This seems to repeat the confusion in the original post. In what sense are "you" sampling different times?
There is a difference between the standard "timeless physics" of the 4D block universe, and the specific kind of "timeless physics" advocated by Julian Barbour, which Eliezer was referencing in that post. Your explanation was a good reply within the context of the 4D block universe. And there, there is no issue of "sampling".
However, under Barbour's account, what exists is bigger than a single 4D block universe. What exists is a configuration space (called "Platonia") in which each point represents a possible state for a 3D spatial universe. Conversely, every such possible state is represented by a point in Platonia. In addition, this configuration space supports a static complex scalar field (something like a stationary state solution to the Schrödinger equation in quantum mechanics). Using the Born rule, this scalar field can be interpreted as a fixed probability distribution assigning high probability to some regions in Platonia and low probability to others. Barbour does appeal to this probability distribution to explain why we never "find ourselves" in highly "improbable" configurations of the macroscopic universe.
For example, Platonia contains a point (i.e., a 3D universe) containing people just like us, except that they are looking up into the sky and seeing two suns where their memories say that there was only one sun moments before. That is, they are witnessing what appears to be a flagrant violation of the laws of physics. Barbour's explanation for why we never have this experience is that such configurations get practically no probability mass from the complex scalar field.
So, there is a sort of sampling going on in Barbour's account. He would admit, I think, that this "sampling" is just as mysterious as the Born rule is in the usual many-worlds interpretation of quantum mechanics.
Related to: lesswrong.com/lw/qp/timeless_physics/
Why do I find myself at this point in
time, configuration space, rather than another point? In other words, why do I have certain expectations rather than others?I don't expect the U.S. presidential elections to have happened but to happen next, where "to happen" and "to have happened" internally marks the sequential order of steps indexed by consecutive timestamps. But why do I find myself to have that particular expectation rather than any other, what is it that does privilege this point?
My question is why I find myself to remember that the particle went left and then right rather than left but not yet right?
Yes, but why does my version experience this point of my branch and not any other point of my branch?
I understand that if this universe was a giant simulation and that if it was to halt and then resume, after some indexical measure of causal steps used by those outside of it, then I wouldn't notice it. Therefore if you remove the notion of an outside world there ceases to be any measure of how many causal steps it took until I continued my relational measure of progression.
But that's not my question. Assume for a moment that my consciousness experience is not a causal continuum but a discrete sequence of causal steps from 1, 2, 3, ... to N where N marks this point. Why do I find myself at N rather than 10 or N+1?