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

docs/ARCHITECTURE.md

@@ -467,6 +467,18 @@ disguised robot stays indistinguishable from a person.
   game is `open` the starter may move on their turn, but resign, chat and nudge are
   refused (no opponent yet) and the lobby and opponent card show a "searching for
   opponent" placeholder.
+- **Simultaneous-game cap**: a player may hold at most `game.MaxActiveQuickGames`
+  (**10**) active quick games. `game.Service.CountActiveQuickGames` counts the games
+  seating the account in status `active` or `open` **without** a linked
+  `game_invitations` row — friend games are excluded, and hidden games still occupy a
+  slot, so it is a dedicated count rather than a filter over the lobby list. The backend
+  **gate** (`Server.ensureUnderGameLimit`) refuses **both** new-game entry points at the
+  cap — `POST /lobby/enqueue` and `POST /invitations` — with **409 `game_limit_reached`**;
+  **accepting** an invitation (`POST /invitations/:id/accept`) is never gated, so friend
+  games are capped only at initiation. The lobby learns the state from a boolean
+  **`at_game_limit`** carried on the `games.list` response — the lobby already re-fetches
+  that on entry and on every game event, so the flag needs no separate request or
+  per-event payload; while it is set the client disables **New Game** and shows a notice.
 - **Friends**: two add paths over one `friendships` table. A **one-time
   code** the to-be-added player issues (a `friend_codes` row: 6-digit numeric,
   SHA-256-hashed, **12 h** TTL, one live code per issuer, single-use, redeem

docs/FUNCTIONAL.md

@@ -110,6 +110,15 @@ is shareable as a Telegram deep link that opens it directly): the inviter choose
 settings and the game starts once every invitee has accepted — any decline cancels it, and an unanswered invitation
 expires after seven days.
 
+**Simultaneous-game cap.** A player may hold at most **10 active quick games** at once —
+counting both in-progress and still-searching auto-match/AI games, but **not** friend games
+created by invitation. At the cap the **New Game** button is disabled (greyed) and a plain,
+low-emphasis line under the lists reads *"You've reached the simultaneous games limit."*; both
+clear automatically once an active game finishes and the count drops below ten. The cap blocks
+**starting** any game (a quick game or a friend invitation, which share the New Game entry), but
+**accepting an incoming invitation is never blocked** — so friend games are capped from the other
+end (you cannot initiate one while at the cap) while you can always join a game you are invited to.
+
 ### Playing a game
 Place tiles, pass, exchange, or resign. Pass and exchange share one control —
 choosing no tiles passes the turn, choosing tiles exchanges them. Tiles are laid

docs/FUNCTIONAL_ru.md

@@ -115,6 +115,16 @@ nudge) приходят от бота **этой партии** — по язы
 выбирает настройки, и партия стартует, когда приняли все приглашённые — любой отказ отменяет приглашение, а без
 ответа приглашение протухает через семь дней.
 
+**Лимит одновременных партий.** Игрок может держать одновременно не более **10 активных
+быстрых партий** — считаются и идущие, и ещё ищущие соперника авто-подбор/AI-партии, но **не**
+игры с друзьями, созданные приглашением. На лимите кнопка **«Новая игра»** становится неактивной
+(серой), а под списками появляется ненавязчивая строка *«Вы достигли лимита одновременных
+партий»*; и кнопка, и надпись снимаются автоматически, как только активная партия завершается и
+счётчик опускается ниже десяти. Лимит запрещает **создавать** любую партию (быструю или
+приглашение в игру с друзьями — у них общий вход «Новая игра»), но **принимать входящее
+приглашение никогда не запрещено** — так игры с друзьями ограничиваются «с другого конца» (на
+лимите их нельзя инициировать), а присоединиться к партии по приглашению можно всегда.
+
 ### Игровой процесс
 Выкладывание фишек, пас, обмен или сдача. Пас и обмен — один элемент управления:
 без выбранных фишек — пас, с выбранными — обмен. Фишки кладутся без выбора

docs/UI_DESIGN.md

@@ -240,6 +240,13 @@ opponent's-turn change is silent. Each card's blink is keyed by game id, so over
 events animate in isolation, and the newest bottom toast cancels the previous one and replays
 its entrance (`components/Toast.svelte`, re-keyed on a per-message sequence).
 
+**Simultaneous-game cap.** When the player is at the active-quick-game cap (`games.list`
+`at_game_limit`), the lobby's **New Game** tab is **disabled** (greyed via the shared
+`.tab:disabled` opacity) and a plain, low-emphasis line — muted, centred, no accent — sits
+under the lists with the localized "limit reached" notice. Both follow the flag carried in the
+cached lobby snapshot, so they render in their last-known state on entry (the button stays
+enabled on the first, uncached load) and flip in place when an event refreshes the lobby.
+
 ## Social, account & history surfaces
 
 - **Settings hub** (`screens/SettingsHub.svelte`, the lobby ⚙️ tab): one nav bar + a bottom