1. Checking boundary conditions is currently done (and should be) with machines (experimental mathematics).
2. When referring to a boundary condition, we often mean the one stemming from well-ordering of the number system (1) or ones coming from the task itself (2). (1) is easily checked with a machine. (2) is also rel. manageable using a machine.
3. More important (for large parameter spaces) boundary condition should, however, arise not 1 and 2, but from the logic of the task. As soon as we treat the domain of definition we are working on as a dataset, and the logic of the task as the inner structure of elements of the solution, we should be able to select the right data model to describe the data in the most effective manner.
4. Given that we have model selection done, we can transform the data exploiting the relevant model, and then we gonna be able to see the boundary conditions again.
5. So, the very boundary conditions would be connected to the data set in question, which implies that finding the good data descriptor could allow us to transform data and then have an easier job doing the boundary condition analysis.
Lets take FLT as an example. Given the fact that a common divisor among in would imply turning the task into a smaller task of the same kind, it is definitely informative. Beal conjecture exploits that by hazarding a guess about a more general case. From this example we also learn one other thing, i.e. we learn that being able to solve the problem by solving the same problem for different variables is extremely powerful.
So, now the big question. Does there for every task with variables and constraints , for exist a function s.t. the task with variables and constraints can be shown in recursive form?