Theorem: Referenced two-pulse profile stability

At any fixed selected packet interface, with the same independently prepared background,

The same bound compares the chosen complete complements when their full environmental identification is retained.

This is a source-input channel estimate. It is not unrestricted bosonic diamond continuity of Weyl unitaries.

Proof. Align the scalar Weyl phases in each comparison. For the first controlled factor on a product input $|\psi {⟩}_{RS}\otimes \Omega$, the spectral inequality

gives a uniform vector bound.

Telescope the two endpoint factors, pricing the second-factor difference on the ideal first-pulse output. Because the second source projectors are orthogonal, its squared norm is bounded by the squared norm of $X({\delta }_2)$ on that joint output. The first-pulse conjugation identity is

Thus its norm on the initial vector is at most $r_2+|s(a_1,{\delta }_2)|$. This argument is uniform in the source-reference vector.

Trace distance of pure output vectors is bounded by their vector-norm difference after phase alignment. Partial traces contract it, proving both channel comparisons. Finally, $|s(a_1,{\delta }_2)|\le 2\Vert a_1{\Vert }_0{r}_2$.