> If after these steps the program hasn’t halted it means it’ll never halt (by the definition of the Sigma(n)) This argument doesn't quite work since Sigma(n)

> If after these steps the program hasn’t halted it means it’ll never halt (by the definition of the Sigma(n)) This argument doesn't quite work since Sigma(n) is the maximum number of characters printed before halting rather than the maximum finite number of steps of computation. The argument would work for the other flavor of Busy Beaver though.