feat(ops): add usage_logs partition status to ops dashboard

Add partition management integration to the smart ops system:
- Backend: Add GetUsageLogsPartitionStatus endpoint in OpsHandler
- Backend: Add partition query methods in OpsRepository
- Backend: Add UsageLogsPartitionStatus type in OpsService
- Frontend: Add OpsPartitionStatusCard component
- Frontend: Add partition status display in OpsDashboard
- i18n: Add Chinese and English translations

The partition status card shows:
- Whether usage_logs is partitioned
- Current row count vs threshold (100K)
- Partition count (if partitioned)
- Warning message when partitioning is recommended

This allows administrators to monitor partition status directly
from the ops dashboard without checking server logs.

backend/internal/repository/ops_repo.go

@@ -1526,3 +1526,78 @@ func opsNullInt16(v *int16) any {
 	}
 	return sql.NullInt64{Int64: int64(*v), Valid: true}
 }
+
+// ==================== Usage Logs Partition Management ====================
+
+// IsUsageLogsPartitioned checks if usage_logs table is partitioned.
+func (r *opsRepository) IsUsageLogsPartitioned(ctx context.Context) (bool, error) {
+	if r == nil || r.db == nil {
+		return false, fmt.Errorf("nil ops repository")
+	}
+
+	var isPartitioned bool
+	err := r.db.QueryRowContext(ctx, `
+		SELECT EXISTS(
+			SELECT 1
+			FROM pg_partitioned_table pt
+			JOIN pg_class c ON c.oid = pt.partrelid
+			WHERE c.relname = 'usage_logs'
+		)
+	`).Scan(&isPartitioned)
+	if err != nil {
+		return false, fmt.Errorf("check usage_logs partitioned: %w", err)
+	}
+	return isPartitioned, nil
+}
+
+// GetUsageLogsRowCount returns the approximate row count of usage_logs table.
+func (r *opsRepository) GetUsageLogsRowCount(ctx context.Context) (int64, error) {
+	if r == nil || r.db == nil {
+		return 0, fmt.Errorf("nil ops repository")
+	}
+
+	var rowCount int64
+	// Use pg_class.reltuples for fast approximate count (avoid slow COUNT(*) on large tables)
+	err := r.db.QueryRowContext(ctx, `
+		SELECT COALESCE(
+			(SELECT reltuples::bigint FROM pg_class WHERE relname = 'usage_logs'),
+			0
+		)
+	`).Scan(&rowCount)
+	if err != nil {
+		// Fallback to actual COUNT if pg_class estimate fails
+		err = r.db.QueryRowContext(ctx, `SELECT COUNT(*) FROM usage_logs`).Scan(&rowCount)
+		if err != nil {
+			return 0, fmt.Errorf("get usage_logs row count: %w", err)
+		}
+	}
+	return rowCount, nil
+}
+
+// GetUsageLogsPartitionCount returns the number of partitions for usage_logs table.
+func (r *opsRepository) GetUsageLogsPartitionCount(ctx context.Context) (int, error) {
+	if r == nil || r.db == nil {
+		return 0, fmt.Errorf("nil ops repository")
+	}
+
+	var count int
+	err := r.db.QueryRowContext(ctx, `
+		SELECT COUNT(*)
+		FROM pg_class c
+		WHERE c.relkind = 'r'
+		AND c.relname LIKE 'usage_logs_%'
+		AND EXISTS (
+			SELECT 1 FROM pg_inherits i
+			WHERE i.inhrelid = c.oid
+			AND EXISTS (
+				SELECT 1 FROM pg_class parent
+				WHERE parent.oid = i.inhparent
+				AND parent.relname = 'usage_logs'
+			)
+		)
+	`).Scan(&count)
+	if err != nil {
+		return 0, fmt.Errorf("get usage_logs partition count: %w", err)
+	}
+	return count, nil
+}