Theorem: Local probability comparison for G outputs

For every two-pulse source output, the local probe satisfies

where

The transfer and noise quantities are explicitly specified in Appendix app:local. For fixed source and target they can meet every positive mathematical probability tolerance by first enlarging the shell and then reducing $ϵ$, with the stated preparation costs retained.

For the two source $X$-eigenstates, the local gap is at least

Proof. The source-output moment hypothesis is Theorem thm:moments. Remove the induced attenuation at cost $(1-c_{ϵ})/2$, then compare its coefficient with $l$ using the packet and transfer errors and the preceding Weyl estimate. The gap follows by the reverse triangle inequality. The finite-parameter schedule is proved in the appendix.