TUVWXYZ-Ring
A TUVWXYZ-Ring is the double-linked form of the TUVWXYZ-Wing: six almost-locked-set cells and a bivalue pivot where both pivot candidates are restricted commons. The seven cells form a locked set, so any external cell sharing a house with all Ring cells of a given value can be stripped of that value.
What is a TUVWXYZ-Ring?
A TUVWXYZ-Ring extends the TUVWXYZ-Wing from a single restricted common to two. The pattern consists of a bivalue pivot cell and six almost-locked-set cells, where both of the pivot's candidates are restricted commons — each appears in at least two ALS cells that see each other, so both are locked within the set. The six ALS cells collectively hold seven candidates for six cells, and the pivot's two candidates tie them to the pivot from both sides. Because both links close, the seven cells form a locked set containing exactly seven values, and any candidate seeing all Ring cells of its value can be eliminated. The single-linked counterpart is the TUVWXYZ-Wing, with only one restricted common and a single elimination value.
How to spot it
Start from a bivalue cell — the pivot — and look for six cells that all see the pivot and whose combined candidates include both pivot candidates plus at least five extras. Both pivot candidates must be restricted commons: each must appear in at least two ALS cells that see each other, so both are locked within the set. The six cells form an almost-locked set with seven candidates for six cells, and the two restricted commons link them to the pivot from both sides, closing the ring. TUVWXYZ-Rings are among the rarest patterns in Sudoku and appear only in the most diabolical puzzles once every smaller ALS pattern has been exhausted.
The logic
The pivot has two candidates, say X and Y, and both are restricted commons. If the pivot were X, then X would be removed from all six ALS cells (they see the pivot), and the ALS would have only five candidates for six cells — impossible. If the pivot were Y, the same contradiction arises. So the pivot must be one of X or Y, and in either case the seven cells form a locked set containing exactly seven values. Any external cell that shares a house with all Ring cells holding a particular value cannot contain that value, because exactly one Ring cell must hold it. Both pivot candidates produce eliminations, unlike the single-linked Wing where only one does.