Review of the solution
The first and foremost task would of course be to review the solution carefully and (preferably) with scientific accuracy. Since the "established" science (as we all happen to know) does not really like to deal with such topics (which, to be honest, after the "theories" of Däniken and consorts is a bit understandable), at first perhaps we, the Internet community should inspect and examine the solution, and then (if it has proven to be true or at least justifiable) popularize it in a way that at some point the academic world would finally be compelled to take it as a hypothesis seriously and analyze it thoroughly.
Theoretical framework
In order to review and (potentially) confirm the hypothesis, apparently a new theoretical framework needs to be created first, since the encoding method seems not to be covered by the existing theories. The new framework should enable us to compute exactly what probability it has to come to a state like that in the control phase from an arbitrary initial layout, and it would be great, if we also had an algorithm to construct similar encoding procedures. Obviously, given a coordinate range and an item set, calculating the total number of possible layouts is a trivial task, but determining which layouts should be considered to be well-arranged is not self-evident at all (at least to me it was not), and certainly requires a more advanced mathematical approach.
To illustrate the requirements and the difficulties with regard to a theoretical framework, let's try to compute the probability for a very basic scenario. Let's have a coordinate range from (−25, −25) to (25, 25) and two 3×3-sized symbols: a Sun placed at (−11, −20) and a star symbol placed anywhere else arbitrarily (but not colliding with the Sun). The question is: how probable is it to have such a position for the star symbol that after performing the rotational and reflectional operations the two symbols and their reflections all will be placed on the edges of a square?
The vertices of the square that should contain the elements on its edges are (determined by the location of the Sun symbol): (−20, −20), (−20, 20), (20, 20) and (20, −20). Obviously, the star symbol must already be placed on the square before the rotations, as these operations do not move the elements out of the square they were positioned on initially (which square is defined by the larger one of the absolute coordinate values). There are 160 different possibilities to locate a star symbol (or any other item) on the edges of a 41×41 square (39 on each side + 4 corners), but we must eliminate those cases, where the two symbols (or any of their copies) collide with each other. By a single Sun there are 7 invalid positions (e.g. from (−14, −20) to (−8, −20) by the original one), and so 8 × 7 = 56 altogether (since the Sun symbols are far from the corners, there are no overlaps we should consider). Subtracting this from 160 gives 104 possible positions for the star symbol, and if we accept only those initial arrangements, where the two symbols are located on a different edge, i.e. the star symbol cannot be placed on the edge from (−20, −20) to (20, −20), then the number of the valid positions decreases further by 27 to 77.
Computing the total number of the possible placements for the star symbol is much easier. It can be placed (almost) anywhere inside the 49×49-sized square from (−24, −24) to (24, 24), which gives 49 × 49 = 2401 different locations (naturally, we cannot place the symbol directly on the edges because of its size). But we must eliminate those cases, where it would be too close to the Sun symbol, which has a 7×7-sized "fence" around it, resulting in 2401 − 7 × 7 = 2352 possibilities altogether. So, we can finally calculate the probability for this (very simple) scenario: 104 / 2352 ≈ 4.42%, and if the star symbol cannot be located initially on the same line as the Sun, then this probability decreases further to: 77 / 2352 ≈ 3.27%.
Of course, for a full-fledged probability computation the triplets must also be taken into account (thus reducing further the calculated probability significantly), the Sun's position should be modifiable, and many many other enhancements are needed, which would require a more advanced mathematical apparatus that I personally can come up with. The best would be to have a definite probability value at the end of the calculations, which could be a clear indication about the randomness or intentionality of the original design, concerning the apparently well-arranged state of the control phase.
We should recognize that this analysis and the framework could be completely separated from the (much more fantastical) hypothesis about the interstellar travel and the interpretation of the final figure (as it was already mentioned here). This part of the solution is purely mathematical, and a well-established theory for it could have the potential of (dis)proving its correctness with an almost absolute certainty. Because of the problem's complexity it is surely not an easy task to construct such a theoretical framework, but it is definitely feasible, and its existence could benefit (at least in my opinion) the science or even the whole mankind.
It should also be noticed that they are two completely different topics, whether the solution (or its geometric part) itself is provable and comprehensible (which is a code breaking issue), and whether the ancient Mayan (or any other) culture could indeed have the knowledge required for constructing a coding system of this kind (which is a historical question). Unfortunately, until we have an almost hundred percent certain proof in our hand that some ancient culture did really possess the necessary intelligence, the historical considerations will always be strong enough to discredit the efforts to crack such supposed coding systems. Therefore, the best strategy could be to separate these two approaches entirely, and to examine the solution (or primarily its geometrical part) at first exclusively on its own, as if it were a present day code breaking problem, and only after we have already analyzed thoroughly the procedure on this assumption, should we turn our attention to the historical question and to the issues around it.
As to the encoding algorithm, as a first step it could be sufficient to have one that would be able to construct a normalized initial layout from a state similar to that in the control phase. Obviously, it would be much more difficult to build this initial layout from the very final state, as in step 11, because in this case we would have to define operations and procedures for recognizing similarities between shapes, which task could easily turn out to be too heuristic for the computational technology of today. But, of course, it would be great to have such an algorithm, and we could even use it or one of its modified versions for encoding and decoding messages, which can be represented by shapes and layout configurations in some analogous way.
Astronomical examinations
If the solution proves to be true, it could then be advisable to examine the nearby solar systems even more carefully, and search for such binary systems whose distance from the Sun is around 13 light years, and have three inner and three outer planets (as depicted on the supposed diagram of the other system). Ideally, only one system would satisfy all the conditions, and then it could be an obvious candidate to search for extra-terrestrial life.