Add break time feature to schedule blocks

- Admin: per-block "Break Time" checkbox + duration (min) setting; new
  Break Activities section (global list, same pattern as Morning Routine)
- Dashboard: break timer section appears on blocks with break enabled;
  Start/Pause/Resume/Reset controls work independently of the main timer
- TV: left column switches to amber break badge + countdown during break;
  center column shows configurable Break Activities list
- Backend: break_time_enabled/break_time_minutes columns on schedule_blocks
  (auto-migrated on startup); break_activity_items table + CRUD router;
  break timer events (break_start/pause/resume/reset) stored as TimerEvents
  and broadcast via WebSocket; break_activities included in dashboard
  snapshot and session_update broadcast

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

backend/app/utils/timer.py

@@ -42,3 +42,36 @@ async def compute_block_elapsed(
 
     is_paused = bool(tick_events) and tick_events[-1].event_type == "pause"
     return int(elapsed), is_paused
+
+
+async def compute_break_elapsed(
+    db: AsyncSession, session_id: int, block_id: int
+) -> tuple[int, bool]:
+    """Return (break_elapsed_seconds, is_break_paused) for a block's break timer."""
+    tick_result = await db.execute(
+        select(TimerEvent)
+        .where(
+            TimerEvent.session_id == session_id,
+            TimerEvent.block_id == block_id,
+            TimerEvent.event_type.in_(["break_start", "break_resume", "break_pause", "break_reset"]),
+        )
+        .order_by(TimerEvent.occurred_at)
+    )
+    tick_events = tick_result.scalars().all()
+
+    elapsed = 0.0
+    last_start = None
+    for e in tick_events:
+        if e.event_type == "break_reset":
+            elapsed = 0.0
+            last_start = e.occurred_at
+        elif e.event_type in ("break_start", "break_resume"):
+            last_start = e.occurred_at
+        elif e.event_type == "break_pause" and last_start:
+            elapsed += (e.occurred_at - last_start).total_seconds()
+            last_start = None
+    if last_start:
+        elapsed += (datetime.utcnow() - last_start).total_seconds()
+
+    is_paused = bool(tick_events) and tick_events[-1].event_type == "break_pause"
+    return int(elapsed), is_paused