feat(chat): unread read-receipts with lobby/game dot and history-open ack

Persist per-message read state as a chat_messages.unread_seats bitmask
(migration 00008): a text message seeds every recipient seat's bit, a nudge
only the awaited seat's. A seat's bit clears when the player opens the move
history or chat (POST /games/:id/chat/read, sent only when something is
unread), and a nudge additionally clears when its recipient answers by moving
(a wired game NudgeClearer, dependency-inverted so game keeps off social).

UI shows a per-viewer unread dot in the lobby (next to the opponent) and the
game score bar — the unread_chat game-view flag seeds it from authoritative
REST views, live chat/nudge events raise it. Opening the move history counts
as reading (even without entering chat): the 💬 fade-blinks twice and the
client acks. Admin Messages gains an unread-only filter, a read/unread column,
and a per-message card with the per-seat read breakdown. Observability:
chat_read_duration histogram + chat_unread_messages gauge + social tracing.

gateway/internal/backendclient/api.go

@@ -134,6 +134,7 @@ type GameResp struct {
 	EndReason            string     `json:"end_reason"`
 	LastActivityUnix     int64      `json:"last_activity_unix"`
 	Seats                []SeatResp `json:"seats"`
+	UnreadChat           bool       `json:"unread_chat"`
 }
 
 // MoveResultResp is the outcome of a committed move. Rack carries the actor's refilled rack as
@@ -468,6 +469,13 @@ func (c *Client) Nudge(ctx context.Context, userID, gameID string) (ChatResp, er
 	return out, err
 }
 
+// MarkChatRead acknowledges that the caller has read the game's chat (sent when they open
+// the move history or chat), so the backend clears their unread bits and records the
+// publish-to-read latency.
+func (c *Client) MarkChatRead(ctx context.Context, userID, gameID string) error {
+	return c.do(ctx, http.MethodPost, c.gamePath(gameID, "/chat/read"), userID, "", struct{}{}, nil)
+}
+
 // GamesList returns the caller's active and finished games.
 func (c *Client) GamesList(ctx context.Context, userID string) (GameListResp, error) {
 	var out GameListResp