How can Turing complete machines exist theoretically if the halting problem is undecidable As the question says, if I input on the tape of a Turing complete machine a program that solves the halting problem with the correct inputs the program will never end its execution regardless of memory and time. Isn't the halting problem a computational problem that can't be executed by a Turing complete machine so that it's halts sometime?
The halting problem is the problem of deciding whether a given Turing machine halts on a given input or not.

This problem is undecidable, meaning that no program running on a Turing machine can solve this problem for all possible inputs. Hence, your assumption that such a program exists is wrong.