fix(router): update route rune synchronously in navigate()

navigate() wrote location.hash and left router.route to the asynchronous
hashchange event. bootstrap flips app.ready in the same tick right after
navigate('/login') on an unauthenticated cold start, so for one frame the app
rendered with app.ready=true and the stale route ('lobby', from the empty hash)
— briefly mounting the lobby shell (tab bar + a doomed games.list) under the new
login screen before hashchange settled it.

Update router.route synchronously inside navigate(); the later hashchange
re-parses to the same value. This removes the boot lobby-flash (and its spurious
games.list 401) and, as the root cause, the offline e2e flake: enterLobby could
latch that transient lobby tab-bar instead of clicking through login, then the
Settings-tab click at line 44 fought the tab detaching into the login slide
(~7% of runs, both engines).

Also:
- offline.spec enterLobby: wait for the guest button and click it, instead of a
  point-in-time count() that a pre-login splash frame sampled as 0 (skipping the
  click and hanging / latching the transient).
- New e2e router.spec pins the synchronous-route property (RED on the old code,
  which returned the previous route; GREEN now) via a mock-only __router seam.

Verify: check 0 / unit 490 / build / bundle 114.3/115 / full e2e 200 / the
offline+router stress at 40x on both engines 160/160 (was 6/80 failing).

ui/e2e/offline.spec.ts

@@ -13,10 +13,12 @@ async function forceStandalone(page: Page): Promise<void> {
   });
 }
 
-// After a load or reload, land in the lobby (dismiss the login screen if it is shown).
+// After a load, land in the lobby. The mock cold-starts on the login screen (no seeded session), so
+// click through as guest. Waiting for the button (rather than a point-in-time count()) is what makes
+// this deterministic: a count() sampled during the pre-login splash frame returned 0, skipped the
+// click, and then hung — or latched a transient lobby tab-bar — instead of opening the real lobby.
 async function enterLobby(page: Page): Promise<void> {
-  const guest = page.getByRole('button', { name: /guest|гост/i });
-  if (await guest.count()) await guest.first().click();
+  await page.getByRole('button', { name: /guest|гост/i }).first().click();
   // The lobby tab bar has three tabs in a fixed order: New (0), Stats (1), Settings (2). nth() is
   // robust to locale, emoji variation selectors and the coachmark anchors (which the first-run
   // onboarding strips after it completes).

ui/e2e/router.spec.ts

@@ -0,0 +1,26 @@
+import { expect, test } from './fixtures';
+
+// The hash router must update its reactive `route` rune synchronously inside navigate(), not defer it
+// to the asynchronous `hashchange` event. Bootstrap flips `app.ready` in the same tick right after
+// `navigate('/login')` on an unauthenticated cold start; a route that trailed the hash for one frame
+// let App.svelte render the stale route (the empty-hash lobby) under the new login screen — a visible
+// lobby-shell flash plus a doomed games.list. It also made the offline e2e flaky: enterLobby could
+// latch that transient lobby tab-bar instead of clicking through the login screen.
+
+test('navigate updates the reactive route synchronously (no hashchange lag)', async ({ page }) => {
+  await page.goto('/');
+  // The bundle has loaded and installed the mock __router seam once the login screen is up.
+  await expect(page.getByRole('button', { name: /guest/i })).toBeVisible();
+
+  // Read the route in the SAME microtask as the navigate: the fix makes it the new route at once;
+  // the pre-fix code returned the previous route (it waited for the async hashchange).
+  const routeAfterNavigate = await page.evaluate(() => {
+    const r = (window as unknown as { __router: { navigate(p: string): void; route(): string } }).__router;
+    r.navigate('/settings');
+    return r.route();
+  });
+  expect(routeAfterNavigate).toBe('settings');
+
+  // The hash was written too, so a reload/back still resolves the same route.
+  expect(new URL(page.url()).hash).toBe('#/settings');
+});