I am quite certain that I am not the first to entertain the idea that I am going to present. However, it would be helpful if I can find any literature related to the idea.
The idea is to construct a Turing Machine M with the property that if P=NP then M will solve 3-SAT in polynomial time. (The choice of 3-SAT is arbitrary. It could be really any problem in NP).
Just to be clear, this is not a claim that P=NP. In fact, I believe the opposite. I merely state that if P=NP, then M will provide a polynomial-time solution. If you are looking for an efficient solution, I should warn that this is far from efficient.
M is constructed as follows: first, assume a canonical encoding for all Turing Machines, and apply a numbering to these machines. So, there is a Turing Machine number 1, a number 2, etc. The idea of a Universal Turing Machine that can read the format for a provided machine and then simulate that machine's running on separate input is pretty well known. M will employ a Universal Turing Machine to construct and simulate each Turing Machine in turn.
It first simulates the running of Turing Machine 1 for a single step.
It then looks at the output of Turing Machine 1.
It the simulates the running of Turing Machine 1 for two steps and looks at output, then proceeds to simulate Turing Machine 2 for 2 steps. It continues and loops in this fashion, in turn running Turing Machine 1 for k steps then 2 for k steps ... then eventually machine k for k steps.
After each simulation run, it examines the output of the run. If the output is an assignment of variables satisfying the 3-SAT problem instance, M halts in an accept state. If on the other hand, the output is a proof-string in some verifiable proof-language with the proven result that the problem instance is not satisfiable, M halts in a reject state. (For a proof-language, we could for example, use the Peano Axioms with Second-Order logic and the basic Hilbert-style logical axioms. I leave it as an exercise for the reader to figure out that if P=NP, a valid proof-language exists and is polynomial-time verifiable).
I will claim here that M will solve 3-SAT in polynomial time if and only if P=NP. Eventually, the algorithm will find some magical Turing Machine with number K, which just so happens to be an efficient solver for the 3-SAT problem, and is able to provide a proof of its results for either success or failure. K will eventually be simulated running poly(strlen(input)) steps for some polynomial. The polynomial for M is roughly the square of the polynomial for k in the largest factor, but with some terrible constants in the polynomial.
To reiterate my question here: I want to know if there is a literature source that employs this idea. I am somewhat less interested in discussing the idea itself.