ATClean: A Novel Method for Detecting Low-Luminosity Transients and Application to Pre-explosion Counterparts from SN 2023ixf

In an effort to search for faint sources of emission over arbitrary timescales, we present a novel method for analyzing forced photometry light curves in difference imaging from optical surveys. Our method "ATLAS Clean'' or ATClean, utilizes the reported fluxes, uncertainties, and fits to the point-spread function from difference images to quantify the statistical significance of individual measurements. We apply this method to control light curves across the image to determine whether any source of flux is present in the data for a range of specific timescales. From ATLAS $o$-band imaging at the site of the Type II supernova (SN) 2023ixf in M101 from 2015--2023, we show that this method accurately reproduces the 3$\sigma$ flux limits produced from other, more computationally expensive methods. We derive limits for emission on timescales of 5~days and 80-300~days at the site of SN\,2023ixf, which are 19.8 and 21.3~mag, respectively. The latter limits rule out variability for unextinguished red supergiants (RSG) with initial masses $>$22~$M_{\odot}$, comparable to the most luminous predictions for the SN 2023ixf progenitor system. We also compare our limits to short timescale outbursts, similar to those expected for Type IIn SN progenitor stars or the Type II SN 2020tlf, and rule out outburst ejecta masses of $>$0.021~$M_{\odot}$, much lower than the inferred mass of circumstellar matter around SN 2023ixf in the literature. In the future, these methods can be applied to any forced point-spread function photometry on difference imaging from other surveys, such as Rubin optical imaging.