Stage 8: UI social/account/history surfaces

Wire the deferred Stage 7 surfaces end-to-end (UI -> gateway transcode ->
backend REST -> existing domain services): friends (incl. one-time friend
codes), per-user blocks, friend-game invitations, profile editing + email
binding, the statistics screen, and the in-game history + GCG export.

Friends gain two add paths (interview decision, a deliberate plan change):
one-time 6-digit codes (friend_codes table, 12h TTL, single-use, rate-limited
redeem); and play-gated requests (shared game required) where an explicit
decline is permanent, an ignored request lapses after 30 days, and a code
bypasses a decline. Migration 00006 widens friendships_status_chk and adds
friend_codes.

Lobby notification badge is poll + push: a new generic `notify` event drives
it live; the client polls on open/focus. Language stays a single Settings
control that writes through to the durable account's preferred_language. GCG
export is finished-only (game.ErrGameActive) and shares/downloads the .gcg file.

Tests: backend unit + inttest (friend gate/decline/code, ListInvitations,
GetStats, GCG gate), gateway transcode round-trips + notify constructor, UI
vitest (codecs, win-rate, share choice) + Playwright social specs. Docs: PLAN
(Stage 8 done + refinements + TODO-5), ARCHITECTURE, FUNCTIONAL(+ru), UI_DESIGN,
TESTING, module READMEs.

backend/internal/game/store.go

@@ -135,6 +135,24 @@ func (s *Store) GetGame(ctx context.Context, id uuid.UUID) (Game, error) {
 	return projectGame(grow, srows)
 }
 
+// SharedGameExists reports whether accounts a and b are both seated in at least
+// one game (active or finished). It backs the social package's "befriend an
+// opponent" gate via a self-join on game_players.
+func (s *Store) SharedGameExists(ctx context.Context, a, b uuid.UUID) (bool, error) {
+	other := table.GamePlayers.AS("other")
+	stmt := postgres.SELECT(table.GamePlayers.GameID).
+		FROM(table.GamePlayers.INNER_JOIN(other, other.GameID.EQ(table.GamePlayers.GameID))).
+		WHERE(
+			table.GamePlayers.AccountID.EQ(postgres.UUID(a)).
+				AND(other.AccountID.EQ(postgres.UUID(b))),
+		).LIMIT(1)
+	var rows []model.GamePlayers
+	if err := stmt.QueryContext(ctx, s.db, &rows); err != nil {
+		return false, fmt.Errorf("game: shared game exists: %w", err)
+	}
+	return len(rows) > 0, nil
+}
+
 // ListGamesForAccount loads every game the account is seated in (active and
 // finished), newest first, each joined with its ordered seats. It backs the lobby's
 // "my games" lists.