fix(server): winner_id on completed games + stats idempotency latch

Two issues in the GAME_OVER broadcast path:

1. log_game_end called update_game_completed with winner_id=None default,
   so games_v2.winner_id was NULL on all 17 completed staging rows. The
   denormalized column existed but carried no information. Compute winner
   (lowest total; None on tie) in broadcast_game_state and thread through.

2. _process_stats_safe had no idempotency guard. log_game_end was already
   self-guarding via game_log_id=None after first fire, but nothing
   stopped repeated GAME_OVER broadcasts from re-firing stats and
   double-counting games_played/games_won. Add Room.stats_processed latch;
   reset it in handle_start_game so a re-used room still records.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

server/services/game_logger.py

@@ -143,20 +143,22 @@ class GameLogger:
                 )
             )
 
-    async def log_game_end_async(self, game_id: str) -> None:
+    async def log_game_end_async(
+        self,
+        game_id: str,
+        winner_id: Optional[str] = None,
+    ) -> None:
         """
-        Mark game as ended.
-
-        Args:
-            game_id: Game UUID.
+        Mark game as ended. winner_id is the player who finished with the
+        lowest total — None when tied or when the caller doesn't have it.
         """
         try:
-            await self.event_store.update_game_completed(game_id)
-            log.debug(f"Logged game end: {game_id}")
+            await self.event_store.update_game_completed(game_id, winner_id)
+            log.debug(f"Logged game end: {game_id} winner={winner_id}")
         except Exception as e:
             log.error(f"Failed to log game end: {e}")
 
-    def log_game_end(self, game_id: str) -> None:
+    def log_game_end(self, game_id: str, winner_id: Optional[str] = None) -> None:
         """
         Sync wrapper for log_game_end_async.
 
@@ -166,8 +168,8 @@ class GameLogger:
             return
 
         try:
-            loop = asyncio.get_running_loop()
-            asyncio.create_task(self.log_game_end_async(game_id))
+            asyncio.get_running_loop()
+            asyncio.create_task(self.log_game_end_async(game_id, winner_id))
         except RuntimeError:
             # Not in async context - skip (simulations don't need this)
             pass