I believe that Dumbledore’s horcrux is Fawkes. When Fawks gives 2 tail feathers for Harry’s and Voldemort’s wands a piece of his soul strengthens the connection beetween their wands. I know you are thinking “so why is Dumbledore not alive.” Well when Vodemort’s curse backfires the part of his soul in Voldemort’s wand goes into Harry. The part in Harry is too small to rwvive Dumbledore. Then when Voldemort kills harry in the forest that part if the soul dies. Harry’s wand snaps in half so that part is gone. Fawks swallows a killing curse so that part dies. This explains how Fawks knew how to go to Harry in the Chamber of Secretes. It sensed that anouther part of Dumbledore (the oart in Harry) was in distress. Please tell me of you see any loopholes in this theory.