XY-Chain
An XY-Chain is an open AIC whose strong-inference nodes are bivalue cells, of any length. Write its alternating signature as (a=b)-(b=c)-...-(y=z): each equals sign is the exactly-one relation inside a cell and each dash is a weak inference between equal digits in peer cells. The endpoint theorem, not a trial assumption, supplies the conclusion.
What is an XY-Chain?
In a bivalue cell (a=b), the strong link (XOR) makes exactly one candidate true. Consecutive nodes connect only when the outgoing occurrence of a digit and the same digit in the next cell see each other, giving a weak inference (NAND): they cannot both be true. The digit changes inside a node and stays the same across a weak inference. Merely sharing a digit label without peer visibility does not connect the chain.
Walkthrough example
A walkthrough of one XY-Chain written as bivalue strong-inference nodes joined by same-digit weak inferences, proving an inclusive OR between its endpoints.
Step 1
Consider these cells.
Step 2
The highlighted candidates form the open XY-Chain (D4:1=2)-(D7:2=9)-(D1:9=4)-(C1:4=3)-(C5:3=1). Each equals sign is a strong link (XOR); each dash is a weak inference (NAND). The endpoint theorem proves D4=1 OR C5=1.
Step 3
These are Type 1 same-digit eliminations: A4 and D5 see both endpoints D4=1 and C5=1, and the chain proves D4=1 OR C5=1 for 1.
An XY-Chain alternates bivalue-cell strong links (XOR) with same-digit weak inferences (NAND) and proves an inclusive OR between its endpoints without a trial assumption. That endpoint relation justifies the common-peer elimination. This is the Type 1 same-digit elimination. You can play sudoku online to practise with the current candidate state visible.
How to spot it
Trace bivalue cells so that each consecutive pair shares one digit and the two occurrences see each other. Record the cell nodes with equals signs and the intercell weak inferences with dashes. The exposed endpoint candidates may carry different digits and the endpoint theorem still holds; the common-peer elimination taught here is the same-digit case, where a target sees both endpoint cells and loses that digit.
The logic
If the first endpoint is false, its bivalue strong link makes the other candidate true. The next weak inference makes the matching candidate in the peer cell false, and the alternation continues until the far endpoint is true. Thus at least one endpoint contains the elimination digit, possibly both, so any candidate seeing both endpoints can be removed. This is the Type 1 same-digit elimination and an AIC endpoint proof, not a forcing-chain contradiction.