Investigating the chemical link between H$_2$CO and CH$_3$OH within the CMZ of NGC 253

Formaldehyde (H$_2$CO) and methanol (CH$_3$OH) have served as traditional tracers of the star formation process for decades. Studies of the environments which produce these species, though, have pointed to significant differences in the physical environments within which each molecule resides. In this paper we investigate the physical and chemical conditions which give rise to formaldehyde and methanol emission in the nearby starburst galaxy NGC 253. We employ high spatial (1.$''$6 or $\sim28$ pc) and spectral ($\sim10$ km/s) imaging of the NGC 253 central molecular zone (CMZ) from the ALCHEMI Large Program to constrain radiative transfer models of the dense gas volume density, and temperature, molecular species column density, and source filling factor within eight giant molecular clouds (GMCs). We also measure the relative abundances of the two nuclear spin isomers of CH$_3$OH to investigate its formation history. The physical and chemical conditions derived clearly indicate that H$_2$CO and CH$_3$OH originate from distinct physical environments. H$_2$CO traces low volume density and high kinetic temperatures, while CH$_3$OH traces high volume density and low kinetic temperatures. The H$_2$CO abundances are constant, though poorly constrained, within the eight NGC 253 GMCs analyzed, while the CH$_3$OH abundance shows a radial gradient from low to high values within the NGC 253 CMZ. Our findings highlight the complex chemical and physical differentiation of CH$_3$OH and H$_2$CO in the starburst environment of NGC 253. Methanol formation appears to be influenced by warm, dynamic processes rather than cold cloud chemistry, while formaldehyde primarily forms via gas-phase reactions. These results challenge the assumption of a direct chemical link between CH$_3$OH and H$_2$CO and underscores the impact of starburst-driven shocks, turbulence, and cosmic rays on molecular gas chemistry.