feat(lobby): cap simultaneous quick games at 10

Limit a player to 10 active quick games (auto-match + AI); friend games created
by invitation are not counted. At the cap the backend refuses both new-game
entry points — quick enqueue and invitation creation — with 409
game_limit_reached, while accepting an incoming invitation stays allowed, so
friend games are capped from the other end. The lobby disables "New Game" and
shows a low-emphasis notice, driven by a new at_game_limit flag on games.list
(no per-event payload: a turn change does not move the count, and the lobby
already re-fetches games.list on entry and every game event).

- game.MaxActiveQuickGames + Store/Service.CountActiveQuickGames (active/open
  seats, no game_invitations row; hidden games still count -> dedicated count)
- Server.ensureUnderGameLimit gating handleEnqueue + handleCreateInvitation;
  game.ErrGameLimitReached -> 409 game_limit_reached
- FB GameList.at_game_limit (regenerated Go + TS) through the gateway transcode
  and UI codec; gameListDTO + lobbycache snapshot + Lobby.svelte + i18n
- tests: integration count rule + HTTP gate + accept bypass; server error map;
  gateway transcode round-trip; UI codec + lobbycache unit; e2e gamelimit
- docs: PRERELEASE (GL), FUNCTIONAL(+ru), ARCHITECTURE 8, UI_DESIGN, backend README

ui/src/lib/codec.test.ts

@@ -241,6 +241,7 @@ describe('codec', () => {
     const games = fb.GameList.createGamesVector(b, [game]);
     fb.GameList.startGameList(b);
     fb.GameList.addGames(b, games);
+    fb.GameList.addAtGameLimit(b, true);
     b.finish(fb.GameList.endGameList(b));
 
     const gl = decodeGameList(b.asUint8Array());
@@ -250,6 +251,7 @@ describe('codec', () => {
     expect(gl.games[0].seats[0].score).toBe(13);
     expect(gl.games[0].lastActivityUnix).toBe(1717000000);
     expect(gl.games[0].vsAi).toBe(true);
+    expect(gl.atGameLimit).toBe(true);
   });
 
   it('decodes an OutgoingRequestList of account refs', () => {

ui/src/lib/codec.ts

@@ -445,7 +445,7 @@ export function decodeGameList(buf: Uint8Array): GameList {
     const g = gl.games(i);
     if (g) games.push(decodeGameView(g));
   }
-  return { games };
+  return { games, atGameLimit: gl.atGameLimit() };
 }
 
 export function decodeMatchResult(buf: Uint8Array): MatchResult {

ui/src/lib/gateway.ts

@@ -24,11 +24,17 @@ if (isMock && typeof window !== 'undefined') {
   // attaches a robot on a timer).
   (
     window as unknown as {
-      __mock?: { joinOpponent(): void; joinOpponentSilently(): void; adminReply(): void };
+      __mock?: {
+        joinOpponent(): void;
+        joinOpponentSilently(): void;
+        adminReply(): void;
+        setGameLimit(v: boolean): void;
+      };
     }
   ).__mock = {
     joinOpponent: () => (gateway as MockGateway).joinPendingOpponent(),
     joinOpponentSilently: () => (gateway as MockGateway).joinPendingOpponentSilently(),
     adminReply: () => (gateway as MockGateway).mockAdminReply(),
+    setGameLimit: (v: boolean) => (gateway as MockGateway).setGameLimit(v),
   };
 }

ui/src/lib/i18n/en.ts

@@ -33,6 +33,7 @@ export const en = {
   'lobby.finishedGames': 'Finished games',
   'lobby.noActive': 'No active games yet.',
   'lobby.noFinished': 'No finished games yet.',
+  'lobby.limitReached': "You've reached the simultaneous games limit.",
   'lobby.new': 'New',
   'lobby.stats': 'Stats',
   'lobby.profile': 'Profile',

ui/src/lib/i18n/ru.ts

@@ -34,6 +34,7 @@ export const ru: Record<MessageKey, string> = {
   'lobby.finishedGames': 'Завершённые игры',
   'lobby.noActive': 'Пока нет активных игр.',
   'lobby.noFinished': 'Пока нет завершённых игр.',
+  'lobby.limitReached': 'Вы достигли лимита одновременных партий',
   'lobby.new': 'Новая',
   'lobby.stats': 'Статы',
   'lobby.profile': 'Профиль',

ui/src/lib/lobbycache.test.ts

@@ -41,7 +41,7 @@ beforeEach(() => clearLobby());
 
 describe('patchLobbyGame', () => {
   it('replaces the matching game by id and leaves the others untouched', () => {
-    setLobby({ games: [gameView('a', 'active', 0), gameView('b', 'active', 0)], invitations: [], incoming: [] });
+    setLobby({ games: [gameView('a', 'active', 0), gameView('b', 'active', 0)], invitations: [], incoming: [], atGameLimit: false });
     // The player's own move flipped game "a" to the opponent's turn.
     patchLobbyGame(gameView('a', 'active', 1));
     const snap = getLobby();
@@ -52,14 +52,14 @@ describe('patchLobbyGame', () => {
 
   it('preserves invitations and incoming when patching a game', () => {
     const incoming: AccountRef[] = [{ accountId: 'u9', displayName: 'Nine' }];
-    setLobby({ games: [gameView('a')], invitations: [], incoming });
+    setLobby({ games: [gameView('a')], invitations: [], incoming, atGameLimit: false });
     patchLobbyGame(gameView('a', 'finished'));
     expect(getLobby()?.incoming).toEqual(incoming);
     expect(getLobby()?.games[0].status).toBe('finished');
   });
 
   it('adds the game when it is not yet in the cached lobby (a game started elsewhere)', () => {
-    setLobby({ games: [gameView('a')], invitations: [], incoming: [] });
+    setLobby({ games: [gameView('a')], invitations: [], incoming: [], atGameLimit: false });
     patchLobbyGame(gameView('z', 'active'));
     // Order is irrelevant — the lobby re-groups and re-sorts on render — so assert membership.
     expect(getLobby()?.games.map((g) => g.id).sort()).toEqual(['a', 'z']);
@@ -72,21 +72,29 @@ describe('patchLobbyGame', () => {
 
   it('does not mutate the previous snapshot array', () => {
     const games = [gameView('a', 'active', 0)];
-    setLobby({ games, invitations: [], incoming: [] });
+    setLobby({ games, invitations: [], incoming: [], atGameLimit: false });
     patchLobbyGame(gameView('a', 'active', 1));
     expect(games[0].toMove).toBe(0); // the original array/object is left intact
   });
+
+  it('preserves the at-game-limit flag across a game patch', () => {
+    // A finished-game patch arriving while the lobby is unmounted must not drop the flag —
+    // the lobby renders the "New Game" button from the cached snapshot before the refresh.
+    setLobby({ games: [gameView('a')], invitations: [], incoming: [], atGameLimit: true });
+    patchLobbyGame(gameView('a', 'finished'));
+    expect(getLobby()?.atGameLimit).toBe(true);
+  });
 });
 
 describe('patchLobbyInvitation', () => {
   it('adds a new pending invitation to the cached lobby', () => {
-    setLobby({ games: [], invitations: [], incoming: [] });
+    setLobby({ games: [], invitations: [], incoming: [], atGameLimit: false });
     patchLobbyInvitation(invitation('i1', 'pending'));
     expect(getLobby()?.invitations.map((x) => x.id)).toEqual(['i1']);
   });
 
   it('replaces a pending invitation already in the list (e.g. an updated response)', () => {
-    setLobby({ games: [], invitations: [invitation('i1', 'pending'), invitation('i2', 'pending')], incoming: [] });
+    setLobby({ games: [], invitations: [invitation('i1', 'pending'), invitation('i2', 'pending')], incoming: [], atGameLimit: false });
     const updated = { ...invitation('i1', 'pending'), turnTimeoutSecs: 999 };
     patchLobbyInvitation(updated);
     expect(getLobby()?.invitations.map((x) => x.id)).toEqual(['i1', 'i2']);
@@ -95,14 +103,14 @@ describe('patchLobbyInvitation', () => {
 
   it('removes an invitation that reached a terminal status', () => {
     for (const terminal of ['declined', 'cancelled', 'started', 'expired']) {
-      setLobby({ games: [], invitations: [invitation('i1', 'pending'), invitation('i2', 'pending')], incoming: [] });
+      setLobby({ games: [], invitations: [invitation('i1', 'pending'), invitation('i2', 'pending')], incoming: [], atGameLimit: false });
       patchLobbyInvitation(invitation('i1', terminal));
       expect(getLobby()?.invitations.map((x) => x.id)).toEqual(['i2']);
     }
   });
 
   it('is a no-op for a terminal invitation that is not in the list', () => {
-    setLobby({ games: [], invitations: [invitation('i2', 'pending')], incoming: [] });
+    setLobby({ games: [], invitations: [invitation('i2', 'pending')], incoming: [], atGameLimit: false });
     patchLobbyInvitation(invitation('i1', 'declined'));
     expect(getLobby()?.invitations.map((x) => x.id)).toEqual(['i2']);
   });
@@ -114,7 +122,7 @@ describe('patchLobbyInvitation', () => {
 
   it('preserves games and incoming when patching an invitation', () => {
     const incoming: AccountRef[] = [{ accountId: 'u9', displayName: 'Nine' }];
-    setLobby({ games: [gameView('g1')], invitations: [], incoming });
+    setLobby({ games: [gameView('g1')], invitations: [], incoming, atGameLimit: false });
     patchLobbyInvitation(invitation('i1', 'pending'));
     expect(getLobby()?.games.map((g) => g.id)).toEqual(['g1']);
     expect(getLobby()?.incoming).toEqual(incoming);

ui/src/lib/lobbycache.ts

@@ -10,6 +10,9 @@ interface LobbySnapshot {
   games: GameView[];
   invitations: Invitation[];
   incoming: AccountRef[];
+  // atGameLimit rides the snapshot so the lobby renders the "New Game" button in its last
+  // known enabled/disabled state instantly (no flicker), then refreshes it in the background.
+  atGameLimit: boolean;
 }
 
 let snapshot: LobbySnapshot | null = null;

ui/src/lib/mock/client.ts

@@ -98,6 +98,8 @@ export class MockGateway implements GatewayClient {
   private feedbackReplyUnread = false;
   // The most recently opened auto-match game still awaiting an opponent, for the e2e join hook.
   private openGameId: string | null = null;
+  // gameLimit forces the lobby's at_game_limit flag for the e2e (window.__mock.setGameLimit).
+  private gameLimit = false;
   private friends: AccountRef[] = MOCK_FRIENDS.map((f) => ({ ...f }));
   private incoming: AccountRef[] = MOCK_INCOMING.map((f) => ({ ...f }));
   private outgoing: AccountRef[] = [];
@@ -152,7 +154,10 @@ export class MockGateway implements GatewayClient {
     return { blocked: false, permanent: false, until: '', reason: '' };
   }
   async gamesList(): Promise<GameList> {
-    return { games: [...this.games.values()].map((g) => structuredClone(g.view)) };
+    return {
+      games: [...this.games.values()].map((g) => structuredClone(g.view)),
+      atGameLimit: this.gameLimit,
+    };
   }
 
   // --- lobby ---
@@ -257,6 +262,15 @@ export class MockGateway implements GatewayClient {
     if (this.openGameId) this.seatOpponent(this.openGameId);
   }
 
+  // setGameLimit forces the lobby's at_game_limit flag (the e2e hook
+  // window.__mock.setGameLimit), then nudges the lobby to re-fetch games.list so the
+  // "New Game" button and the notice update in place. The lobby refetches on any
+  // non-heartbeat event; an unrecognised notify sub triggers only that refetch.
+  setGameLimit(v: boolean): void {
+    this.gameLimit = v;
+    this.emit({ kind: 'notify', sub: 'game_limit' });
+  }
+
   async lobbyPoll(): Promise<MatchResult> {
     if (this.pendingMatch) {
       const g = this.games.get(this.pendingMatch);

ui/src/lib/model.ts

@@ -293,6 +293,9 @@ export interface History {
 
 export interface GameList {
   games: GameView[];
+  /** True when the caller has reached the simultaneous quick-game cap; the lobby then
+   *  disables "New Game" and shows a notice. */
+  atGameLimit: boolean;
 }
 
 /**