fix(agent): task completion safety net — stop tasks sticking at Running

Implements fixes 1-3 of plans/2026-07-11-task-completion-safety-net.md.
Confirmed live that 50/50 production sessions never reached a terminal
status because the model almost never calls complete_task, even for
trivial single-tool Q&A turns SOUL.md explicitly calls out as needing it.

- Inline safety net (agent.go): a session that never called set_goal never
  framed itself as a structured task, so its first plain-text turn-end IS
  the task ending — auto-complete it there instead of leaving status stuck
  at its creation default forever.
- Idle sweep (continue.go, new completion_nudges column): goal-bearing
  sessions that stall get one nudge, then auto-close with outcome=partial
  if the nudge goes unanswered, mirroring the pattern resumeSession already
  uses for a different stuck-session failure mode.

Fix 4 (backfill of the 50 already-stuck live sessions) is deliberately
separate — deferred until this is deployed and verified live, per the
plan's implementation order.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

cmd/nomos/store.go

@@ -508,6 +508,60 @@ func (s *store) completeTask(ctx context.Context, sessionID, outcome, summary st
 	return nil
 }
 
+// staleGoalSession is a goal-bearing task that's gone idle without reaching
+// a terminal state — the idle-sweep worker's work list (fix 2+3 of
+// plans/2026-07-11-task-completion-safety-net.md).
+type staleGoalSession struct {
+	ID               string
+	Goal             string
+	CompletionNudges int
+}
+
+// staleGoalSessions finds sessions that framed themselves as a real task
+// (goal != '', so the inline safety net in agent.go intentionally left them
+// alone) but have sat non-terminal past idleThreshold. completion_nudges
+// tells the caller whether to nudge (0) or give up and auto-close (>=1) —
+// see processIdleSweep in continue.go.
+func (s *store) staleGoalSessions(ctx context.Context, idleThreshold time.Duration, limit int) []staleGoalSession {
+	if s == nil {
+		return nil
+	}
+	rows, err := s.pool.Query(ctx, `
+		SELECT id, goal, completion_nudges
+		FROM agent_sessions
+		WHERE goal <> ''
+		  AND status IN ('active', 'planning', 'executing')
+		  AND last_active_at < now() - ($1 * interval '1 second')
+		ORDER BY last_active_at
+		LIMIT $2`, idleThreshold.Seconds(), limit)
+	if err != nil {
+		return nil
+	}
+	defer rows.Close()
+	var out []staleGoalSession
+	for rows.Next() {
+		var s staleGoalSession
+		if err := rows.Scan(&s.ID, &s.Goal, &s.CompletionNudges); err == nil {
+			out = append(out, s)
+		}
+	}
+	return out
+}
+
+// bumpCompletionNudge records that the idle sweep nudged a stalled session,
+// stamping last_active_at so it isn't picked up again until it's genuinely
+// idle again (a fresh nudge shouldn't fire every tick while the model is
+// mid-response to the previous one).
+func (s *store) bumpCompletionNudge(ctx context.Context, sessionID string) error {
+	if s == nil {
+		return nil
+	}
+	_, err := s.pool.Exec(ctx, `
+		UPDATE agent_sessions SET completion_nudges = completion_nudges + 1, last_active_at = now()
+		WHERE id = $1`, sessionID)
+	return err
+}
+
 // planStep is a persisted plan step, as returned to the frontend for hydration
 // (the panel otherwise only sees steps live via plan.proposed/plan.step.*).
 type planStep struct {