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alyssavance comments on The Graviton as Aether - Less Wrong
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Is there a single example of this that you can think of? There are the different ways of computing classical mechanics (Newtonian, Lagrangian, Hamiltonian), but these were known to be just different ways of doing the same math at the time of discovery.
The original formulations of QM were famously shown to be equivalent, though I'm not sure they were ever expected to be incompatible. QFT and S-matrix theory were originally politically opposed (though QFT produces an S-matrix), but I have heard that recently people advocate widening QFT to the point that it appears to cover all of S-matrix theory.
No, it's just a theoretical property of the 'A, B, X' abstraction that you mention. In fact, it would not surprise me if the general problem of proving whether or not two theories are exactly equivalent is intractable in a similar way to the halting problem.