Projection Theorems for $\Phi$-Intermediate Dimensions

$\Phi$-intermediate dimensions interpolate between Hausdorff and box-counting dimensions by restricting admissible coverings to scale windows of the form $[\Phi(r),r]$. Using a family of $\Phi$-dependent kernels, we develop a potential-theoretic framework that characterizes these dimensions in terms of capacities and leads to associated $\Phi$-dimension profiles. This framework provides effective tools for obtaining lower bounds from uniform potential estimates. As an application, we prove Marstrand--Mattila type projection theorems, showing that for $\gamma_{n,m}$-almost all $m$-dimensional subspaces $V$, the $\Phi$-intermediate dimensions of $\pi_V E$ coincide with deterministic profile values depending only on $E$ and $m$. We also discuss consequences for continuity at the Hausdorff end-point and for the box dimensions of typical projections.