Previously asked and bountied at MSE without success:
Let $\sim$ denote the "no-disagreement" relation between partial functions: $f\sim g$ iff there is no $x$ such that $f(x)$ and $g(x)$ are each defined and are distinct. For a differential field $\mathcal{K}=(K;0,1,+,\cdot,\partial)$, let $C(\mathcal{K})=\{f\in K: \partial f=0\}$ be the constant subfield.
Say that a differential field $\mathcal{K}$ is fully concrete iff there is some way to assign, to each $\alpha\in K$, a partial function $\pi_\alpha:C(\mathcal{K})\rightarrow C(\mathcal{K})$ such that:
For $\gamma\in C(\mathcal{K})$, $\pi_\gamma$ is the constant function $\lambda x. \gamma$.
$\pi$ is injective-up-to-$\sim$, in the sense that $\pi_\alpha\not\sim\pi_\beta$ whenever $\alpha\not=\beta$.
$\mathcal{K}$ is closed-up-to-$\sim$ under composition, in the sense that for each $\alpha,\beta\in K$ there is a unique $\gamma\in K$ such that $\pi_\gamma\sim\pi_\alpha\circ\pi_\beta$. (Since the functions involved are partial, this falls short of $\pi_\gamma=\pi_\alpha\circ\pi_\beta$.) I'll write "$\alpha\circ\beta$" for this unique $\gamma$.
The field operations commute-up-to-$\sim$ with $\pi$, in the sense that $$\lambda x.\pi_{\alpha\star\beta}(x)\sim\lambda x.[\pi_\alpha(x)\star\pi_\beta(x)]$$ for $\star\in\{+,\cdot\}$.
The chain rule holds, in the sense that $\partial(\alpha\circ\beta)=[(\partial\alpha)\circ\beta]\cdot\partial\beta$.
Finally, say that a differential field is concrete iff it is a sub-differential field of a fully concrete differential field with the same constant subfield. (To motivate concreteness as opposed to full concreteness, consider the example - pointed out to me by Qiaochu Yuan - of $\mathbb{C}(f)$ equipped with the obvious derivation satisfying $\partial(f)=f$. Clearly this differential field should be concrete via $f\mapsto \lambda x.e^x$, but we don't have closure under composition within $\mathbb{C}(f)$ itself.)
Concreteness seems like a natural property to consider for differential fields, but I haven't run into it before (I'm very new to differential fields though) and it's not obvious to me how common or rare it is. To get things started:
Suppose $\mathcal{K}$ is a differential field with the same cardinality as its constant subfield. Must $\mathcal{K}$ be concrete?
I'm also generally interested in any sources about concreteness, or any variant thereof - generally, when can a differential field be "faithfully" thought of as consisting of partial functions over its constant subfield?
EDIT: The most relevant result I've found so far is the embedding theorem from Singer's The model theory of ordered differential fields. However, it is at least in statement more specific than what I'm looking for, and it's not clear to me that the argument there can be applied to this more general question.
