You're emotionally committed to that, and any time anything proves it wrong, you start attacking it and trying to show that it doesn't exist or doesn't mean anything.

You're wrong. Give it up.

You are not going to find a magical way of eliminating the fact that there really are multiple states over time in Conway's life, and that means that you can't get away from the fact that determinism does not imply reversibility. And that means your entire argument as to why the end state causes the earlier states, rather than vice versa, is wrong.

No matter how cool it would be, no matter what a great short story it would make (I read a lovely time travel story based on it), it isn't actually true. And it won't become true no matter how many times you try to find new ways of asking the question "but doesn't that mean that everything anyone has ever told me about functions or mathematics is wrong and I'm right".