Capitalize new construction introduced in mathematical proof

Let’s say that I am trying to prove a mathematical theorem. As part of that proof, I may want to introduce a construction that I will use repeatedly throughout the proof (and potentially combine many instantiations of the same construct). To that end, let’s say that I name the construct Named Thing. I may say something like the following:

We will create a Named Thing. A Named Thing has the following structure etc.

Should Named Thing be capitalized in both cases here? It’s used somewhat like a proper noun here, although it’s referring to a class of objects, as opposed to referring to a single specific object from this class.

Furthermore, let’s now consider the following sentence:

We construct named things for each of the five cases.

Here I believe it should be lower case, as its usage is (I believe) as a common noun. Is that correct?