Is there a conjecture suggesting if some other conjecture is true for all $x<n$, then it's true for all $x$?

Question

So everybody here probably knows if numerical testings support a conjecture, that conjecture isn't necessarily true for larger numbers. In fact,there have been many times where a conjecture was proved to be wrong by large counterexamples. But is there any conjecture that suggests if a conjecture is true for all $x<n$ for some $n$, that conjecture holds for all values of $x$?

Please note that I'm not looking for just one specific conjecture that's proven to be true for all $x>n$ for some $n$ and we just need to prove it for all $x<n$ such as Goldbach's weak conjecture(I know it's been proved that it's true for all odd numbers but if I recall correctly, someone proved it's true after a very large number but we couldn't test all the odd numbers less than that number). I'm looking for something that could be applied to any conjecture(that could be numerically tested) or something that at least could be used for a large variety of conjectures.

Thanks in advance!

Answer 1

I doubt it. For every integer $N$ there are clearly conjectures that are true up to $N$ but false eventually. Most of them are dumb (like "every number is less than $2N$") but they do exist.

So your meta conjecture is really asking for a description of what you hope is "a large variety of conjectures" that need only be verified in finitely many instances. There are conjectures like that. For example, the four color theorem was proved by reducing it to the task of checking just finitely many cases (and that was done with a computer program). But I think deciding which conjectures can be so verified is not a well posed problem.

Answer 2

It's not clear what you mean by "numerically testable", but presumably it would contain at least the $\Pi_1$ statements which are formulas (of PA) that are logically equivalent to formulas of the form $\forall n.Q(n)$ where $Q$ is a formula that only uses bounded quantifiers, i.e. $\exists a < m$ or $\forall a < m$. Such statements can be computationally evaluated for each instantiation of the outermost universal quantifier. Your conjecture applied to $\Pi_1$ formulas would mean that every $\Pi_1$ formula is $\Sigma_1$ (which is the same thing just for existential quantifiers instead of universal quantifiers). Explicitly, you'd be saying that $$\forall n.Q(n)\iff\exists N.\forall n < N.Q(n)$$

We know neither of $\Sigma_1$ and $\Pi_1$ include the other, though they overlap in $\Delta_1$ formulas. So this conjecture is false, and any very generic version seems unlikely.

Nevertheless, we have decision procedures for large chunks of mathematics that often reduce to testing many cases. An example you are already familiar with is testing equality of polynomials. Two polynomials with coefficients in a field with characteristic $0$ (e.g. $\mathbb Q$) are equal iff they coincide on some finite set of points. The book A=B goes into several more complicated decision procedures which handle various recurrences and hypergeometric series which covers many combinatorial identities.