Theoretical notion of white-box
Caution: White Box Cryptography is out of my comfort zone, and I have not been following the latest developments in the field. The following is my current opinion, which I present in hope of seeing it challenged, and learning in the process.

As I see it, the traditional goal of White Box Cryptography (and the one the industry would like most) is to design software that computes a standard public keyed cryptographic transformation (such as a keyed PRP, ideally AES) with a particular instance of the secret key embedded in the software, such that leaking the software does not leak the key.
I know no serious claim that this is even close to be achieved. On the contrary, WBC competitions that focused on the above goal with AES as the PRP have AFAIK all ended with quick key extraction.
With respect to that goal, it seems we do not even have practically secure WBC, and thus much less theoretically secure WBC. Contrast with the better situation in symmetric crypto, where we have practically secure PRPs and PRFs, but arguably no theoretically secure construction from first principles.

Yet a variant of WBC as defined above is practically feasible if we allow to construct the keyed transformation with WBC in mind: for example it's trivial to make a practically secure WBC implementation of the AES-256 lookalike WES-256, defined as:
$$\begin{align}
\text{WES-256}:\quad&amp;\{0,1\}^{256}\times\{0,1\}^{128}\to\{0,1\}^{128}\\
&amp;P\mapsto\text{WES-256}(K,P)\underset{\text{def}}=\text{AES-256}(\text{SHA-256}(K),P)
\end{align}$$
If we take that variant definition, security of WBC follows from that of more traditional constructs.

My conclusion is that I know no good definition of the theoretical goal of WBC, thus can't answer the question!