Stage 17 round 6 (#16/#17, PR C): lobby sort + server-derived in-game friend state

Lobby: group the my-games list into your-turn / opponent-turn / finished
(empty sections hidden), ordered by last activity (your-turn oldest-first,
the other two newest-first), as a compact line-separated list. gameDTO and
FB GameView gain last_activity_unix (turn start while active, finish time
once finished); a pure lib/lobbysort.ts holds the grouping/ordering.

Friends: the in-game 'add to friends' item is now server-derived via a new
GET /user/friends/outgoing (+ friends.outgoing op), returning addressees with
a pending OR declined request (both read as 'request sent'), so it is correct
across reloads; it shows a disabled '✓ in friends' once accepted. It
live-updates when the opponent answers: RespondFriendRequest now publishes
friend_added (accept) / friend_declined (new notify sub-kind, decline) to the
original requester, whose open game re-derives its friend state.

Tests: lobbysort unit test; gateway outgoing + last_activity transcode tests;
backend integration ListOutgoingRequests + respond-publishes-to-requester;
e2e updated for the new lobby section labels + a non-friend active opponent.
Docs: ARCHITECTURE notify catalog, FUNCTIONAL(+ru) lobby/friends, PLAN.

backend/internal/server/dto.go

@@ -83,16 +83,19 @@ type seatDTO struct {
 
 // gameDTO is the shared game summary.
 type gameDTO 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"`
-	Seats           []seatDTO `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"`
+	MoveCount       int    `json:"move_count"`
+	EndReason       string `json:"end_reason"`
+	// LastActivityUnix is the lobby sort key: the current turn's start for an active
+	// game, the finish time once finished (Stage 17).
+	LastActivityUnix int64     `json:"last_activity_unix"`
+	Seats            []seatDTO `json:"seats"`
 }
 
 // moveResultDTO is the outcome of a committed move.
@@ -189,17 +192,22 @@ func gameDTOFromGame(g game.Game) gameDTO {
 			IsWinner:  s.IsWinner,
 		})
 	}
+	last := g.TurnStartedAt
+	if g.FinishedAt != nil {
+		last = *g.FinishedAt
+	}
 	return gameDTO{
-		ID:              g.ID.String(),
-		Variant:         g.Variant.String(),
-		DictVersion:     g.DictVersion,
-		Status:          g.Status,
-		Players:         g.Players,
-		ToMove:          g.ToMove,
-		TurnTimeoutSecs: int(g.TurnTimeout.Seconds()),
-		MoveCount:       g.MoveCount,
-		EndReason:       g.EndReason,
-		Seats:           seats,
+		ID:               g.ID.String(),
+		Variant:          g.Variant.String(),
+		DictVersion:      g.DictVersion,
+		Status:           g.Status,
+		Players:          g.Players,
+		ToMove:           g.ToMove,
+		TurnTimeoutSecs:  int(g.TurnTimeout.Seconds()),
+		MoveCount:        g.MoveCount,
+		EndReason:        g.EndReason,
+		LastActivityUnix: last.Unix(),
+		Seats:            seats,
 	}
 }