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