Based on the manifold hypothesis, which states that data in a high-dimensional space reside in a low-dimensional submanifold, the diffusion model (DM) implicitly learns the local intrinsic dimension (LID) for each data point in this submanifold. Kamkari et al. (2024b) proposed FLIPD, which estimates the LID through the rate of change of the logarithmic neighborhood density of the DM. This paper formally proves the theoretical basis of FLIPD under realistic conditions and shows that similar results are obtained by replacing Gaussian convolution with uniform convolution.