Corollary: When a gap survives a continuation

If a continuation leaves the previously reported task, success/failure, and outcome record unchanged, the final joint output distance is at least the corresponding gap in (eq:dictionaryTV). The identity report transport below preserves it.

If the record is discarded or overwritten, no such general lower bound follows.

Proof. The retained record is a marginal of the final output, so contraction bounds its distance by the final joint distance. For the negative statement, a replacement channel erases every input dependence.