Errors and retained calibration boundary

For compatible complete-channel errors,

For a common packet embedding and common feasible protocol class, the optimized errors differ by at most $e_{\mathrm{model}}$. Both statements follow from triangle inequalities and CPTP contraction.

For common isometry embeddings,

For changed branch probabilities,

Such stepwise errors telescope. The supplement discards no amplitude by a nonzero threshold. Entropy eigenvalue tolerances and numerical solver tolerances are not trajectory truncations.

The optional benchmark spectral comparison retains its pre-barrier convention:

These external scalar-rate inputs are motivated by Hawking,BlackHawk, not computed here. Transmission is applied once. For weight mismatch $e_x$, transmission mismatch ${\eta }_{jx}$, frequency Lipschitz constant $L_x$, mass derivative bound $K_x$, and mass variation $\Delta \mathcal{M}$, the selected two-rail count error is bounded by

The proof is weight comparison, midpoint integration, and the mass mean-value bound. Omitted target number expectation adds separately. This does not calibrate the repeated model or determine coherent channel error.