feat(db): add backend shutdown hook for graceful runtime shutdown

Add Database::shutdown() with a default no-op implementation for backward compatibility.

Implement libSQL shutdown via flush_replicator(), treating SyncNotSupported as non-fatal.

Implement Postgres shutdown by closing the pool.

src/db/libsql/mod.rs

@@ -326,6 +326,18 @@ impl Database for LibSqlBackend {
         libsql_migrations::run_incremental(&conn).await?;
         Ok(())
     }
+
+    async fn shutdown(&self) -> Result<(), DatabaseError> {
+        match self.db.flush_replicator().await {
+            Ok(Some(frame_no)) => {
+                tracing::debug!("libSQL replicator flushed at frame {}", frame_no);
+                Ok(())
+            }
+            Ok(None) => Ok(()),
+            Err(libsql::Error::SyncNotSupported(_)) => Ok(()),
+            Err(error) => Err(DatabaseError::from(error)),
+        }
+    }
 }
 
 // ==================== Row conversion helpers ====================

src/db/mod.rs

@@ -523,6 +523,13 @@ pub trait Database:
 {
     /// Run schema migrations for this backend.
     async fn run_migrations(&self) -> Result<(), DatabaseError>;
+
+    /// Shutdown hook for backend-specific drain/flush behavior.
+    ///
+    /// Default implementation is a no-op so existing backends remain compatible.
+    async fn shutdown(&self) -> Result<(), DatabaseError> {
+        Ok(())
+    }
 }
 
 #[cfg(test)]

src/db/postgres.rs

@@ -61,6 +61,11 @@ impl Database for PgBackend {
     async fn run_migrations(&self) -> Result<(), DatabaseError> {
         self.store.run_migrations().await
     }
+
+    async fn shutdown(&self) -> Result<(), DatabaseError> {
+        self.store.pool().close();
+        Ok(())
+    }
 }
 
 // ==================== ConversationStore ====================