Turing completeness and P completeness are completely different things. There is no sense in which P-completeness is a "more specific" version of Turing-completeness.
My understanding of this is that P-completeness for a problem implies that any problem in P can be transformed into it with a polynomial-time reduction. Deterministic Turing machines (more precisely, the problem of determining the future state of a deterministic Turing machine) are in P.
Related How to Build an Origami Computer (63 points, 2024, 15 comments) https://news.ycombinator.com/item?id=39191627
Honestly wild how you can get Turing completeness outta folding paper, never thought I'd read that today.
That's why I have always prefered Church approach to computation to Turing machines.
The lambda calculus, by its simplicity as just a rewriting language, makes it "obvious" how effective computability emerges from very little.
> we prove that flat origami, when viewed as a computational device, is Turing complete, or more specifically P-complete
...aren't those mutually exclusive?
I feel a mix of "those are obviously different complexity levels" and "is it like C pre-processor turing-completeness situation?"
Turing completeness and P completeness are completely different things. There is no sense in which P-completeness is a "more specific" version of Turing-completeness.
My understanding of this is that P-completeness for a problem implies that any problem in P can be transformed into it with a polynomial-time reduction. Deterministic Turing machines (more precisely, the problem of determining the future state of a deterministic Turing machine) are in P.