The output direction can move away from the target
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Two data/target wires already suffice. A SWAP is implemented by three legal CNOTs. Let \(P\) be that SWAP, and let \(Q\) be NOT on the data wire followed by another SWAP. Then
The whole circuit is a restored constant-one oracle with zero Toffolis. At the cut after the first SWAP,
The target discrepancy is zero on every input, while the data discrepancy is one on every input. Reading only the target at this cut would reject the constant-one function.
| \(x\) | \(A\) | \(B\) | \(H\) | named target discrepancy |
|---|---|---|---|---|
| 0 | \((0,0)\) | \((1,0)\) | \((1,0)\) | 0 |
| 1 | \((0,1)\) | \((1,1)\) | \((1,0)\) | 0 |
This example uses the target as a control during each SWAP. That behavior is expressly allowed by the model. Its relevance is logical: completeness belongs to the full register comparison, not to an arbitrarily chosen wire. The coordinate packets remain perfectly safe even though the detection direction has moved.
Constructive formal counterexample to target-only reconstruction: outputActiveOracle_restores, moved_discrepancy_data, moved_discrepancy_target.