Proposition: Public conditioning and aborts

For the outputs in Theorem thm:bit-gap:

1. Conditioning on a measurable public-transcript event with probability $r>0$ preserves the $1/5$ gap for the normalized accepted output.

2. If a completed output includes local abort symbols with joint abort probability $a<1$, and ${\eta }_{\mathrm{ok}}$ is its normalized non-aborting distance from an ideal bit on a common Eve interface, then

   $$(1-a){\eta }_{\mathrm{ok}}+2a\ge \frac{1}{5}.$$

   In particular, vanishing abort probability cannot coexist with vanishing non-aborting key error.

Proof. For (1), restrict the transcript measure to the public event and divide every branch density by $r$. In (eq:transcript), absorb $1/r$ into both Alice effects. The four-effect inequality is homogeneous, so the normalized accepted output has the same certificate.

For (2), replace each party's own abort symbol by zero. This is a local completed bit protocol. Its distance from the chosen ideal bit is at most $(1-a){\eta }_{\mathrm{ok}}+2a$, since the distance between normalized states in unhalved trace norm is at most two. Apply Theorem thm:bit-gap.