On the questionable use of CIE L * to infer geometric properties of achromatic perception

In the paper “The non-Riemannian nature of perceptual color space,” the authors analyze the data of visual experiments conducted using triad of CIE $L^{*}$ achromatic stimuli. The statistical analysis of their results leads them to claim that the $L^{*}$ axis cannot be endowed with a Riemannian metric coherent with observers' responses. In this communication we provide a mathematical analysis which, in our opinion, refutes the main claim of the quoted paper. Furthermore, we raise questions on the soundness of using an achromatic perceptual coordinate to describe non-chromatic perception. We inform about an alternative in which achromaticity emerges from the very act of observing colors (Provenzi E. SIAM J Imag Sci, 15(4):1944–1976, 2022; Berthier M, Provenzi E. Proc R Soc A, 478(2258):20210508, 2022).