One Turing machine can simulate the machine described by part of its input. I want the notation, the mechanism, and the failure mode visible at the same time.
The mathematical object
A decision problem is computable when a Turing machine \(M_e(x)\) halts on every input with the correct answer. A set \(A\subseteq\mathbf N\) is computably enumerable when a machine can list its members.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
One explicit computation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Why the identity matters
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
Where it can fail
Enumerability is weaker than decidability. A search may confirm membership eventually without ever certifying nonmembership.
I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.