refactor: use centralized player ownership in session-manager

Replace inline ownership building with buildPlayerOwnershipMap() utility.

Changes:
- Import buildPlayerOwnershipMap and PlayerOwnershipMap types
- Replace manual DB query + Object.fromEntries with utility call
- Simplify code: 7 lines → 1 line
- Use shared type for consistency

Part of player ownership centralization plan (Phase 2/7).

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

apps/web/src/lib/arcade/session-manager.ts

@@ -5,6 +5,7 @@
 
 import { eq } from 'drizzle-orm'
 import { db, schema } from '@/db'
+import { buildPlayerOwnershipMap, type PlayerOwnershipMap } from './player-ownership'
 import { type GameMove, type GameName, getValidator } from './validation'
 
 export interface CreateSessionOptions {
@@ -215,7 +216,7 @@ export async function applyGameMove(
   })
 
   // Fetch player ownership for authorization checks (room-based games)
-  let playerOwnership: Record<string, string> | undefined
+  let playerOwnership: PlayerOwnershipMap | undefined
   let internalUserId: string | undefined
   if (session.roomId) {
     try {
@@ -229,13 +230,8 @@ export async function applyGameMove(
         }
       }
 
-      const players = await db.query.players.findMany({
-        columns: {
-          id: true,
-          userId: true,
-        },
-      })
-      playerOwnership = Object.fromEntries(players.map((p) => [p.id, p.userId]))
+      // Use centralized ownership utility
+      playerOwnership = await buildPlayerOwnershipMap(session.roomId)
       console.log('[SessionManager] Player ownership map:', playerOwnership)
       console.log('[SessionManager] Internal userId for authorization:', internalUserId)
     } catch (error) {