## 1 Introduction  
Medium-carbon steels, containing approximately 0.30 wt.% to 0.60 wt.% carbon and 0.60 wt.% to 1.65 wt.% manganese, are extensively used in various structural applications, particularly as load-bearing components [1]. It is widely utilized in large components, forging, and automotive parts due to its balanced ductility and strength, along with good wear resistance [2]. A significant amount of research is being conducted on these steels to explore their performance and enhance their properties [3, 4, 5]. Martensite start (Ms) temperature is crucial for steels during the manufacturing process, considering the significant impact of phase transformation and heat treatment on the mechanical properties of steel [6]. Consequently, substantial progress has been made in estimating or predicting Ms temperature by numerous researchers through numerical methods over the years [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Many researchers have proposed at least one formula for Ms temperature, based on their research or as an improvement to earlier formulations. From these studies, a generalized linear formula was derived to describe the relationship between the Ms temperature and the chemical composition, as expressed in Equation (1). \[y=\theta+\sum_{i}\omega_{i}x_{i} \tag{1}\]  
Here, \(y\) represents the calculated Ms temperature for the target steel. The offset parameter \(\theta\) is a constant, representing the Ms temperature of steel without any alloying elements--essentially pure iron, derived from extrapolating the Ms temperatures of iron-based alloys as adopted by researchers or from the databases and references they utilized. The weight coefficient \(\omega_{i}\) for alloying element \(i\), is adjusted iteratively to achieve the best-fit prediction of Ms temperature as a dependent variable. The amount of the alloying element \"\(i\)\" is denoted by \(x_{i}\). In essence, each input \(x_{i}\) representing the content of an alloying element is multiplied by its corresponding weight coefficient \(\omega_{i}\), and the resulting products are summed and then added to a temperature offset constant \"\(\theta\)\". The output y is the predictive Ms temperature value. Table S1 in Supplementary Materials presents the calculation formulae for Ms temperature specifically applicable to medium-carbon steels. It includes 21 empirical equations from 17 research studies, spanning from earlier works to more recent investigations. The value of \(\theta\) varies between 693 K and 834 K across different formulae. Beyond the common linear form, some formulae incorporate unique forms or transformed terms. For instance, [PERSON]'s formula includes a product formula \(\prod[(1-\omega_{i}x_{i})]\)[29]. [PERSON] formulae [11] feature product terms \(\Sigma\omega_{ij}x_{i}x_{j}\), and [PERSON] et al. 's formula [26] includes the square roots of product terms \(\Sigma\omega_{ij}\sqrt{x_{i}x_{j}}\), both of which aim to account for the co-action by alloying elements on Ms temperature. Additionally, [PERSON] and [PERSON]'s method [25] introduces a chemical composition independent item \(11.67\ln(\mathrm{d}\gamma)\) to reflect the influence of average austenite grain size on Ms temperature. However, as indicated by the researchers, the existing prediction models or empirical formulae for Ms temperature are limited by the types of alloying elements included, the range of chemical composition content, or their inability to align with experimental results when values exceed the established ranges. Fe-based materials have been widely studied due to their diverse applications in structural and functional materials. For instance, Fe oxides have been explored for memory devices due to their unique electronic properties [30], while Fe-based Heusler alloys have shown great potential for thermoelectric energy conversion [31]. These studies highlight the versatility of Fe-based systems in advanced technologies.