During Relius' arcade mode in BlazBlue: Continuum Shift Extend, if Bang Shishigami is fought and defeated, Relius will comment "Truly… Staggering. But that thing you carry… Is that the lynchpin…!? "This implies that Bang's nail, Phoenix Wing: Rettenjō, that was inherited from his master Tenjō, is the lynchpin.
It is meant to be powered by Celica Ayatsuki Mercury (and would have killed her), and as of BlazBlue: Phase 0, it was in Shūichirō's old laboratory. It went missing at the beginning of 2107, the UN, OLN and Magic Association initiated separate investigations in order to retrieve it. The OLN was also observing Celica, as she was mentioned in Ayatsuki's reports. The Lynchpin was actually retrieved by Nine. It was hidden and sealed deep below her workshop, that was located in the basement of her house in Ishana. Four months before the Black Beast's reappearance, Nine unsealed it in order to borrow it's power. 
Interestingly enough, before removal of the core (which was turned into Rettenjo), the Lynchpin was described as being silver, with lines emanating green light.
As the Lynchpin requires a constant, controlled energy source – which would work where electricity and magic doesn't – the only viable option is a living Soul. What's more, a simple soul isn't enough; only a soul of someone adept at controlling the flow of life can be used as its key. The more adept a person is at it, the more effective and lasting its results would be.
After the soul is merged with the core, it is being constantly drained of its life. After the two bodies are merged, they will never come apart again. The person themselves remain conscious, alone, slowly watching as one's life is being eaten away. After the soul is consumed, Kushinada turns itself off, and the world is again filled with magic element.
Tenjō Amanohokosaka's soul being compatible with the Lynchpin would indicate that she herself had at least minor powers regarding control over the flow of life. As the extent of her power is unknown, it can't be specified how long it will remain active.