Theoretical notion of white-box Is there any theoretically safe notion of white-box?

I have been reading about white-box [1, 2, 3] recently. It seems white-box deals with code obfuscation and its not known whether a perfect white-box exists.
I am curios about theoretical notion behind this. For a stream cipher, the theoretical notion is PRF. For block cipher it is PRP. For a hash function, it is one-way function. Is there anything comparable for white-box?
I am looking for something which is not necessarily practical, but we can actually show this achieves perfect white-box, like a really scrambled code or something.
UPDATE. I vaguely remember reading something on the line of

White-box would be possible if a super-huge memory exists. The memory could be loaded as a look-up table with the plaintexts as the look-up-keys and the ciphertexts the look-up-values. Since the encryption-key is not involved in the look-up table (it has been used previously to create the look-up table), it cannot be recovered but encryption would be still possible.

But I could not retrieve the source. Does anybody know anything about it?
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!