Two-Node Same-Digit AIC Ring
A Two-Node Same-Digit AIC Ring — the four-cell ring traditionally listed as Generalized X-Wing / X-Cycle — closes two single-digit strong nodes with weak inferences verified independently of the path. The candidate propositions and links define the deduction; a count of highlighted cells or a familiar drawing does not.
What is a Two-Node Same-Digit AIC Ring?
A representative ring is (A=B)-(C=D) closed by the two remaining weak inferences between the nodes' sides. Each equals sign is a strong link (XOR) — at least one of the two positions of the digit holds; each dash is a weak inference (NAND) — not both. The closing weak inferences must be verified independently of the path.
Walkthrough example
A walkthrough of one Two-Node Same-Digit AIC Ring: two single-digit strong links (XOR) closed by weak inferences verified independently of the path.
Step 1
Consider these cells.
Step 2
The highlighted candidates form the closed Same-Digit AIC Ring (B5=C5)-(C1=B1). Each equals sign is a strong link (XOR); each dash is a weak inference (NAND). The last link, B1=5 back to B5=5, is a weak inference that holds independently of the path — it closes the loop, and the endpoint theorem supplies its complementary strong inference, so every weak edge of the ring becomes an exact-one pair. The numbered implication trace walks the ring in one direction to display the alternating links; it illustrates the closed structure and is not an extra forcing assumption. 1. If B1 contains the value 5, then B5 cannot contain the value 5 (the value can occur only once in the column) 2. If B5 does not contain the value 5, then C5 must contain the value 5 (only remaining possible position in the row) 3. If C5 contains the value 5, then C1 cannot contain the value 5 (the value can occur only once in the column) 4. If C1 does not contain the value 5, then B1 must contain the value 5 (only remaining possible position in the block)
Step 3
So value 5 can be removed: closure turns every weak edge of the ring into an exact-one pair, so exactly one of its two alternating assignments holds — and each red candidate conflicts with a true ring proposition in both.
A two-node same-digit ring alternates strong links (XOR) with weak inferences (NAND) and closes through weak inferences verified independently of the path. The endpoint theorem saturates the closing pair, so the digit can be removed from any cell outside the ring that sees both sides of a weak edge. You can play sudoku online to practise with the current candidate state visible.
How to spot it
Choose one digit and list two strong nodes — for example, two rows where the digit is confined to the same two columns. Verify both weak connections between their sides, including the closing one; an X-shaped drawing alone proves nothing.
The logic
With the closing weak inferences verified, the endpoint theorem supplies the missing strong inference between the closing pair, so every weak edge around the ring gains a complementary strong inference. Remove the digit from any cell outside the ring that sees both sides of one of those saturated pairs; the ring itself never selects which alternating assignment is the solution.