feat(ui): single-word rule indicators + auto-match select redesign

Surface the per-game "single word" rule to the client and refine the
random-opponent New Game screen.

- Wire: thread multiple_words_per_turn into the GameView and Invitation
  FlatBuffers tables (Go + TS regenerated), through pkg/wire builders and both
  the backend push-event and gateway REST paths.
- In-game indicators (single-word games only): a small 1 in the status bar's
  score-preview slot (yields to the live preview) and a centred "One word per
  turn" label in the history-drawer header. Standard games show neither.
- Invitation card gains a "One word per turn" line for single-word invitations.
- Auto-match redesign: variant plaques are mutually-exclusive selects (highlight
  on tap, no longer enqueue); a lone offered variant is pre-selected; a bottom
  "Start game" button (disabled until a variant is chosen) confirms. The rule
  toggle appears once a Russian variant is selected.
- Tests: e2e for the new auto flow and the in-game indicator (mock g3 is a
  single-word game); mock/data + fixtures carry the new field. Docs: UI_DESIGN.

gateway/internal/backendclient/api.go

@@ -93,17 +93,18 @@ type SeatResp struct {
 
 // GameResp is the shared game summary.
 type GameResp struct {
-	ID               string     `json:"id"`
-	Variant          string     `json:"variant"`
-	DictVersion      string     `json:"dict_version"`
-	Status           string     `json:"status"`
-	Players          int        `json:"players"`
-	ToMove           int        `json:"to_move"`
-	TurnTimeoutSecs  int        `json:"turn_timeout_secs"`
-	MoveCount        int        `json:"move_count"`
-	EndReason        string     `json:"end_reason"`
-	LastActivityUnix int64      `json:"last_activity_unix"`
-	Seats            []SeatResp `json:"seats"`
+	ID                   string     `json:"id"`
+	Variant              string     `json:"variant"`
+	DictVersion          string     `json:"dict_version"`
+	Status               string     `json:"status"`
+	Players              int        `json:"players"`
+	ToMove               int        `json:"to_move"`
+	TurnTimeoutSecs      int        `json:"turn_timeout_secs"`
+	MultipleWordsPerTurn bool       `json:"multiple_words_per_turn"`
+	MoveCount            int        `json:"move_count"`
+	EndReason            string     `json:"end_reason"`
+	LastActivityUnix     int64      `json:"last_activity_unix"`
+	Seats                []SeatResp `json:"seats"`
 }
 
 // MoveResultResp is the outcome of a committed move. Rack carries the actor's refilled rack as

gateway/internal/backendclient/api_social.go

@@ -66,17 +66,18 @@ type InvitationInviteeResp struct {
 
 // InvitationResp is a friend-game invitation with its settings and invitees.
 type InvitationResp struct {
-	ID              string                  `json:"id"`
-	Inviter         AccountRefResp          `json:"inviter"`
-	Invitees        []InvitationInviteeResp `json:"invitees"`
-	Variant         string                  `json:"variant"`
-	TurnTimeoutSecs int                     `json:"turn_timeout_secs"`
-	HintsAllowed    bool                    `json:"hints_allowed"`
-	HintsPerPlayer  int                     `json:"hints_per_player"`
-	DropoutTiles    string                  `json:"dropout_tiles"`
-	Status          string                  `json:"status"`
-	GameID          string                  `json:"game_id"`
-	ExpiresAtUnix   int64                   `json:"expires_at_unix"`
+	ID                   string                  `json:"id"`
+	Inviter              AccountRefResp          `json:"inviter"`
+	Invitees             []InvitationInviteeResp `json:"invitees"`
+	Variant              string                  `json:"variant"`
+	TurnTimeoutSecs      int                     `json:"turn_timeout_secs"`
+	HintsAllowed         bool                    `json:"hints_allowed"`
+	HintsPerPlayer       int                     `json:"hints_per_player"`
+	MultipleWordsPerTurn bool                    `json:"multiple_words_per_turn"`
+	DropoutTiles         string                  `json:"dropout_tiles"`
+	Status               string                  `json:"status"`
+	GameID               string                  `json:"game_id"`
+	ExpiresAtUnix        int64                   `json:"expires_at_unix"`
 }
 
 // InvitationListResp is the caller's open invitations.