Unpossible
Your function is impossible:
Starting from an initial garden of Eden configuration at time t = 0 every configuration of the universe at a future point in time can be calculated by a function applied to the configuration of the previous point in time.
The reason is that "previous" is an infinitely long time away once you introduce a loop. Therefore, there is no "Garden of Eden" configuration from which you can bootstrap your configuration.
Counter Example
Marty meets Doc Brown in 1985, and then jumps back to 1955. The sequence up to 1985 is causal and deterministic. However, as soon as Marty jumps back, you now have a problem. Future Marty is a new cause of events between 1955 and 1985 that you already computed. Now, let us suppose that one of Marty's hobbies in 1985 is solving the Rubik's Cube. On his jump back to 1955, he happens to have a Rubik's Cube in his pocket. Suppose further that Dr. Rubik happened to be visiting Hill Valley for a conference, and walked past Marty sitting on a park bench, working on his cube. He pauses and watches Marty for a few minutes, then returns to Hungary, working out the mathematics to convince himself that a solution was possible. Since he is an architect, not a mathematician, he is unable to prove that the cube is solvable, and gives up on it for 25 years. Then, in 1980, while replacing a light bulb in his toilet, he slips and falls and bumps his head. At this point, the solution to the puzzle appears to him like a dream, and he visualizes the mechanism of the cube in a flash, which he quickly draws and subsequently licenses to toy manufacturers. What we didn't say is why Marty jumped back to 1955. When Doc Brown was showing Marty the flux capacitor, Marty was solving the cube idly without looking at it (Marty was able to solve the cube blindfolded). Doc wasn't ready to use the DeLorean yet, but Marty's hand slips and he almost drops the cube. As he reaches to catch it, he bumps a switch which activates the flux capacitor. Doc realizes that his precious plutonium will be consumed if he doesn't go now, so he tells Marty to drive the car up to 88 mph, at which point they jump to the past.
Analysis
Now, the version of determinism you describe can be modeled as a function, $F(C_t) = C_{t+1}$. That is, the function $F$ applied to the configuration $C$ at time $t$ produces the configuration at time $t+1$. In particular, the proximate cause of an event in configuration $C_t$ must exist in $C_{t-1}$, and the ultimate cause in $C_s$, for $s<t$. Furthermore, you have defined $C_0$ as the "Garden of Eden".
We know that there are configurations in which the Rubik's Cube exists. Therefore, it is natural to ask: "Which state contains the ultimate cause of the Rubik's Cube?" We know that Dr. Rubik invented it, and so we might say that the incident on the toilet is the ultimate cause. But we know he was thinking about it for 25 years, and he actually saw it before he invented it. So we could say that the cause was when he passed by Marty. But Marty only had a cube because Dr. Rubik had invented it, so we could say that the ultimate cause was when Marty jumped back in time. But Marty had the cube before he jumped back, so we should say that the invention of the cube was the cause of the invention of the cube.
Let us call the moment of invention in the bathroom $C_i$. The point at which Marty jumps back to the past is $C_j$. The moment when Dr. Rubik is exposed to the idea of a cube by Marty is $C_e$. We know that $i<j$ because Marty only jumps back to the past because he bobbled his cube. And we know that $e<i$ because Dr. Rubik did not invent the cube until he had already seen it. And we know that $j<e$ because Dr. Rubik didn't see the cube until Marty had jumped to the past.
If history is well-formed, then for any configuration $C_x$, there should be a finite number of applications of $F$ starting from $C_0$ which arrives at $C_x$. Or, going backwards, we should be able to find the cause of any event by searching backwards through the timeline until we find $C_c$ such that $F(F(F(...(F(C_c))...))) = C_x$. However, we cannot find $C_i$ above, the moment of invention, because we are stuck with $i<j<e<i$. The configuration $C_i$ must occur before itself, and thus be its own cause!
Conclusion
And herein lies the rub: time loops allow self-caused objects to exist. Since time travel itself violates conservation of energy, there is no limit to what objects may self-cause. This theme is well-explored in the German Netflix drama Dark. A wise, powerful dragon may suddenly appear because it whispers in the ear of a bioengineer on how to craft the DNA of the dragon and implant it into an existing creature, which causes the dragon to be born, after which time the dragon uses time travel go to back into the past and cause its own existence.
In particular, time travel breaks your function $F$. When $F$ arrives at $C_e$, the earliest point in the time loop, Future Marty appears out of nowhere. There is nothing the set of configurations $C_{<e}$ which explain Future Marty. Only the future configuration $C_j$ can explain his sudden appearance, but that is many configurations subsequent to $C_e$. And therefore, you have broken determinism at this point. At any moment, $F$ may as well introduce a dragon or a unicorn or raining fish teleported from another dimension. Time travel makes a mockery of determinism. Laplace's Demon becomes infinitely powerful, and also capricious.
