feat(ui): per-kind active-game limit lock on the New Game screen

Carry the caller's per-tier active-game caps and each game's kind on the
wire (Profile.game_limits + GameView.kind, additive FBS + gateway transcode
+ client codec, committed regen). The New Game screen counts the player's
active games per kind from the lobby and locks a capped start: an outline
button with a lock that opens a funnel modal instead of a game -- a sign-in
prompt for a guest, a "finish a current game first" notice for a signed-in
account (native Telegram popup, in-app modal elsewhere). The lock lifts via
the existing profile refetch after a guest->durable upgrade.

Remove the lobby's old at_game_limit New-Game tab disable + notice: the flag
(now the random-kind cap) conflicted with the per-kind lock -- it hid the
screen where the lock lives and wrongly blocked an unfulfilled kind. The New
Game tab is always enabled; the per-kind start lock is the only gate. The
at_game_limit wire field stays (unused by the client) for a later cleanup.

Tests: client lock logic + codec kind/game_limits roundtrip + gateway
transcode encode + native popup builders (unit); a mock e2e for the lock
badge and the modal.

ui/e2e/gamelimit.spec.ts

@@ -1,31 +1,40 @@
 import { expect, test } from './fixtures';
 
-// The simultaneous quick-game cap. When the backend reports the player is at the limit
-// (games.list at_game_limit), the lobby disables the "New Game" tab and shows a plain,
-// low-emphasis notice under the lists; when it clears, the button re-enables and the
-// notice goes. Driven by the mock transport's __mock.setGameLimit seam (no backend), which
-// also nudges the lobby to re-fetch (the lobby refreshes on any live event).
+// The active-game limit lock on the New Game screen. When a start's kind is at the caller's per-kind
+// cap (Profile.game_limits), the start button shows a 🔒 and opens a funnel modal instead of a game;
+// when the cap lifts (the profile refetch a guest→durable upgrade would trigger), the lock clears and
+// the button starts a game again. Driven by the mock's __mock.setGameLimits seam (no backend). The
+// native Telegram popup path is verified on the contour; here the cross-platform in-app modal shows
+// (the mock profile is a durable account, so it is the "finish a current game first" notice).
 
-test('lobby: New Game disables and a notice shows at the simultaneous-game limit', async ({ page }) => {
+type Limits = { vsAi: number; random: number; friends: number };
+function setLimits(l: Limits) {
+  (window as unknown as { __mock: { setGameLimits(l: Limits): void } }).__mock.setGameLimits(l);
+}
+
+test('new game: a start locks at its per-kind cap and the funnel modal opens', async ({ page }) => {
   await page.goto('/');
   await page.getByRole('button', { name: /guest/i }).click();
 
-  // Below the limit: the New Game tab is enabled and no notice is shown.
-  const newGame = page.getByRole('button', { name: /🎲/ });
-  await expect(newGame).toBeEnabled();
-  await expect(page.getByText(/reached the simultaneous games limit/i)).toHaveCount(0);
+  // The New Game button always opens the screen now — the per-kind lock lives on the start button.
+  await page.getByRole('button', { name: /🎲/ }).click();
 
-  // Reach the limit: the button greys out (disabled) and the notice appears under the lists.
-  await page.evaluate(() =>
-    (window as unknown as { __mock: { setGameLimit(v: boolean): void } }).__mock.setGameLimit(true),
-  );
-  await expect(newGame).toBeDisabled();
-  await expect(page.getByText(/reached the simultaneous games limit/i)).toBeVisible();
+  const start = page.getByRole('button', { name: /start game/i });
+  await expect(start).not.toContainText('🔒');
 
-  // Drop back below the limit (a game finished): the button re-enables and the notice clears.
-  await page.evaluate(() =>
-    (window as unknown as { __mock: { setGameLimit(v: boolean): void } }).__mock.setGameLimit(false),
-  );
-  await expect(newGame).toBeEnabled();
-  await expect(page.getByText(/reached the simultaneous games limit/i)).toHaveCount(0);
+  // Cap every kind: the AI start (the default opponent) locks.
+  await page.evaluate(setLimits, { vsAi: 0, random: 0, friends: 0 });
+  await expect(start).toContainText('🔒');
+
+  // Tapping the locked start opens the modal instead of a game, and does not navigate away.
+  await start.click();
+  const notice = page.getByText(/reached the limit of simultaneous games/i);
+  await expect(notice).toBeVisible();
+  await expect(page).toHaveURL(/\/new$/);
+  await page.getByRole('button', { name: /^ok$/i }).click();
+  await expect(notice).toHaveCount(0);
+
+  // Lift the cap (the same profile-refetch path a registration takes): the lock clears.
+  await page.evaluate(setLimits, { vsAi: 10, random: 10, friends: 10 });
+  await expect(start).not.toContainText('🔒');
 });