Lemma: Reference-preserving chain identity

Install $n+1$ equal payload registers on a supplied regular chain. Initialize all except the source in one fixed blank. Successive adjacent SWAPs transport the source factor exactly to the final site and return the previous sites to blank. The identity holds on all payload-reference operators.

Proof. Each SWAP exchanges tensor factors, not merely their marginal states. Inductively the complete unknown factor and all its correlations move one step. Every factor left behind equals the original blank. Matrix-unit linearity proves the referenced identity.