Cycle Completion

A cycle completes in the very moment it could start again.

More precisely:
A non-empty move sequence is called cycle, and its last move is said to complete it, if that move sequence could indeed complete and thereafter be repeated forever wouldn’t rules prevent either. Two moves are considered to be the same if they are performed by the same player and target and clear the same intersections, if any (e.g. ({},{}) passes, ({A1},{}) occupies corner, ({A1},{A2,B1}) captures two, and ({A1},{A1,A2,B1}) commits suicide).^[1][2]

Example

does not complete a cycle, does.

Does sending two and returning one complete a cycle?

7×2, B’s turn  

If Black ataris, White will take two and Black will retake one, repeating the position. However, the cycle isn’t yet complete since for that to be, Black would have to be able to atari again. But he can’t — it’s White’s turn. Only if she now passes, the cycle completes. Before that, it’s nothing but an almost cycle.

The completing move, of course, belongs to the cycle, even if it happens to be a pass. This cycle’s length therefore is 4, not 3. Who denies passes first-class citizenship compares to old Romans denying zero to join their odd number system.

[edit]

Does a single pass complete a cycle?

No, it does not. If he passes, he certainly can’t immediately pass again — it’s her turn. Before he can push down his clock button again, she has to raise it. But couldn’t it be translated to some cycle, just like 1, 2, 3, 4, 5, . . . can be translated to 1, 0, 1, 0, 1, . . . via modulo 2? Well, the only translation imaginable is to ignore the player to move, but then nothing changes. Not really a cycle.

[edit]

Rules can focus on cycles

Note that advocates of positional superko do not depend on a false use of the term cycle. They can elegantly define as follows:

The game ends with completing a cycle. There are three cases:

• completing the trivial cycle ends the game and leads to scoring
• completing any non-trivial cycle with a pass wins the game
• completing any cycle with a non-pass loses the game

Advocates of situational superko only have to delete the immediate-win clause. (Remember, cycles have an even lenght).

However, this reveals what is wrong with those two superko rules: they treat the same cycle depending on where it started differently. How to exploit that is shown in Rules Beast 1. If we wouldn’t care about the start, the flaw disappears:

The game ends with completing a cycle. There are two cases:

• completing a cycle that includes at least one pass ends the game and leads to scoring
• completing a cycle that does not include a pass loses the game

I’m still not satisfied with, say, sound superko, but that’s another story . . .

— Robert Pauli

[1] RP: Note that the number of moves was not explicitly required to be even — it just miraculously turns out to be. :-)

[2] RP: Note that we inaccurately ignore the prisoner difference in case of Japanese rules, possibly stopping a what there would be foolish cycle.