I'll give a bounty of 500 rep to any answer which successfully proves the Turing completeness of their answer to this challenge.

I like the idea of the challenge, but I feel it's extremely hard to make a language that isn't gimmicky. I can't wait to see what others answer.

@ConorO'Brien Gimmicky answers are fine (most esolangs have a gimmick) but they should not be obvious patches of existing language like the language jokingly suggested by CalculatorFeline.

If a language can only solve problems in P, would it be TC iff P = NP?

@Challenger5 I don't think so. That seems to imply that it will always terminate which is certainly not true of a Turing Machine.

@VoteToClose Assuming, of course, that the answer is valid, right? Because if not... I could do with 500 more rep :P

Brainfuck, but memory locations are only valid if they are positive and the collatz sequence of its index ends at 1.

@Challenger5 Yes. :P

I have a design that I like, but it will be a while before I will be at a computer that I could write the parser on. Would it be alright to post the specification before the parser was complete? Or, do you want both at the same time?

My language goes against the spirit of the challenge so I'm posting it as a comment. It's a Turing machine but all the cells are labeled with a number. The initial cell is 0 and the labels increase as you move in one direction and decrease as you move in the other direction. The label does not affect the value of the cell. You can only read/write on cells whose label has an absolute value equal to a Mersenne prime. This language is only Turing complete if there are an infinite number of Mersenne primes, which is currently unknown.

I think the simplest possible language (or at least one of the simplest) that fulfills this criterion is a BF variant where the only valid memory positions are integers n where |n| and |n|+2 are both prime. That goes against the spirit of the challenge, though, so I wouldn't suggest actually making it :P

Is the language required to be TC iff (if and only if) the conjecture is true, or is one way implication sufficient?

@trichoplax Iff is the requirement. I hadn't though to put it like that so I will update the question.

Does it have to possible for one to create a compiler/interpreter for the language without solving the conjecture?

@ghosts_in_the_code Yes, you must provide one.

So CalculatorFeline's answer is invalid. :p