P12: persistent live session across navigation

- Player room connection lifted out of the /play route to an app-level hook
  (usePlayerConnection in RootLayout), keyed by a persisted joinIntent — so a
  player stays connected while browsing their sheet, the compendium, anywhere,
  and a reload reconnects automatically.
- Header "● Live: CODE · Leave" badge (PlayerSessionBadge); leaving is explicit
  (leaveRoom clears intent + disconnects). /play renders the table from the store.
- Realtime e2e: player navigates away, badge persists, returns to a synced table.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

e2e-realtime/realtime.spec.ts

@@ -38,6 +38,14 @@ test('GM hosts a session; a player on another device sees live combat', async ({
   // enemy exact HP must NOT be shown to players
   await expect(player.getByText('7/7')).toHaveCount(0);
 
+  // Player navigates away — the live connection persists (badge stays) and
+  // returning still shows the synced table.
+  await player.getByLabel('Primary').getByRole('link', { name: 'Dice' }).click();
+  await expect(player.getByTestId('player-live')).toBeVisible();
+  await expect(player.getByTestId('player-live')).toContainText(code);
+  await player.getByTestId('player-live').getByRole('link').click();
+  await expect(player.getByText('Lia the Brave').first()).toBeVisible();
+
   await gmCtx.close();
   await playerCtx.close();
 });