Theorem: Finite-execution stability

Suppose two executions use the same label dictionaries and satisfy

Their terminal channels obey

The same bound holds for their complete geometry-history channels.

For a common input, including an arbitrary reference, the trace distance of the corresponding outputs, and the total-variation distance of their geometric history laws, are at most

Proof. Insert and subtract one implemented step at a time. Each term has the difference of one pair of steps sandwiched between channels. Channels have diamond norm one, and the diamond norm is submultiplicative Watrous2018. The triangle inequality gives (eq:telescoping).

For history retention, a current history is a classical input label. Its next-step column is $D_{g_t}$, with the preceding history copied into the output label. Copying that fixed classical prefix does not change the norm of the column. Proposition prop:diamond-column therefore gives the same one-step bound, now on the history algebra. Apply the same telescoping argument.

A common normalized input gives trace-norm difference at most the channel diamond bound. Trace distance contributes the factor $1/2$. Reading only the geometric record is a channel, so it cannot increase trace distance; on diagonal states that distance is total variation. Distances between states are at most one.