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

backend/internal/game/service.go

@@ -1178,6 +1178,14 @@ func (svc *Service) ListForAccount(ctx context.Context, accountID uuid.UUID) ([]
 	return svc.store.ListGamesForAccount(ctx, accountID)
 }
 
+// CountActiveQuickGames reports how many in-progress quick games the account holds —
+// the count the simultaneous-game limit (MaxActiveQuickGames) is checked against. It
+// counts active and still-open quick games (including honest-AI ones) and excludes
+// friend games created by invitation and finished games. See Store.CountActiveQuickGames.
+func (svc *Service) CountActiveQuickGames(ctx context.Context, accountID uuid.UUID) (int, error) {
+	return svc.store.CountActiveQuickGames(ctx, accountID)
+}
+
 // HideGame hides a finished game from accountID's own lobby (it stays visible to the other
 // players); it is irreversible by design. Only a player of a finished game may hide it
 // (ErrNotAPlayer / ErrGameActive otherwise); hiding an already-hidden game is a no-op.

backend/internal/game/store.go

@@ -450,6 +450,28 @@ func (s *Store) ListGamesForAccount(ctx context.Context, accountID uuid.UUID) ([
 	return out, nil
 }
 
+// CountActiveQuickGames counts the account's in-progress quick games — the ones the
+// simultaneous-game limit (MaxActiveQuickGames) is checked against. It includes both
+// active and still-open (awaiting-opponent) games, the honest-AI ones among them, and
+// excludes friend games (those linked to a game_invitations row) and finished games.
+// A hidden game still occupies a slot, so this is a dedicated count rather than a
+// filter over ListGamesForAccount (which drops hidden games). Joining on the account's
+// own seat counts each game once (an open game's empty opponent seat has no account).
+func (s *Store) CountActiveQuickGames(ctx context.Context, accountID uuid.UUID) (int, error) {
+	// The status literals are game.StatusActive / game.StatusOpen, matching the
+	// games.status CHECK in the baseline migration.
+	const q = `
+SELECT COUNT(*) FROM backend.games g
+JOIN backend.game_players gp ON gp.game_id = g.game_id
+LEFT JOIN backend.game_invitations gi ON gi.game_id = g.game_id
+WHERE gp.account_id = $1 AND g.status IN ('active', 'open') AND gi.game_id IS NULL`
+	var n int
+	if err := s.db.QueryRowContext(ctx, q, accountID).Scan(&n); err != nil {
+		return 0, fmt.Errorf("game: count active quick games: %w", err)
+	}
+	return n, nil
+}
+
 // HideGame hides a game from the account's own lobby list (idempotent). The caller validates the
 // game is finished and the account is a player.
 func (s *Store) HideGame(ctx context.Context, accountID, gameID uuid.UUID) error {

backend/internal/game/types.go

@@ -65,6 +65,9 @@ var (
 	ErrNoHintsLeft = errors.New("game: no hints remaining")
 	// ErrNoHintAvailable is returned when the player has no legal play to reveal.
 	ErrNoHintAvailable = errors.New("game: no legal move to suggest")
+	// ErrGameLimitReached is returned when an account already holds MaxActiveQuickGames
+	// simultaneous quick games and tries to create another game (quick or by invitation).
+	ErrGameLimitReached = errors.New("game: simultaneous game limit reached")
 )
 
 // AllowedTurnTimeouts is the set of per-game move clocks a game may be created
@@ -87,6 +90,15 @@ const DefaultTurnTimeout = 24 * time.Hour
 // one of AllowedTurnTimeouts (never offered in the creation UI).
 const AIInactivityTimeout = 7 * 24 * time.Hour
 
+// MaxActiveQuickGames is the cap on a player's simultaneous quick games (human
+// auto-match and honest-AI), counting both in-progress (StatusActive) and
+// still-open, awaiting-opponent (StatusOpen) games. Friend games created by
+// invitation are not counted. At the cap the backend refuses to create a new game
+// of any kind — quick or by invitation — and the lobby disables "New Game";
+// accepting an incoming invitation is always allowed. See
+// Store.CountActiveQuickGames.
+const MaxActiveQuickGames = 10
+
 // aiPlayerName labels the robot seat in an honest-AI game's GCG export, so a downloaded
 // game file shows a clean "AI" rather than the robot's human-like pool name (the in-app
 // UI shows 🤖 from the game's vs_ai flag).

backend/internal/inttest/game_limit_test.go

@@ -0,0 +1,188 @@
+//go:build integration
+
+package inttest
+
+import (
+	"context"
+	"encoding/json"
+	"fmt"
+	"net/http"
+	"net/http/httptest"
+	"strings"
+	"testing"
+	"time"
+
+	"github.com/google/uuid"
+	"go.uber.org/zap/zaptest"
+
+	"scrabble/backend/internal/account"
+	"scrabble/backend/internal/engine"
+	"scrabble/backend/internal/game"
+	"scrabble/backend/internal/server"
+)
+
+// The game-limit suite covers the simultaneous quick-game cap (game.MaxActiveQuickGames):
+// the counting rule (Store/Service.CountActiveQuickGames) and the HTTP gate that refuses a
+// new game once the cap is reached, while accepting an incoming invitation stays allowed.
+
+// TestCountActiveQuickGames checks the count includes active and open quick games (the
+// honest-AI ones among them) and excludes finished games, friend games (created by
+// invitation) and games the account is not seated in.
+func TestCountActiveQuickGames(t *testing.T) {
+	ctx := context.Background()
+	clearOpenGames(t)
+	games := newGameService()
+	human := provisionAccount(t)
+	opp := provisionAccount(t)
+
+	if n := mustCount(t, games, human); n != 0 {
+		t.Fatalf("fresh account count = %d, want 0", n)
+	}
+
+	// An open (awaiting-opponent) quick game counts.
+	if _, _, err := games.OpenOrJoin(ctx, human, game.CreateParams{
+		Variant: engine.VariantEnglish, TurnTimeout: 24 * time.Hour,
+	}, time.Now().Add(time.Minute)); err != nil {
+		t.Fatalf("open quick game: %v", err)
+	}
+	// An active quick game and an honest-AI quick game both count (neither has an invitation row).
+	mustCreateQuick(t, games, []uuid.UUID{human, opp}, false, 1)
+	mustCreateQuick(t, games, []uuid.UUID{human, opp}, true, 2)
+
+	// A finished quick game does NOT count.
+	fin := mustCreateQuick(t, games, []uuid.UUID{human, opp}, false, 3)
+	if _, err := testDB.ExecContext(ctx, `UPDATE backend.games SET status='finished' WHERE game_id=$1`, fin); err != nil {
+		t.Fatalf("finish game: %v", err)
+	}
+	// A game the human is not seated in does NOT count.
+	mustCreateQuick(t, games, []uuid.UUID{opp, provisionAccount(t)}, false, 4)
+	// A friend game (created by invitation) does NOT count, even though it is active.
+	startFriendGame(t, human)
+
+	if n := mustCount(t, games, human); n != 3 {
+		t.Fatalf("active quick count = %d, want 3 (active + AI + open)", n)
+	}
+}
+
+// TestGameLimitGate drives the cap through the assembled HTTP server: under the cap the lobby
+// reports at_game_limit false; at the cap it flips to true and both new-game endpoints (quick
+// enqueue and invitation creation) are refused with 409 game_limit_reached, while accepting an
+// incoming invitation is still allowed.
+func TestGameLimitGate(t *testing.T) {
+	ctx := context.Background()
+	games := newGameService()
+	srv := server.New(":0", server.Deps{
+		Logger:      zaptest.NewLogger(t),
+		DB:          testDB,
+		Accounts:    account.NewStore(testDB),
+		Games:       games,
+		Matchmaker:  newMatchmaker(t, newRobotService(t, newGameService()), time.Minute, 0),
+		Invitations: newInvitationService(),
+	})
+
+	human := provisionAccount(t)
+	opp := provisionAccount(t)
+
+	// Under the cap: the lobby reports the player is not limited.
+	if gamesListAtLimit(t, srv, human) {
+		t.Fatal("a fresh account must be under the game limit")
+	}
+
+	// Reach the cap with active quick games seating the human (no invitation row → quick).
+	for i := 0; i < game.MaxActiveQuickGames; i++ {
+		mustCreateQuick(t, games, []uuid.UUID{human, opp}, false, int64(i+1))
+	}
+
+	// At the cap: the lobby flags it and both create paths are refused with the stable code.
+	if !gamesListAtLimit(t, srv, human) {
+		t.Fatalf("at %d games at_game_limit must be true", game.MaxActiveQuickGames)
+	}
+	if rec := userPost(t, srv, "/api/v1/user/lobby/enqueue", human, `{"variant":"scrabble_en"}`); rec.Code != http.StatusConflict || errorCode(t, rec) != "game_limit_reached" {
+		t.Fatalf("enqueue at limit = (%d, %q), want (409, game_limit_reached)", rec.Code, errorCode(t, rec))
+	}
+	invBody := fmt.Sprintf(`{"variant":"scrabble_en","invitee_ids":[%q]}`, opp.String())
+	if rec := userPost(t, srv, "/api/v1/user/invitations", human, invBody); rec.Code != http.StatusConflict || errorCode(t, rec) != "game_limit_reached" {
+		t.Fatalf("invitation at limit = (%d, %q), want (409, game_limit_reached)", rec.Code, errorCode(t, rec))
+	}
+
+	// Accepting an incoming invitation is never blocked, even at the cap: another player invites
+	// the capped human, who accepts over HTTP and the game starts (friend games do not count).
+	inviter := provisionAccount(t)
+	inv := newInvitationService()
+	invitation, err := inv.CreateInvitation(ctx, inviter, []uuid.UUID{human}, englishInvite())
+	if err != nil {
+		t.Fatalf("create invitation: %v", err)
+	}
+	if rec := userPost(t, srv, "/api/v1/user/invitations/"+invitation.ID.String()+"/accept", human, ""); rec.Code != http.StatusOK {
+		t.Fatalf("accept at limit = %d (%s), want 200 — accept must bypass the cap", rec.Code, rec.Body.String())
+	}
+}
+
+// mustCount returns the account's active-quick-game count, failing on error.
+func mustCount(t *testing.T, games *game.Service, id uuid.UUID) int {
+	t.Helper()
+	n, err := games.CountActiveQuickGames(context.Background(), id)
+	if err != nil {
+		t.Fatalf("count active quick games: %v", err)
+	}
+	return n
+}
+
+// mustCreateQuick creates an active quick game (no invitation) seating the given accounts and
+// returns its id; vsAI flags an honest-AI game (the service then applies the 7-day clock).
+func mustCreateQuick(t *testing.T, games *game.Service, seats []uuid.UUID, vsAI bool, seed int64) uuid.UUID {
+	t.Helper()
+	p := game.CreateParams{Variant: engine.VariantEnglish, Seats: seats, TurnTimeout: 24 * time.Hour, Seed: seed}
+	if vsAI {
+		p.VsAI = true
+		p.TurnTimeout = 0 // the service applies the 7-day AI inactivity clock for vs_ai games
+	}
+	g, err := games.Create(context.Background(), p)
+	if err != nil {
+		t.Fatalf("create quick game (vsAI=%v): %v", vsAI, err)
+	}
+	return g.ID
+}
+
+// startFriendGame has a fresh inviter invite the given account to a friend game and the invitee
+// accept it, so the (active) friend game exists with its game_invitations row.
+func startFriendGame(t *testing.T, invitee uuid.UUID) {
+	t.Helper()
+	ctx := context.Background()
+	inv := newInvitationService()
+	inviter := provisionAccount(t)
+	invitation, err := inv.CreateInvitation(ctx, inviter, []uuid.UUID{invitee}, englishInvite())
+	if err != nil {
+		t.Fatalf("create invitation: %v", err)
+	}
+	if _, err := inv.RespondInvitation(ctx, invitation.ID, invitee, true); err != nil {
+		t.Fatalf("accept invitation: %v", err)
+	}
+}
+
+// gamesListAtLimit fetches /api/v1/user/games and returns its at_game_limit flag.
+func gamesListAtLimit(t *testing.T, srv *server.Server, id uuid.UUID) bool {
+	t.Helper()
+	rec := userGet(t, srv, "/api/v1/user/games", id)
+	if rec.Code != http.StatusOK {
+		t.Fatalf("games.list status = %d, want 200", rec.Code)
+	}
+	var body struct {
+		AtGameLimit bool `json:"at_game_limit"`
+	}
+	if err := json.Unmarshal(rec.Body.Bytes(), &body); err != nil {
+		t.Fatalf("decode games.list: %v", err)
+	}
+	return body.AtGameLimit
+}
+
+// userPost issues an authenticated JSON POST (X-User-ID) against the assembled server.
+func userPost(t *testing.T, srv *server.Server, path string, id uuid.UUID, body string) *httptest.ResponseRecorder {
+	t.Helper()
+	rec := httptest.NewRecorder()
+	req := httptest.NewRequest(http.MethodPost, path, strings.NewReader(body))
+	req.Header.Set("X-User-ID", id.String())
+	req.Header.Set("Content-Type", "application/json")
+	srv.Handler().ServeHTTP(rec, req)
+	return rec
+}

backend/internal/server/dto_test.go

@@ -34,6 +34,7 @@ func TestStatusForError(t *testing.T) {
 		"chat forbidden": {social.ErrForbiddenContent, http.StatusUnprocessableEntity, "chat_rejected"},
 		"no hint move":   {game.ErrNoHintAvailable, http.StatusConflict, "no_hint_available"},
 		"no hints left":  {game.ErrNoHintsLeft, http.StatusConflict, "hint_unavailable"},
+		"game limit":     {game.ErrGameLimitReached, http.StatusConflict, "game_limit_reached"},
 		"unknown -> 500": {context_deadline, http.StatusInternalServerError, "internal"},
 	}
 	for name, tc := range cases {

backend/internal/server/handlers.go

@@ -174,6 +174,8 @@ func statusForError(err error) (int, string) {
 		return http.StatusConflict, "no_opponent_yet"
 	case errors.Is(err, game.ErrGameActive):
 		return http.StatusConflict, "game_active"
+	case errors.Is(err, game.ErrGameLimitReached):
+		return http.StatusConflict, "game_limit_reached"
 	case errors.Is(err, account.ErrInvalidProfile):
 		return http.StatusBadRequest, "invalid_profile"
 	case errors.Is(err, account.ErrAlreadyConfirmed):

backend/internal/server/handlers_game.go

@@ -44,9 +44,14 @@ type historyDTO struct {
 	Moves  []moveRecordDTO `json:"moves"`
 }
 
-// gameListDTO is the caller's games (active and finished) for the lobby.
+// gameListDTO is the caller's games (active and finished) for the lobby. AtGameLimit
+// reports whether the caller has reached the simultaneous quick-game cap
+// (game.MaxActiveQuickGames); while it is true the lobby disables "New Game" and shows a
+// notice. It rides the lobby response — which the lobby re-fetches on every game event —
+// instead of a separate request.
 type gameListDTO struct {
-	Games []gameDTO `json:"games"`
+	Games       []gameDTO `json:"games"`
+	AtGameLimit bool      `json:"at_game_limit"`
 }
 
 // chatListDTO is a game's chat history.
@@ -388,6 +393,34 @@ func (s *Server) handleSaveDraft(c *gin.Context) {
 	c.JSON(http.StatusOK, okResponse{OK: true})
 }
 
+// atGameLimit reports whether uid already holds the maximum number of simultaneous
+// quick games (game.MaxActiveQuickGames). It backs both the lobby's at_game_limit flag
+// and the new-game gate; friend games created by invitation are not counted.
+func (s *Server) atGameLimit(ctx context.Context, uid uuid.UUID) (bool, error) {
+	n, err := s.games.CountActiveQuickGames(ctx, uid)
+	if err != nil {
+		return false, err
+	}
+	return n >= game.MaxActiveQuickGames, nil
+}
+
+// ensureUnderGameLimit aborts the request with 409 game_limit_reached when uid is at the
+// simultaneous quick-game cap, and reports whether the caller may proceed. It guards
+// every new-game entry point — quick auto-match/AI and invitation creation; accepting an
+// incoming invitation is deliberately exempt.
+func (s *Server) ensureUnderGameLimit(c *gin.Context, uid uuid.UUID) bool {
+	atLimit, err := s.atGameLimit(c.Request.Context(), uid)
+	if err != nil {
+		s.abortErr(c, err)
+		return false
+	}
+	if atLimit {
+		s.abortErr(c, game.ErrGameLimitReached)
+		return false
+	}
+	return true
+}
+
 // handleListGames returns the caller's active and finished games for the lobby.
 func (s *Server) handleListGames(c *gin.Context) {
 	uid, ok := userID(c)
@@ -400,6 +433,11 @@ func (s *Server) handleListGames(c *gin.Context) {
 		s.abortErr(c, err)
 		return
 	}
+	atLimit, err := s.atGameLimit(c.Request.Context(), uid)
+	if err != nil {
+		s.abortErr(c, err)
+		return
+	}
 	memo := map[string]string{}
 	out := make([]gameDTO, 0, len(games))
 	for _, g := range games {
@@ -407,7 +445,7 @@ func (s *Server) handleListGames(c *gin.Context) {
 		s.fillSeatNames(c.Request.Context(), &dto, memo)
 		out = append(out, dto)
 	}
-	c.JSON(http.StatusOK, gameListDTO{Games: out})
+	c.JSON(http.StatusOK, gameListDTO{Games: out, AtGameLimit: atLimit})
 }
 
 // handleChatList returns a game's chat history for the viewer.

backend/internal/server/handlers_invitations.go

@@ -130,6 +130,9 @@ func (s *Server) handleCreateInvitation(c *gin.Context) {
 		}
 		inviteeIDs = append(inviteeIDs, id)
 	}
+	if !s.ensureUnderGameLimit(c, uid) {
+		return
+	}
 	inv, err := s.invitations.CreateInvitation(c.Request.Context(), uid, inviteeIDs, settings)
 	if err != nil {
 		s.abortErr(c, err)

backend/internal/server/handlers_user.go

@@ -161,6 +161,9 @@ func (s *Server) handleEnqueue(c *gin.Context) {
 		abortBadRequest(c, "unknown variant")
 		return
 	}
+	if !s.ensureUnderGameLimit(c, uid) {
+		return
+	}
 	enter := s.matchmaker.Enqueue
 	if req.VsAI {
 		enter = s.matchmaker.StartVsAI