Publish how many episodes shared each break, and plan the rest

The server has computed and stored Partners since ad detection shipped --
state.json carries it on every break -- but breakRange never put it on the
wire, so the phone's new tap-a-break popup could only ever say "the server
did not say how many other episodes carried this audio". That count is the
one piece of evidence a listener can weigh before judging a break, and the
popup exists to show it.

My brief for the popup asserted the server already sent it. It did not.
The implementing agent checked rather than taking my word, which is why
this is a two-line fix now instead of a wrong "why" line forever.

Also lands the handover plan for the correction channel and the signature
library, written before a context compaction, including the gesture design
Peter specified for the notification button and the jump-size argument he
correctly rejected.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_018AwnALZL6bD4VBtATDVv3B

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