The theory of the hidden Horcrux is based upon the idea that Albus Dumbledore created a Horcrux at one point during his life. 

At first when this theory entered my mind, it seemed almost unbelievable. Horcruxes always seemed evil to me. But almost unbelievable means it can still be believable, and there are certain clues within the Harry Potter books that give way to this theory.

During my research I got a reminder of what a horcrux is and I like to share this with you before we dive head first into the theory.

"Horcrux - An object within which a wizard [or witch] hides a piece of his [or her] soul. This is the Darkest of magic and can only be performed after a wizard [or witch] splits his [or her] soul by committing murder, a pure act of evil. A Horcrux protects the bit of soul encased within from anything that might happen to a wizard's [or witch's] body. Once a Horcrux has been created, the wizard [or witch] who made it cannot be killed until the object is destroyed." -The Lexicon (Steve Vander Ark)

Based upon this definition it is deemed necessary that a witch or wizard must take someone’s  life in order for a Horcrux to then be created. When reading this, or thinking about a Horcrux in general, it seems difficult to believe the beloved headmaster would ever commit an act of murder. However, there are clues in the books that point towards a death caused by Dumbledore...

Clue number one:  this theory is the death of Ariana Dumbledore. It is during the King's Cross chapter of Harry Potter and the Deathly Hallows when Harry Potter learns more about the death of Ariana Dumbledore. When Albus Dumbledore is confessing to Harry about his youthful plans for power and quest for the Deathly Hallows with his friend Gellert Grindelwald, Dumbledore claims that after Grindelwald fled from a duel with Albus and Aberforth, Dumbledore feared Grindelwald; not out of fear but in fear of the truth.

"You see [Harry], I never knew which one of us, in that horrific fight, had actually cast the curse that killed my sister. [...] I dreaded beyond all things the knowledge that it had been I who brought about her death, not merely through my arrogance and stupidity, but that I actually struck the blow that snuffed out her life." - Albus Dumbledore (DH-Chapter 35-Pg.718)

Dumbledore might have fallen out with his friend. Gellert Grindelwald was a dark wizard and Dumbledore doesn’t really fit that bill. So something might have triggered him and during a duel there might have been an accident. Like a certain killing curse hitting another target: Ariana.

I’ve found a clue in another book. During The Cave chapter in Harry Potter and the Half-Blood Prince when Dumbledore is drinking the potion that causes him to be 'driven out of his mind.' Dumbledore is sobbing and moaning phrases of remorse and plea.

"It's all my fault, all my fault. Please make it stop, I know I did wrong, oh please make it stop and I'll never, never again...Don't hurt them, don't hurt them, please, please, it's my fault, hurt me instead..." -Albus Dumbledore (HBP-Chapter 26- Pg. 572)

A year later when Harry Potter is in the Hog's Head and conversing with Aberforth Dumbledore, Harry mentions the night when Dumbledore died.

"[Albus] drank a potion that drove him out of his mind. He started screaming, pleading with someone who wasn't there. 'Don't hurt them, please...hurt me instead.' He thought he was back there with you and Grindelwald, I know he did. He thought he was watching Grindelwald hurting you and Ariana...It was torture to him, if you'd seen him then, you wouldn't say he was free." -Harry Potter-DH-Chapter 28-Pg.567-568)

If what Harry Potter says is true, and all those years ago Dumbledore did watch Grindelwald hurting his fammily, tortured by the sight of what he was seeing Grindelwald do to his brother and sister, Albus Dumbledore could have had murderous intentions toward Grindelwald. I know I would have! When the fight broke out between Albus, Grindelwald, and Aberforth, Albus could have cast a spell with murderous intentions at Grindelwald. Yet, rather than that murderous-intended spell hitting Grindelwald, it accidentally struck Ariana, thus causing her untimely death.

From these two clues, there does exist plausibility that Albus Dumbledore did kill his sister, Ariana, with a spell that had murderous intent. If this is true, then a question to follow would be, "Why would Dumbledore even create a Horcrux?" 

Dumbledore would have been feeling guilty. In the case of Albus Dumbledore creating a Horcrux, it would be a very literal use of wanting to rip a feeling or part of his body that he no longer wanted. Perhaps  this is his way of no longer having to live with the part of his soul that now forever embeds itself with the guilt of killing his baby sister.

Among the many papers that Dumbledore willed to the Hogwarts Archives were notes regarding The Tales of Beedle the Bard. Within his notes about The Warlock's Hairy Heart, Dumbledore mentions Horcuxes and their magical consequences.

Though it is not directly stated within any of the Harry Potter books, one example of a consequence when creating a Horcrux is becoming 'less than human,' both physically and emotionally. Being that the only known person within the Wizarding World to create more than one Horcrux is Lord Voldemort, it is easily noticeable to observe the changing human nature of Tom Marvolo Riddle into Lord Voldemort.

"Voldemort had entered the room, [...] yet he was no longer handsome Tom Riddle. It was though his features had been burned and blurred; they were waxy and oddly distorted, and the whites of his eyes now had a permanently bloody look [...]" -HBP-Chapter 20-Pg. 441

If Dumbledore did create a Horcrux, though there is no observable physical change (possibly being because he only created a single Horcrux), there would still be the emotional aspect that would change, and this change, no matter how minute, would be deemed as being 'less than human.'

Now, the ultimate question regarding the theory of Dumbledore's Horcrux : What is Dumbledore's Horcrux? There have been many theories stating it would be Fawkes the Phoenix. But I am not supporting this. After long and hard thinking I got a different view and we will work with my answer instead:

The Elder Wand!

the Elder Wand is to be considered as the Horcrux of Dumbledore (that is, if you have read this article up to this point), then two questions come to mind: Why would Dumbledore choose the Elder Wand as his Horcrux? as well as, What significant proof is there that the Elder Wand is indeed the Horcrux of Albus Dumbledore?

In response to the first question, one plausible explanation would be that at the timing of winning the Elder Wand from Grindelwald, Dumbledore thought that the wand would no longer have any more 'masters' over it. Dumbledore, as did probably most (if not all) of the Wizarding population at the time thought that after the defeat of the Dark Wizard Grindelwald, there would not be any more Dark Wizards that would attempt to take over the Wizarding World. Even upon meeting Tom Riddle, Dumbledore did not know what he would one day become.

"Did I know that I had just met the most Dark wizard of all time? No, I had no idea that he was to grow up to be what he is." -Albus Dumbledore (HBP-Chapter 13-Pg.276)

Another plausible explanation to add in response to the question would be the intention for which the Elder Wand would be used by Dumbledore. When he won the wand, Dumbledore then defined his use of it.

"I was fit to own the Elder Wand, and not to boast of it, and not to kill with it. I was permitted to tame and use it, because I took it, not for gain, but to save others from it." -Albus Dumbledore (DH-Chapter 35-Pg.720)

By using the wand to 'save others from it,' Dumbledore is portraying the idea that no other person will be the 'master' of the Elder Wand, thus causing the power of the wand to be broken. Dumbledore even appreciates this aspect when Harry Potter says to the portrait of Dumbledore in the Headmaster's / Headmistress' Office his intentions for the Elder Wand.

"'If I die a natural death like Ignotus, [the Elder Wand], its power will be broken, won't it? The previous master will never have been defeated. That'll be the end of it.'

Dumbledore nodded. They smiled at each other." -Harry Potter (DH-Chapter 36-Pg749)

With the Elder Wand losing its power, no person would seek to control the wand again, and as it is custom to bury one's wand with the deceased, no one would ever have to be in contact with the Horcrux ever again.

This way of thinking would not last once Lord Voldemort came to power and successfully retrieved the Elder Wand from the tomb of Dumbledore. However, it is because of Lord Voldemort and his search for the Elder Wand that offers a possible answer, specifically by 'killing' Harry Potter with the Elder Wand, to the question of what significant proof there is to suggest that Dumbledore did create a Horcrux using the Elder Wand.

King's Cross, Chapter 35 of Deathly Hallows, is the chapter that J.K. Rowling herself was waiting for so many years to write just so that she could write one sentence...

"Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real?" -Albus Dumbledore (DH,Chapter 35,Pg.723)

Individuals, such as John Granger and Gregory Bassham, contributors to the book The Ultimate Harry Potter and Philosophy, have commented on this chapter as Harry Potter being in 'limbo' when he meets Albus Dumbledore. The term 'limbo,' in this context, is used to describe the existence of Harry between the 'real world' and the 'spirit world,' or the afterlife. The use of the word 'limbo' is a well used word to describe the situation in which Harry resides in that particular moment.

However, rather than Dumbledore coming from the afterlife to 'greet Harry as an old friend and together depart from this life as equals,' what if the experience of Harry was him encountering the soul of Dumbledore that had been kept within the Elder Wand, and was then transferred to Harry when Lord Voldemort cast the Killing Curse at Harry?

While Harry is in 'limbo,' Dumbledore reveals everything regarding his mysterious past, the errors of an old man, and how Harry is the 'true master of death.' What Dumbledore reveals to Harry is information that Harry could never have known already up to this point, meaning that Harry received this information from Dumbledore himself, or at least his memories. Yes, Dumbledore is dead for almost a year at this point, but not his soul. 

In a paper written by Gregory Bassham, entitled Harry Potter and the Metaphysics of Soul-Splitting, Bassham makes a connection between the soul and memories, specifically those of Lord Voldemort and Albus Dumbledore.

"...Voldemort survives the death of his body because his memories and personality survive intact. Likewise, the postmortem Dumbledore that Harry encounters in the limbo-like King's Cross Station (DH p. 707) is still Dumbledore because the requisite psychological continuity [(memory)] is preserved." -Gregory Bassham (Reason Papers Vol. 34, no. 1)

The idea of souls preserving memories is an explanation of how Dumbledore is able to recant to Harry his thoughts during the King's Cross chapter in Deathly Hallows. But if Dumbledore created a Horcrux within the Elder Wand, and created it prior to the rise of Lord Voldemort, then there must exist a connection between parts of souls that exist outside the body within Horcruxes and the part of the soul that still remains within the body of the living person. The evidence within the Harry Potter books to support this claim resides in observing the knowledge obtained by Lord Voldemort between his demise in 1981 and his 'rebirth' in 1995.

In Chamber of Secrets, Chapter 17, The Heir of Slytherin, Tom Riddle asks how Harry Potter, as a baby, managed to destroy the powers of Lord Voldemort. In response, Harry answers:

"No one knows why you lost your powers when you attacked me. I don't know myself. But I know why you couldn't kill me. Because my mother died to save me." -Harry Potter (CoS-Chapter 17-Pg.316)

Nearly two years later in the graveyard of Little Hangleton, when Harry Potter meets Lord Voldemort with his new body, Lord Voldemort tells the story to his Death Eaters of how he lost his powers the night he attacked Harry Potter. Something the dark lord did not know before the CoS.

"You all know that on the night I lost my powers and my body, I tried to kill [Harry Potter]. His mother died in the attempt to save him - and unwittingly provided him with a protection I admit I had not foreseen... I could not touch the boy." -Lord Voldemort (GoF-Chapter 33-Pg.652)

If Harry revealed to Tom Riddle in Chamber of Secrets how it was the sacrifice of his mother that saved him from Lord Voldemort, and in Goblet of Fire Lord Voldemort revealed to his Death Eaters this same exact information, then there exists evidence to support the idea that parts of the same soul are still connected. This is why, Harry is able to encounter Dumbledore in his 'limbo-like' state and learn of the memories of his former Headmaster as a result of Albus Dumbledore creating a Horcrux within the Elder Wand.

Hopefully, the evidence that I have presented is enough to consider the possibility that Albus Dumbledore may have created a Horcrux using the Elder Wand. Let me know what you think!