@Evil If you are available, In essence, I found out my problem is like completing partial vertices on a n^2xn^2 board.

Color coded vertices for readability

I wrote you answer. Yes, these are equivalent. If you do not aim at faster sudoku solving by pattern tagging, this concludes the proof.

Wow, all this time I should've known that this is the vertex cover problem in General Sudoku. I knew, intutively that this has to be equally as hard. I should've thought of that before posting.

@TravisWells during tagging in unsolved instance you have a graph with possible solutions. If there are vertices with degree 3 or greater it is similar to complete colouring

Well, at least I found a way of Sudoku <p Vertex Cover

Or, like you said, vertex cover of some kind

Yes, pretty cool. The lack of formal knowledge (and I'm working on it, because I found a cool free online discrete math course) is what hurt my sharing of ideas

For future: state what you want to solve/prove, give clear definitions (step by step algorithm to solve subproblems is crucial), then what you have tried.

Solving Vertex Cover = Solving General Sudoku. That would be the result of something more formal

I'm gonna edit a few things to leat people know that its a vertex cover on n^2 x n^2 boards

No, you do not have a proof or reduction

Make proper reduction first

Otherwise it will not lift the lock and you will still write intuition instead of formal proof

proof of contradiction can be just as proof as reduction. May seem dirty, and it can be converted into predicate logic(this would be hard for me)

The strange reason for the contradiction proof was because I didn't realize that it was a vertex cover problem

Otherwise, i would have aimed for reduction