Modern deep learning has revealed a surprising statistical phenomenon known as benign overfitting, with high-dimensional linear regression being a prominent example. This paper contributes to ongoing research on the ordinary least squares (OLS) interpolator, focusing on the partial regression setting, where only a subset of coefficients is implicitly regularized. On the algebraic front, we extend Cochran's formula and the leave-one-out residual formula for the partial regularization framework. On the stochastic front, we leverage our algebraic results to design several homoskedastic variance estimators under the Gauss-Markov model. These estimators serve as a basis for conducting statistical inference, albeit with slight conservatism in their performance. Through simulations, we study the finite-sample properties of these variance estimators across various generative models.