refactor: extract GameModeIndicator component from PageWithNav

Extract mode badge display logic into dedicated component to reduce PageWithNav complexity. Includes config for none/single/battle/tournament modes with proper colors and emojis.

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

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

apps/web/src/components/nav/GameModeIndicator.tsx

@@ -0,0 +1,46 @@
+import React from 'react'
+
+type GameMode = 'none' | 'single' | 'battle' | 'tournament'
+
+interface GameModeIndicatorProps {
+  gameMode: GameMode
+  shouldEmphasize: boolean
+  showFullscreenSelection: boolean
+}
+
+const gameModeConfig = {
+  none: { label: 'Select Players', emoji: '👥', color: '#6b7280' },
+  single: { label: 'Solo', emoji: '🎯', color: '#3b82f6' },
+  battle: { label: 'Battle', emoji: '⚔️', color: '#8b5cf6' },
+  tournament: { label: 'Tournament', emoji: '🏆', color: '#f59e0b' }
+}
+
+export function GameModeIndicator({ gameMode, shouldEmphasize, showFullscreenSelection }: GameModeIndicatorProps) {
+  const modeInfo = gameModeConfig[gameMode]
+
+  return (
+    <div style={{
+      display: 'flex',
+      alignItems: 'center',
+      gap: shouldEmphasize ? '10px' : '4px',
+      padding: shouldEmphasize ? '12px 24px' : '4px 8px',
+      background: shouldEmphasize
+        ? `linear-gradient(135deg, ${modeInfo.color}25, ${modeInfo.color}35)`
+        : `${modeInfo.color}20`,
+      border: `${shouldEmphasize ? '3px' : '2px'} solid ${modeInfo.color}${shouldEmphasize ? '70' : '40'}`,
+      borderRadius: shouldEmphasize ? '16px' : '6px',
+      fontSize: shouldEmphasize ? '20px' : '12px',
+      fontWeight: 'bold',
+      color: modeInfo.color,
+      boxShadow: shouldEmphasize ? `0 6px 20px ${modeInfo.color}40, inset 0 1px 0 rgba(255,255,255,0.3)` : 'none',
+      transition: 'all 0.4s cubic-bezier(0.4, 0, 0.2, 1)',
+      transform: shouldEmphasize ? 'scale(1.05)' : 'scale(1)'
+    }}>
+      <span style={{
+        fontSize: shouldEmphasize ? '28px' : '12px',
+        transition: 'font-size 0.4s cubic-bezier(0.4, 0, 0.2, 1)'
+      }}>{modeInfo.emoji}</span>
+      <span>{modeInfo.label}</span>
+    </div>
+  )
+}