Four+ Node X-Chain
A Four+ Node X-Chain is an open same-digit AIC with four or more strong-inference nodes — the general member of the X-Chain family beyond the two- and three-node forms. Write its alternating signature as (A=B)-(C=D)-...-(Y=Z): equals signs are strong inferences and dashes are weak inferences. The chain proves the inclusive endpoint relation A OR Z without assuming either endpoint first.
What is an X-Chain?
Every node concerns one digit and is a strong link (XOR) joining exhaustive alternatives: if one side is false, the other is true. Every edge is a weak inference (NAND): its connected sides cannot both be true. The chain may use singleton or properly justified grouped sides. Historical forcing-chain and Nice Loop descriptions are alternate frameworks, not the AIC proof taught here.
Walkthrough example
A walkthrough of one X-Chain written as an alternating sequence of strong and weak inferences that proves an inclusive OR between its endpoints.
Step 1
Consider these cells.
Step 2
The highlighted candidates form the open X-Chain (I6=I1)-(F1=F3)-(C3=C8)-(G8=H9). Each equals sign is a strong link (XOR); each dash is a weak inference (NAND). The endpoint theorem proves I6=1 OR H9=1.
Step 3
This is the Type 1 same-digit elimination: H4 sees both endpoints I6=1 and H9=1, and the chain proves I6=1 OR H9=1 for 1.
An open same-digit AIC alternates strong links (XOR) with weak inferences (NAND) and proves an inclusive OR between its endpoints without a trial assumption. When the endpoints are distinct, that relation justifies the Type 1 same-digit common-peer elimination; a degenerate endpoint OR can instead prove the endpoint itself. You can play sudoku online to practise with the current candidate state visible.
How to spot it
For one digit, list the strong-inference nodes and connect consecutive nodes only through independently valid weak inferences. Verify grouped edges against every represented candidate. The two unconnected outer sides are the endpoints; a target must conflict with every candidate represented by both.
The logic
If the first endpoint A is false, each strong link propagates false to true and each weak inference propagates true to false until the far endpoint Z is true. Therefore A OR Z holds, possibly with both endpoints true. A same-digit candidate conflicting with both endpoints can be removed; when the endpoints are distinct, this is the Type 1 same-digit elimination. An open chain does not select one unique internal assignment.