Aggiunge overlay server-side burn-in e stabilizza diretta live.

Tabellone, badge e brand sono ricodificati nel flusso _air via OverlayRelay;
sync punteggio HTTP, volume condiviso rails/sidekiq, qualità 720p migliorata
e badge CONNECTING in ripresa da pausa.

Co-authored-by: Cursor <cursoragent@cursor.com>

backend/app/services/streams/overlay/badge_label.rb

@@ -0,0 +1,53 @@
+module Streams
+  module Overlay
+    # Etichetta stato broadcast (stessa logica della pagina live / badge HTML).
+    class BadgeLabel
+      Result = Struct.new(:text, :background, :foreground, keyword_init: true)
+
+      def self.for(session)
+        new(session).call
+      end
+
+      def initialize(session)
+        @session = session
+      end
+
+      def call
+        return Result.new(text: "DIRETTA CHIUSA", background: "#374151", foreground: "#dddddd") if @session.terminal?
+        return Result.new(text: "IN PAUSA", background: "#455a64", foreground: "#ffffff") if @session.paused?
+
+        if publisher_online? || (@session.live? && !@session.paused?)
+          return Result.new(text: "IN ONDA", background: "#2e7d32", foreground: "#ffffff")
+        end
+
+        if path_has_output?
+          if @session.reconnecting?
+            return Result.new(text: "RICONNESSIONE", background: "#f57c00", foreground: "#ffffff")
+          end
+          if @session.connecting?
+            return Result.new(text: "CONNECTING", background: "#f57c00", foreground: "#ffffff")
+          end
+          return Result.new(text: "COPERTINA", background: "#455a64", foreground: "#ffffff")
+        end
+
+        Result.new(text: "IN ATTESA", background: "#455a64", foreground: "#ffffff")
+      end
+
+      private
+
+      def path_info
+        @path_info ||= Mediamtx::Client.new.list_paths.find { |i| i["name"] == @session.mediamtx_path_name }
+      end
+
+      def publisher_online?
+        info = path_info
+        info && info["online"]
+      end
+
+      def path_has_output?
+        info = path_info
+        info && (info["ready"] || info["available"] || info["online"])
+      end
+    end
+  end
+end