Shared-Dot Balance

How it works

1. 1Take a 2x2 square of cells and work out every way the lines around it could possibly go, keeping only the ways where each clue in the square comes out exactly right.
2. 2The square's own middle dot is the key: all four of its edges belong to cells inside the square, so its "0 or 2 lines" rule can be checked here — something a two-cell check can never do, because its middle dots always have a fourth edge poking outside.
3. 3Against the board's edge you get even more: the dots along the border have only three edges (two at a corner), so those are fully inside the square too, and their rule applies as well.
4. 4That's what settles these two 3s on the left border. A plain adjacent-pair check can't finish them — "leave the shared edge blank and give each 3 its other three sides" still survives there. Enclosing the dots rules that out.
5. 5The bottom edge of the left 3 is a line in every surviving combination, so it's forced — and from there the rest of the classic adjacent-3s pattern follows.