Yin-Yang
Color every cell black or white. All cells of each color form one orthogonally connected group, and no 2×2 square is a single color. Click a cell to cycle black → white → empty; right-click erases. Two global rules force one local shape: the frontier between the colors — drawn live in purple — is always a single unbroken curve. Every solved board is literally a yin-yang symbol.
The name is a theorem
Read the frontier — the grid edges separating black from white — as a curve on the lattice points. A monochrome 2×2 is an interior lattice point the curve misses; a checkered 2×2 is a point where it crosses itself; a second curve would cut the board into three monochrome regions, two of which share a color they cannot reach. So in every legal board the frontier is a single non-crossing curve through every interior lattice point — a Hamiltonian path of the interior grid, run out to the border at both ends. Measured over 1,500 generated boards and every valid board up to 4×4: one curve, zero exceptions.
Rules that leak
Only one of the window rule's two halves is stated in the rules of the puzzle. The monochrome ban is definitional; the checkered ban is connectivity's local shadow, and the border-arc rule is its planar shadow — two colors cannot alternate around the rim, because the four connecting paths would have to cross. The solver's cheap rules are all borrowed consequences of the one expensive global rule.
Landlocked boards have a parity
The curve may also close into a loop — then one color never touches the border and the board is a pure yin-yang symbol. Exhaustively: 3×3 has exactly 2 such boards (a lone stone in the center), 4×4 and 6×6 have none, and odd sizes have explicit constructions. On even boards the corners pin the curve open.