feat(snake): optimize duels and persist customizations

- Add deadline-aware iterative duel search with bitboards and tuple bodies.

- Reuse transposition bounds and move-order hints across search depths.

- Persist snake colors, heads, and tails in SQLite and PostgreSQL.

- Restore customization metadata when hydrating dashboard replays.

- Cover duel deadlines, cache reuse, schema storage, and replay output.

server/database/backend/SqliteGameplayBackend.py

@@ -112,6 +112,7 @@ class SqliteGameplayBackend(GameplayBackendTemplate):
           snake_id TEXT NOT NULL,
           snake_name TEXT,
           is_you INTEGER NOT NULL DEFAULT 0,
+          customizations_json TEXT NOT NULL DEFAULT '{}',
           PRIMARY KEY (game_id, snake_id),
           FOREIGN KEY (game_id) REFERENCES games(game_id) ON DELETE CASCADE
         );
@@ -140,6 +141,7 @@ class SqliteGameplayBackend(GameplayBackendTemplate):
       self._ensure_column_exists(connection, "games", "your_snake_version", "TEXT")
       self._ensure_column_exists(connection, "games", "game_type", "TEXT")
       self._ensure_column_exists(connection, "snake_turns", "latency", "TEXT")
+      self._ensure_column_exists(connection, "game_snakes", "customizations_json", "TEXT NOT NULL DEFAULT '{}'")
       self._ensure_column_exists(connection, "games", "winner_name", "TEXT")
       self._ensure_column_exists(connection, "games", "has_replay", "INTEGER NOT NULL DEFAULT 1")
       self._ensure_column_exists(connection, "games", "quality_status", "TEXT NOT NULL DEFAULT 'retained'")
@@ -266,14 +268,22 @@ class SqliteGameplayBackend(GameplayBackendTemplate):
 
     with self._connect() as connection:
       connection.executemany("""
-        INSERT INTO game_snakes (game_id, snake_id, snake_name, is_you)
-        VALUES (?, ?, ?, ?)
+        INSERT INTO game_snakes (
+          game_id, snake_id, snake_name, is_you, customizations_json
+        ) VALUES (?, ?, ?, ?, ?)
         ON CONFLICT(game_id, snake_id) DO UPDATE SET
           snake_name = excluded.snake_name,
-          is_you = excluded.is_you
+          is_you = excluded.is_you,
+          customizations_json = excluded.customizations_json
         """,
         [
-          (game_id, snake.get("id"), snake.get("name"), 1 if snake.get("id") == you.get("id") else 0)
+          (
+            game_id,
+            snake.get("id"),
+            snake.get("name"),
+            1 if snake.get("id") == you.get("id") else 0,
+            self._to_json(snake.get("customizations") or {}),
+          )
           for snake in snakes if snake.get("id") is not None
         ],
       )
@@ -545,6 +555,7 @@ class SqliteGameplayBackend(GameplayBackendTemplate):
           COALESCE(gs.snake_name, st.snake_name) AS snake_name,
           st.health, st.length, st.head_x, st.head_y, st.body_json,
           COALESCE(gs.is_you, st.is_you) AS is_you,
+          COALESCE(gs.customizations_json, '{}') AS customizations_json,
           st.inferred_move, st.latency
         FROM snake_turns AS st
         LEFT JOIN game_snakes AS gs