I am not the creator of this mod, so the guide below follows from my own experience: I've tried my best to be accurate, but there's a good chance that some steps will be lacking in detail or inaccurate in future versions of this mod.

After all 4 "Vortical Fairy" titans appear in the galaxy (I think you might be able to make them appear faster by attacking the existing titans, though this might be my imagination), the titans will turn hostile and spawn a small escort of smaller ancient ships. While you can destroy the smaller ships, whenever the titan is about to be destroyed, it will escape to another randomly selected star system.

After fighting the titans a few times, you eventually receive a situation which grants you a technology that allows you to build "Measuring Ships". (These take a while to build and have no weapons, so it might be a good idea to build some, or even many, spares.) You will need to force each of the 4 titans to perform its special escape move while at least 1 Measuring Ship is in the same system. (You can mix some Measuring Ships into your fleets to achieve this.) This will allow the Measuring Ship to measure the Vortical Fairy's "eigenvalues", which will then appear in the Situation Log. After you have measured the eigenvalues of each titan, you can move onto the final step, most frustrating step: calculating the correct settings to enter into the Ancient Gateway Control Panel in order to take control of the Ancient Gateway.

When you click on the Ancient Gateway Control Panel's settings, it will offer you two options: to "enter an access code," or to "test symbols". If you click on either, you will notice that these menus, as well as the Ancient Gateways and Vortical Fairy eigenvalue data, all use the same alphabet of 4 distinct symbols. Namely, the "test symbols" menu will allow you to "add" two symbols (one on the left and one on the right), to produce a third new symbol. Keep these rules in mind:

- There is a "universal symbol".
- Adding a symbol to the universal symbol results in that symbol.
- Any symbol added to itself is the universal symbol. (you can use this to easily figure out which of the 4 symbols is universal).
- Adding any two different non-universal symbols results in the remaining symbol (there are 4 symbols, 1 is universal, so if you add two non-universal symbols there's only one symbol left over that it could be).

Each titan has a color you get interpreting their color value via RGB, and adding the different values corresponding to a titan's color across all 4 titans using the rules above yields that titan's character. (Note: the order that you add the symbols in DOES NOT MATTER. Only that you add all 4 symbols for that Titan's color!)

The symbol that is the universal symbol differs from save file to save file, but you can identify it using the second or third rule above: add two identical symbols together, or find the symbol that always produces the other symbol it is added with.

Once you know which symbol is the universal symbol, you're almost done. Each Vortical Fairy has 5 row vectors of length 3 (5 groups of 3 rows). You may have noticed in the menu to enter the code that clicking on the symbols on the left of each row changes their color. Each Vortical Fairy has been assigned a color that you can figure out by rounding their first row vector and then interpreting the row as an RGB value. This means (1, 0, 0) is red, (0, 1, 0) is green, (1, 1, 0) is yellow, and (0, 0, 1) is blue. Each Vortical Fairy should be a different color.

After figuring out the colors of each Vortical Fairy, we can calculate the final value for each titan after "superposition". Each Titan has 4 remaining row vectors, each of which has a corresponding symbol to the left of it. Using the above RGB interpretation method, you can assign each row a color - and each row should have a different color. Then, for each Titan, you want to add the characters corresponding to that Titan's color from all the Titans together. For example, if the first Titan is blue, and the characters from Titans 1 through 4 corresponding to blue is a, b, u, a respectively, we can find the character of the first titan by simplifying a + b + u + a. This becomes c + u + a = c + a = b. Hence the symbol of the first Vortical Fairy is b in this example. Repeat this process for each Titan and its color and then submit the color and character of each Titan to the Ancient Gateway Control Panel.

(TLDR: Each titan has a color you get interpreting their color value via RGB, and adding the different values corresponding to a titan's color across all 4 titans using the rules above yields that titan's character.)

I got really frustrated at this quest chain and also noticed that other people were as well (in these discussion boards.) Hopefully this guide helps some more people enjoy the content this mod has to offer! If this thread gets enough interest, I'll consider making a video tutorial with diagrams (which probably would clarify these instructions a lot...)

======== APPENDIX ========:
What follows is the rough sketch of a proof positing that the 4 symbols and the operation defined over them in this mod form an Abelian group. This means that you can add the symbols in this mod in any order and get the same result!

Without loss of generality, let us call these symbols a, b, c and u. To dip briefly into abstract algebra, we can think of these 4 symbols as an alphabet over which we define an addition operation with the following properties:

Without loss of generality, let u be the universal symbol.
- Any symbol added to itself is the universal symbol.
- Any symbol added to u (the universal symbol) is itself. This means u is the additive identity.
- Adding any two symbols out of {a, b, c} will result in the remaining symbol. For example, a + b = c, a + c = b, c + b = a.
- Addition is commutative. This follows from the rules above and is left as an exercise to the reader.
- Addition is associative. You can prove this via brute force of all cases, and is left as an exercise to the reader.

If people are interested in the full proof I can typeset it in Latex or a formal proof management system, just lmk.

========== EDITS ==========:
(12/4) Edited "addition" steps for simplicity/clarity.