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Ruling out Method X is generally much harder than Method X itself, so one shouldn't necessarily expect that for any method that won't work a "certificate" can be found establishing its insufficiency. With that said:

There are plenty of environments (rings) where the algebraic equation in FLT makes sense, but has solutions. If the ingredients of Method X apply in one such environment, they aren't using specific enough properties of the integers to prove FLT. It is well known that if Method X = congruences (reduction modulo $t$ for different values of $t$), then this argument works in rings of $p$-adic numbers where FLT has solutions. Similarly, if Method X = inequalities, then it would have to rule out the positive real solutions of $x^n + y^n = z^n$.

Another possibility, requiring much more knowledge, would be to use the close relations between FLT and elliptic curves (Frey, Wiles, etc) to project method X onto the canvas of Wiles' proof and see how much of it can be understood and delimited in those terms. It could be that X is in effect trying to construct nontrivial objects of a certain kind (esp. cohomology classes) and one can see in light of Wiles methods that the relevant classes are zero, necessary field extensions or coverings are trivial, important obstructions are not killed, etc. This approach stays within environments where FLT is true, but tries to subsume Method X within existing lines of attack, and show that it only includes a part of what is needed.

Another idea is "reverse mathematics", to represent Method X as formal derivations in some weak system Y of arithmetic, and show that FLT is of higher proof theoretic strength (because it implies all the theorems of an even stronger system Z). This is unlikely because FLT is too specialized a statement and the proof-theoretic calibrations of strength only work for reasonably generic theorems with a lot of parameters that can be varied.

EDIT: I should add that each of these approaches has been pursued for showing that the P=NP problem doesn't have elementary solutions. The Baker-Gill-Solovay theorem demonstrated that environments (oracles) exist where P=NP has a different answer but simple methods of proof would continue to work. The Razborov-Rudich "natural proofs" paper showed that any proof sharing certain features of all the arguments then known (for proving lower bounds on circuit complexity) couldn't produce bounds growing faster than any polynomial. And there are weak formal systems whose most general class of definable languages or constructible functions is exactly P or NP; Stephen Cook himself has many papers on the logic/formal-systems approach to P=NP.

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Ruling out Method X is generally much harder than Method X itself, so one shouldn't necessarily expect that for any method that won't work a "certificate" can be found establishing its insufficiency. With that said:

There are plenty of environments (rings) where the algebraic equation in FLT makes sense, but has solutions. If the ingredients of Method X apply in one such environment, they aren't using specific enough properties of the integers to prove FLT. It is well known that if Method X = congruences (reduction modulo $t$ for different values of $t$), then this argument works in rings of $p$-adic numbers where FLT has solutions. Similarly, if Method X = inequalities, then it would have to rule out the positive real solutions of $x^n + y^n = z^n$.

Another possibility, requiring much more knowledge, would be to use the close relations between FLT and elliptic curves (Frey, Wiles, etc) to project method X onto the canvas of Wiles' proof and see how much of it can be understood and delimited in those terms. It could be that X is in effect trying to construct nontrivial objects of a certain kind (esp. cohomology classes) and one can see in light of Wiles methods that the relevant classes are zero, necessary field extensions or coverings are trivial, important obstructions are not killed, etc. In this This approach we work in stays within environments where FLT is true, but try tries to subsume Method X within existing approacheslines of attack, and show that it only attains includes a part of what is needed.

Another idea is "reverse mathematics", to represent Method X as formal derivations in some weak system Y of arithmetic, and show that FLT is of higher proof theoretic strength (because it implies all the theorems of an even stronger system Z). This is unlikely because FLT is too specialized a statement and the proof-theoretic calibrations of strength only work for reasonably generic theorems with a lot of parameters that can be varied.

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Ruling out Method X is generally much harder than Method X itself, so one shouldn't necessarily expect that for any method that won't work a "certificate" can be found establishing its insufficiency. With that said:

There are plenty of environments (rings) where the algebraic equation in FLT makes sense, but has solutions. If the ingredients of Method X apply in one such environment, they aren't using specific enough properties of the integers to prove FLT. It is well known that if Method X = congruences (reduction modulo $t$ for different values of $t$), then this argument works in rings of $p$-adic numbers where FLT has solutions. Similarly, if Method X = inequalities, then it would have to rule out the positive real solutions of $x^n + y^n = z^n$.

Another possibility, requiring much more knowledge, would be to use the close relations between FLT and elliptic curves (Frey, Wiles, etc) to project method X onto the canvas of Wiles' proof and see how much of it can be understood and delimited in those terms. It could be that X is in effect trying to construct nontrivial objects of a certain kind (esp. cohomology classes) and one can see in light of Wiles methods that the relevant classes are zero, necessary field extensions or coverings are trivial, important obstructions are not killed, etc. In this approach we work in environments where FLT is true, but try to subsume Method X within existing approaches, and show that it only attains a part of what is needed.

Another idea is "reverse mathematics", to represent Method X as formal derivations in some weak system Y of arithmetic, and show that FLT is of higher proof theoretic strength (because it implies all the theorems of an even stronger system Z). This is unlikely because FLT is too specialized a statement and the proof-theoretic calibrations of strength only work for reasonably generic theorems with a lot of parameters that can be varied.