This may be poorly explained. The point here is that
Tx is supposed to be always well-defined. So each state has a definite next state (since X is finite, this means it will eventually cycle around).
Since T is well-defined and bijective, each x is T(x′) for exactly one x.
We’re summing over everyx, so each x also appears on the list of Txs (by the previous point), and each Tx also appears on the list of xs (since it’s in X)
E.g. suppose X={x1,x2,x3,x4,…xn} and T(xi)=xi+1 when i≤n, and T(xn)=x1. Then ∑xS(x) is S(x1)+S(x2)+⋯+S(xn−1)+S(xn). But ∑xS(T(x))=S(T(x1))+⋯+S(T(xn))=S(x2)+S(x3)+⋯+S(xn)+S(x1) - these are the same number.
Having implicit closed timelike curves seems highly irregular. In such a setup it is doubtful whether stepping “advances” time.
That explains that the math works out. T gives each state a future but unintuitive part is that future is guaranteed to be among the events. Most regular scenarios are open towards the future ie have future edges where causation can run away from the region. One would expect for each event to have a cause and an effect but the cause of the pastest event to be outside of the region and the effect of the most future event to be outside of the region.
Having CTCs probably will not extend to any “types of dynamics that actually show up in physics”
This may be poorly explained. The point here is that
Tx is supposed to be always well-defined. So each state has a definite next state (since X is finite, this means it will eventually cycle around).
Since T is well-defined and bijective, each x is T(x′) for exactly one x.
We’re summing over every x, so each x also appears on the list of Txs (by the previous point), and each Tx also appears on the list of xs (since it’s in X)
E.g. suppose X={x1,x2,x3,x4,…xn} and T(xi)=xi+1 when i≤n, and T(xn)=x1. Then ∑xS(x) is S(x1)+S(x2)+⋯+S(xn−1)+S(xn). But ∑xS(T(x))=S(T(x1))+⋯+S(T(xn))=S(x2)+S(x3)+⋯+S(xn)+S(x1) - these are the same number.
Having implicit closed timelike curves seems highly irregular. In such a setup it is doubtful whether stepping “advances” time.
That explains that the math works out. T gives each state a future but unintuitive part is that future is guaranteed to be among the events. Most regular scenarios are open towards the future ie have future edges where causation can run away from the region. One would expect for each event to have a cause and an effect but the cause of the pastest event to be outside of the region and the effect of the most future event to be outside of the region.
Having CTCs probably will not extend to any “types of dynamics that actually show up in physics”