Of course, it's also possible that they discussed resolving the equity when they split but never put it into writing...
Of course, it's also possible that they discussed resolving the equity when they split but never put it into writing...
Elsewhere in this thread, someone found that the company was incorporated in September 2013.
So a possible timeline is that Kyle creates the company with himself as the sole owner first, and goes looking for a technical partner. He finds Jeremy, they talk, they hit it off, and they decide to apply to YC together, and Kyle offers to split the existing company 50/50. But they don't put anything in writing, so on paper the company is still 100% owned by Kyle. Things go sour between them, and Jeremy leaves/is kicked out without the ownership of the company ever changing on paper. And then things move on, other people are brought into the company, shares and vesting schedules are formalized, and the whole thing is forgotten/buried. Until now.
One of the major questions is whether the YC application and any stated intent for a 50/50 split in it would be that writing.
Also, if there is sufficient evidence to declare that the 50/50 partnership existed, then Jeremy did not have to do anything to explicitly declare or let Kyle know that he (Jeremy) understood himself to still possess a 50% stake. There would be no such thing as an "implicit acceptance" of losing equity.
Basically the points you raise are not at all obviously addressable, and there are many ways they could play out that actually do support a significant award to Jeremy, regardless of whether that is a popular outcome.