The anisotropic local law for sample covariance matrices under quadratic-form concentration

We study sample covariance matrices $K=\frac{1}{N}{\sum }_{i=1}^N{\text{x}}_i{\text{x}}_i^{\ast }\in {\mathbb{R}}^{n\times n}$ in the proportional regime $n\asymp N$. The columns ${\text{x}}_1,\dots ,{\text{x}}_N\in {\mathbb{R}}^n$ are independent and centered, with common covariance $\text{E}{\text{x}}_i{\text{x}}_i^{\ast }=\mathrm{\Sigma }$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $\mathrm{\mid }{\text{x}}_i^{\ast }A{\text{x}}_i-\text{Tr}\mathrm{\Sigma }A\mathrm{\mid }\prec \mathrm{\parallel }A{\mathrm{\parallel }}_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\mathrm{\Sigma }$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta :=\mathrm{\Im }z\ge N^{-1+\tau }$, $$\mid \text{<}\breve{,}((K-z{)}^{-1}-(-z{I}_n-z{\tilde{m}}_0(z)\mathrm{\Sigma }{)}^{-1}\text{bv}\mid \prec \sqrt{\frac{\mathrm{\Im }{\tilde{m}}_0(z)}{N\eta }}+\frac{1}{N\eta }$$ for all deterministic unit vectors $\breve{,}\text{bv}\in {\text{C}}^n$, where ${\tilde{m}}_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.