Do note that the halting problem is fundamentally true, no AGI will realize some new way around, unless are mathematics are flawed to the core.
So we will probably never practically solve the halting for LBAs, but the quest for this AGI that should be able to solve this, is not just day dreaming, it's rooted in the theory.
An LBA is effectively "just" a Turing machine that has a finite tape.
A typical current computer is an LBA only if you disallow all IO of any sort or bound that IO and include it as part of the system you analyse and so fix the values which will be provided as IO, which of course is a very unusual situation, and so that constraint does not really make the halting problem more tractable in situations we usually care about.
If you can represent a program with an LBA, it is by definition a context sensitive language and decidable. You could also show it by making a equivalent gramma for the language, that accepts the same input as the program. This gramma must be constructed in a certain way, and then you know it is a context sensitive language.
We can certainly look for AGI that can do better at deciding the halting of decidable programs, but even for current computers the general halting problem is undecidable without adding artificial constraints.
So, no, you can not construct a machine for every conceivable input without imposing an artificial constraint on the input size.
Put another way: For every tape you construct and analyse, there is a tape one segment longer that might contain a symbol that can alter the outcome.
hailstone :: Integer -> Integer
hailstone n
| n `mod` 2 == 0 = n `div` 2
| otherwise = 3 * n + 1
collatz :: Integer -> Bool
collatz n
| hailstone n == 1 = True
| otherwise = collatz (hailstone n)
But yeah, it is not even trivial to say whether it has a bounded max memory, so in case of an arbitrary precision int type, it may not be LBA, but Turing?
We can actually do it in O(1) space complexity in exchange for higher time complexity.
For i in range 1..n, compute n_i, the ith number in the hailstone iteration, and then continue the hailstone iteration n_(i+1)..n_max to see if n_i is equal to any of them. This takes us to quadratic time complexity, O(n * n_max), or constant complexity depending on how you look at it, but it only requires storing a single n_i at a time for cycle detection.
But then again, if you actually loop for n_max iterations without halting by reaching 1 or crashing, then you had to reuse a number somewhere, so the explicit cycle detection isn’t really important.
Why would it be extremely useful?
Humans are also limited by the Turing model’s limits, we can only ever determine computable functions as well. With all due respect, it is stupid to assign more capabilities to ML than what we know is fundamentally the limit..
Of course ML has use cases where traditional tools are less fit, my gripe is the hype-based anti intellectual nonsense that often surrounds it. They are no magic tools, the fundamental limits these giants of math/CS discovered still apply to them and we can save ourselves from a lot of pain if we don’t bother solving unsolvable problems.
The bitter lesson is that Messy AI is better able to cope with Messy World Problems than Neat AI (by light-years at this point), not that it can hack Neat Problems.