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Translations: Bleach 600 (2)
I've created a profile for White that sums up my theory of what he is, and where he fits into the storyline concerning Ichigo's powers .etc. Please comment and add any more ideas!!! Note that not all of this has been backup up by evidence (yet), and is pure theory.
"White" is a special being created by Aizen, Tousen and Gin that has the ability to create copies of itself by infecting hosts, behaving in a similar manner to a virus. White also possesses the ability to take the core powers of those he possesses and incorporate them into himself, allowing him to continuously evolve. It's possible that White was either derived from or IS the "prototype" that was mentioned in the TBTP arc.
White creates copies of himself similar to a virus, but instead of altering the being's DNA, he alters their core powers to match his own. This consequently results in the being's Zanpakutou and Hollow form also matching those of White. White may also retain some of the host's original core powers if they are beneficial.
Once infected, a host's core powers are manipulated by the infection, causing the host's core powers, Hollow form, and Zanpakutou to slowly transform into those of White. It is unknown if it is necessary, but it seems that the host is forced to transform into a Hollow during the final stages of the infection. The resulting White may then go on to infect another host by selecting a target and exploding in their vicinity. White may also infect more than one host when he self-destructs.
Although it didn't originally seem to be intended, White may also progagate if the host has offspring, as seen in the case of Ichigo. In this case, White will start to transform the host's core powers before they are even born.
The first appearance of White in Bleach was approximately 20 years before the current timeline. The first host's infection had advanced to the point that they had been forcibly Hollowfied and were about to undergo their final transformation (according to Aizen). This White's Hollow form wore a distinctive horned mask, from which it could fire Ceros, as well as black armour that covered nearly all of its body. This Hollow also appeared humanoid, and its arms were blade-like. Interestingly (and understandibly), the first host shared many similarities with Ichigo, in that it wore a similar-looking Hollow mask, was clad in black, it possessed incredible speed, and its arms looked similar to Ichigo's Zanpakutou in its most recent Bankai form. This White attacked Isshinn and Masaki, and although it targeted the latter, it managed to catch both in its explosion and infect them.
Although little is known about Masaki's infection, it seems as though White saw Ishinn's Engetsu as beneficial, and adopted his core powers into himself. Later on, Masaki and Isshin both became the parents of Ichigo, who also contracted the infection.
Isshin's infection, carrying Engetsu's powers, was transferred to the fourth host, Ichigo, when he was conceived, thus granting him with core powers similar to his father (as well as White's other acquired traits, including his mask, immense speed, Zanpakutou's shape, and elements of his black armour). Throughout the series, Ichigo's infection has slowly been transforming him into White, altering his Hollow and Zanpakutou forms to bear similarities to those of the first host. Interestingly, Ichigo's orginal Hollow form (when he first Hollowfied) was more Lizard-like, and may hint towards what Ichigo's core powers would have been had he not contracted the White infection.
The first host that has been shown so far was possibly not the first altogether, and could have easily been a host that was infected by another White. Unfortunately, it's hard to imagine what the original White would have looked like before it started to acquire and combine the abilities of the various hosts that it infected. It may be that the original White is in fact the "prototype" that was mentioned in the TBTP arc. This being would have most likely carried the sole ability to target a host and explode to infect them (after which it would have started to adopt the powers of its hosts into its own form).
Interestingly, if White does have the ability to copy the core powers of its host in order to become more powerful, then it will eventually become the ultimate being (in both Shinigami and Hollow forms)!!! This would explain why such a being was created in the first place, as it was Aizen's goal to do this.
Oh, one interesting point to make is that White seems to share the same method of infection of other beings as Metastacia does (the Hollow who infected Kaien)...Perhaps Metastacia was actually a prototype of White???
^ It wasn't an Arrancar from what I remember, so, I don't think it was an artificially created Hollow.