Turing kicked us out of heaven
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That's just wrong. `while true {}` never halts even with finite memory.
Which, by the way, isn't true if you take into account the fact that a computer may interface with an external, potentially infinite, memory device - such as a memory tape reader-writer, together with a human who responds to the "INSERT NEXT TAPE TO CONTINUE" messages by going out to the store and buying more tape.
The whole "acktshually computer is just a finite state machine" is the most annoying "gotcha!" I've ever heard from computer scientists. There's a reason we invented virtual memory and file systems - it's so we can treat real computers like Turing Machines, and increase the memory when needed without having to re-write our whole systems every time.
That's not recursively enumerable, that's Π₂. And ABC conjecture looks even harder.
the halting problem is either truly solvable, or there is an actual proof which doesn’t rely on weak unconstructable paradox.
At least TFA didn’t regurgitate the bad proof and accept it. However, they did worse, they accepted the bad proof without even attempting to regurgitate it.
There is a real insight in your comment though. There is a weaker version of the halting problem, where the goal is to write a does_halt function that returns yes/no/maybe. That problem is trivially solvable. There are also non-trivial solutions to it that are actually useful.
Do you have any proof of this statement?
* Do you also reject Church's more formal proof?
* Do you more generally reject the rule of the excluded middle and would you generally think of yourself as a Constructivist?