Imagine for a moment there is a system. "System" means something specific here, but for most, the naive definition will work.
OED suggests that a system is "an organized or connected group of things". A solar system would be an organized group of planets connected by gravity to a star, such as our Sun. A central nervous system (CNS) would be an organized group of brain functions connected by neurons.
The system is set up so that any time it is not "locked out" of access to "future sight": it can project forward for some period of time the perfect system state as a function of some free variable.
So, here we are speaking specifically of the CNS, which provides mental functions, such as imagining the likely outcomes (prediction or "future sight") of our possible choices.
For the purposes of simplicity, we will assume the condition that locks something out of future sight is that the particular current range has been the subject of a previous future sight.
I'm guessing that the "current range...subject to previous future sight" would be our current understanding of how things are at the moment. Prior predictions that were incorrect would likely constrain us from making the same prediction. Prior predictions that were correct and useful (self-rewarding) would bias us in favor of those predictions.
All future seeings must* be finite and extend from the moment of initiation forward.
We do usually predict future events. However, we may use what we learned from past events to make future predictions more reliable. So, seeing both the past and the present are key to making accurate predictions (seeing's of the future). So, I'm not sure that you will be able to defend this assertion that suggests we can ignore the past and still make good predictions of the future.
The operation of "future sight" is that for some element of the system treated as "free", all executions of the system upon permutations of the state at the beginning of the initiation, through the variance, and out to the finite endpoint of "future sight" are rendered. Then, the agent may examine all of these results, a large but finite set, pick one, and replace the variable with the calculated result "what it will decide", and then the system will collapse back to the moment of future sight, the agent's decision function on this set determines the value of it's "next behavior", and then it behaves in that way.
It is not clear what "some element of the system treated as 'free'" means, unless you can give us an example of the kind of specific "element" to consider and the constraint that the element is expected to be "free of".
I think you are saying that the decision making process is deterministic. Given the same problem, the same information, and the same person, the decision will be the same. Variations can be introduced through physical, biological, or rational causes. For example, a mosquito is buzzing near my ear and I swat it away, losing my train of thought. Or, I'm hungry and tired, so I make a mental mistake. Or, my information is bogus or my logic is bogus, resulting in a result that is inevitably (but reliably) wrong.
I'm not sure what "the system will collapse back to the moment of future sight" means. Each of the options will be a different future sight. So, we have multiple future sights. After the decision, we follow through on the future sight of the option we chose, and, as you say, it will causally determine our next behavior.
This can be designed in reality as a deterministic mathematical system: the calculation of "future sight" is both perfect, and finite.
I recoil at the notion of a "mathematical system". Neither the solar system nor the central nervous system appears to be a "mathematical" system. We can employ math to describe the quantitative aspects of these systems, but I believe that's about all that math is useful for in systems analysis.
Now it stands that the system will only ever resolve one way: the calculation on the finite set of "possible futures" renders a fixed number as a function of a finite process.
Exactly right. That's what "deterministic" means.
It is deterministic, yet there are literally points that a defined agent may decide for itself what the future will be, on the basis of forward-planning.
In other words, the choosing process is fully deterministic, but it is only performed by intelligent species, like us. There will be reasons for our choice being what it is, but they will be our own reasons, based on the predictions of the outcome of each option.
There also follows from this that in other systems, approximations of this may be executed: by calculating what happens at critical bounds of a decision, one can rule out everything that happens between those bounds.
What "other systems"?
What are the "critical bounds of a decision"? And why would we "rule out everything that happens between those bounds"?
Or, I don't need to know the exact values of a function between two consecutive zeroes to know that all points between them will share the same sign, and if the only determinant of the outcome is sign, I can just say "any of these" and skip a bunch of work.
I don't get how any values between two consecutive zeroes can share the same sign. It would seem that there must be negative signs and positive signs if the intervening values are to produce the second zero.
I can sacrifice precision by saying "most of these get me what I want", and "certain events can block this but they are unlikely" and by accepting error and qualitative outputs, dramatically accelerate the capability until executing it can be done without needing a god power, but it may instead be done between the time the future moment of decision must be made, and the moment that moment is seen looming. It's less perfect and more limited, a pale shadow of this slice of omniscience described by "future sight" but is no less real math operating on the same concept to the best of the ability of the person so operating.
In other words, due to imprecise and incomplete knowledge, our predictions of the outcomes of our choices are only estimates, but we can usually live with that.