Gaussian equivalent model conjecture for generic feature maps

Let $\{\boldsymbol{x}^{\mu}\}_{\mu=1}^{n}$ be independent samples from a data distribution on $\mathcal{X}\subset\mathbb{R}^{D}$, and let the centred teacher and student features be

$$\boldsymbol{u}=\boldsymbol{\varphi}_{t}(\boldsymbol{x})\in\mathbb{R}^{p},\qquad \boldsymbol{v}=\boldsymbol{\varphi}_{s}(\boldsymbol{x})\in\mathbb{R}^{d},$$

for feature maps $\boldsymbol{\varphi}_{t}:\mathcal{X}\to\mathbb{R}^{p}$ and $\boldsymbol{\varphi}_{s}:\mathcal{X}\to\mathbb{R}^{d}$. Gaussian equivalent model conjecture. For a wide class of data distributions and feature maps, the generalisation and training errors of the estimator defined in the source are asymptotically captured by the equivalent Gaussian covariate model in which $[\boldsymbol{u},\boldsymbol{v}]$ are jointly Gaussian variables, and hence by the corresponding closed-form expressions of the stated theorem. The conjecture proposes that this Gaussian replacement captures learning behaviour beyond the exactly Gaussian cases, including deterministic or data-learned feature maps; the source gives no resolution, so its validity remains open.