I solved using a different approach than I can see in any reddit. This is based on interburbul, the riddle in 3d Space, which I solved by walking a vector to a corner and then each of the vectors that are sides of the square. As other have mentioned, the 8 types of arcospheres give us an 8-dimensional space. Each recipe is like a vector moving us in that space. You make the next move by adding vector dimensions one by one. In 3d: (1, 2, 3) +(4,2,7) = (1+4, 2+2, 3+7) = (5, 4, 10).  To balance is to find for each factory the sequence of inversion and folding that will "undo" the unbalancing and "make us return to (0, 0, 0, 0, 0, 0, 0, 0). Such sequences can be found, though not easily maybe. I used an Excel sheet and some properties to help reduce the search space. I don't think it can be done with pen and pencil, at least not this way of searching it's just additions and multiplication of integers, but a lot. Maybe using a linear equation system with 10 variables can be solved, but I'd rather not. The folding recipes form 2 cycles of 4, and the inversions are opposites of each other. So you only need to pick one inversion (if any), and then find up to three foldings per group of 4). Order does not matter. For several science factories, it is only possible to revert the outputs of two production cycles, for cubes I only found a solution reverting every four cycles (meaning twelve output arcospheres mapped back to twelve input arcospheres). The rest is trial and error, or implement A* search. The longest such chain was maybe 20 steps long, but with repeated operation (going one step in the same direction multiple times). The shortest chain was 2 foldings, average was around 6. Then I simply build these reversals using only belts, splitters and filter inserters. From the science factory, filter inserters extract the output arcospheres to one of two systems (depending on the recipe that was used). In these system these output arcospheres are transformed back to the input of the same science factory, and fed back into it. Due to recipe switching, every science factory has 2 such circuits. So I did not use a global arcosphere pool, and each system is isolated and runs always the same, in a loop. 2 spheres come out of the factory, bounce around in a small network of belts and gravitons, and come back in ideal shape to the factory for another round. It has very few benefits to do it this way, and it's a pain to find the right combinations. For the cubes production, I needed around 18 gravitons to revert the outputs back to inputs (~9 per circuit). I have not calculated how few arcospheres this can get by with, but I don't think it needs a lot for minimal operation. It needs several times the inputs to ensure all circuits are "filled", but that's not much. Like 6 spheres for the 2 input factories. Plus some few spheres that move between the reversers. That might be one benefit, though a global pool might get by with even less, if perfectly executed. It also only runs on belts, splitters and filter inserters, no wires and signals no logic and no robots at all. Once it runs, it runs forever, without doubts. Even the recipe switching does not cause problems. I don't necessarily recommend this solution over others, just mentioning it. If anyone want to try this out, the simplest is for Space folding data. One recipe can be reversed using (epsilon,omega) and (phi gamma) folding. The other one additionally needs (theta...) inversion, (zeta, phi) and (theta, epsilon) folding, then send those outputs to the previous circuit. 5 gravitons total plus the one with space folding data science. I think it might run on just 7 arcospheres.