Some initial deductions around the edges already give some regions:
We can break in some more with internal clues:
Some more clever deductions can lead to the next step:
That gives a break-in to the rest of the puzzle:
So the solution to the puzzle is:
Suppose we have a solution, which is a tree graph with vertices of degree 1 and 3 only. This graph has the following property:
This can be proved by induction:
This property has some consequences for the $M\times N$ grids that can be filled with Ts and bulbs.
For square grids $N\times N$ this means that
Here are pictures of general solutions for ...
Let's take one more step, with logic.
Or, look at this picture:
But from here, there's no way to progress by logic, because there are two solutions. (Thanks to @venus in the comments for alerting me to this!)
What an interesting and rewarding puzzle. There is indeed a mistake in the puzzle, but one that is easily rectified. It was a bit tedious doing this on paper, I wish there was an app to integrate something like this together. Anyway, the first step is obviously to start solving the sudoku. I will not post every single step, just the key ones that will allow ...
As the comments say, the smallest number of clues found for a 16x16 sudoku (with 4x4 squares) is 55. It has not been proven that this is the smallest number, but no one's found a better one yet. I suspect that less research into a minimum number has occurred for the larger variants because they are less common and would require more computing power to brute-...