LogFAQs > #962427610

Ok, the trick/break-in is this:

Because the grid is toroidal, there is no reason to think of R5C5 as "the center". There is nothing different between R5C5 and R2C2 or any other of the centers, their only difference is their proximity to the border, and because it's toroidal the whole grid loops and there is no border.

Thus there exists toroidal phistomaphel rings around all 9 centers. Just like the regular phistomaphel rings, this eliminates extra cells that could otherwise be that color. Here are all the cells that could be red:
https://gamefaqs.gamespot.com/a/user_image/5/4/0/AACZqoAAC5b0.png
:
Here are all the cells that could be yellow:
https://gamefaqs.gamespot.com/a/user_image/5/4/1/AACZqoAAC5b1.png

You simply go through every color and label everywhere it can be and you are left with this grid:
https://gamefaqs.gamespot.com/a/user_image/5/4/2/AACZqoAAC5b2.jpg

Then you use the 6x6 killer cages to start eliminating colors, there are no clever tricks needed in this part, just fairly standard sudoku deduction. Using the knights move and the cages you can resolve the entire grid such that every cell is reduced to a single color.

At this point you can finally get to work with the inequalities. and with the 8 inequalities involving 9 colors you can solve the entire puzzle.

I might leave this part up for a day or two before showing those last 2 easy steps in case people want to try now that they know the break-in.

PS: The break-in can also be done without SET logic if you just fill out the grid normally and then notice the squid patterns on the cells that have 6 colors.

PPS: For people not familiar with these terms, none of these are things the solver couldn't figure out on their first solve, though knowing certain solving techniques does help.