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/PostgresqlGameplayBackend.py

@@ -76,6 +76,7 @@ CREATE TABLE IF NOT EXISTS game_snakes (
   snake_id         TEXT        NOT NULL,
   snake_name       TEXT,
   is_you           BOOLEAN     NOT NULL DEFAULT FALSE,
+  customizations   JSONB       NOT NULL DEFAULT '{}'::jsonb,
   PRIMARY KEY (game_id, snake_id)
 );
 
@@ -114,6 +115,7 @@ ALTER TABLE games ADD COLUMN IF NOT EXISTS quality_tier      TEXT;
 ALTER TABLE games ADD COLUMN IF NOT EXISTS quality_reasons   JSONB;
 ALTER TABLE turns ADD COLUMN IF NOT EXISTS my_thinking       JSONB;
 ALTER TABLE snake_turns ADD COLUMN IF NOT EXISTS latency     TEXT;
+ALTER TABLE game_snakes ADD COLUMN IF NOT EXISTS customizations JSONB NOT NULL DEFAULT '{}'::jsonb;
 """
 
 # Force TOAST compression on the large JSONB columns so that even
@@ -458,14 +460,22 @@ class PostgresqlGameplayBackend(GameplayBackendTemplate):
     async with pool.acquire() as conn:
       async with conn.transaction():
         await conn.executemany("""
-          INSERT INTO game_snakes (game_id, snake_id, snake_name, is_you)
-          VALUES ($1,$2,$3,$4)
+          INSERT INTO game_snakes (
+            game_id, snake_id, snake_name, is_you, customizations
+          ) VALUES ($1,$2,$3,$4,$5)
           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 = EXCLUDED.customizations
           """,
           [
-            (game_id, snake.get("id"), snake.get("name"), snake.get("id") == you.get("id"))
+            (
+              game_id,
+              snake.get("id"),
+              snake.get("name"),
+              snake.get("id") == you.get("id"),
+              snake.get("customizations") or {},
+            )
             for snake in snakes if snake.get("id") is not None
           ],
         )
@@ -739,6 +749,7 @@ class PostgresqlGameplayBackend(GameplayBackendTemplate):
           COALESCE(gs.snake_name, st.snake_name) AS snake_name,
           st.health, st.length, st.head_x, st.head_y, st.body AS body_json,
           COALESCE(gs.is_you, st.is_you) AS is_you,
+          COALESCE(gs.customizations, '{}'::jsonb) AS customizations_json,
           st.inferred_move, st.latency
         FROM snake_turns AS st
         LEFT JOIN game_snakes AS gs

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