No, there is no way of proving that a hash exists other than actually finding that hash. (i.e. brute force). The existence of a method of being able to predict the result of a hash other than computing the hash would render the hash function cryptographically unsound. (And so we can infer that no mathematician has found such a method for SHA-256).

That said, as discussed in this other Q&A, there are a number of characteristics of a block that are dynamic so it isn't as if an "insolvable" block would actually cause any network problems.

No, there is no way of proving that a hash exists other than actually finding that hash. (i.e. brute force). The existence of a method of being able to predict the result of a hash other than computing the hash would render the hash function cryptographically unsound. (And so we can infer that no mathematician has found such a method for SHA-256).

That said, as discussed in this other Q&A, there are a number of characteristics of a block that are dynamic so it isn't as if an "insolvable" block would actually cause any network problems.