NOT.4:5.4 - Reconnect an ordered pair after removing two swaps

A diagram carries an ordered pair of integer values on two wires: port 1 carries a and port 2 carries b. A swap exchanges the two values. Its output is (b, a); a second swap restores (a, b). Replace the two swaps by straight connections preserving port numbers. This argument holds for every integer pair.

The enclosing operation subtracts port 2 from port 1. In compact diagram notation:

(1:a, 2:b) -> swap -> swap -> subtract(1, 2)
(1:a, 2:b) -> straight     -> subtract(1, 2)

Both forms return a - b. The replacement makes the source of each subtraction input directly traceable. If the replacement’s outgoing wires are accidentally crossed, the enclosing operation instead receives (b, a) and returns b - a. Inputs (5, 2) then give -3 instead of 3; inputs (2, 5) give 3 instead of -3. Retaining two ports without retaining their correspondence loses the required use.

The diagram describes pure value operations, and the enclosing operation consumes only the ordered pair. That interpretation justifies this replacement in its context. Physical wire length or signal delay would be additional observations requiring a different preservation argument.