feat(tasks): SessionStart reads the claim — a compaction gets its work back (milestone 381 step 3)

The claim now pays rent to the session that set it. The SessionStart hook
sends the host's `source` and the session id; the server renders a claim
section by source (task_claims.render_claims):

- compact / clear: the work this session had claimed, each with its two
  latest log entries, so a compacted session resumes from the record
  rather than from a count of open tasks.
- startup / clear: other sessions' live claims (may still be running) and
  in-progress tasks whose claim went quiet (abandoned mid-task).
- fork: the same, framed as "two sessions may now hold this".
- resume, or no source sent: nothing.

The task sidebar shows a live claim as "Being worked" and a dead one on
open work as "Went quiet", so the operator sees a session die mid-task.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

frontend/src/types/note.ts

@@ -29,6 +29,10 @@ export interface Note {
   due_date: string | null;
   started_at: string | null;
   completed_at: string | null;
+  // Which session is working this task now (milestone 381). Null when no
+  // session ever claimed it; `live` false when the claim's lease ran out —
+  // a session that went quiet mid-task, which is worth seeing, not hiding.
+  claim?: TaskClaim | null;
   recurrence_rule: Record<string, unknown> | null;
   recurrence_next_spawn_at: string | null;
   is_task: boolean;
@@ -53,3 +57,11 @@ export interface NoteListResponse {
   notes: Note[];
   total: number;
 }
+
+export interface TaskClaim {
+  held_by: number | null;
+  session: string | null;
+  since: string | null;
+  touched: string | null;
+  live: boolean;
+}