## 1 Introduction
[PERSON] and team compared traditional and novel designs for two-stage speed reducers, analyzing their implications for stress and motion [18]. [PERSON] et al. showcased the 2K-H internal meshing abnormal cycloidal gear (ACG) planetary reducer's superior transmission properties and reduced stress levels [19], while [PERSON] proposed a distinctive transmission design that optimized stress distribution [20]. The analytical models by [PERSON] et al. and [PERSON] et al. took into account factors like elastic- and heat-induced deformations in cycloid and planetary gear speed reducers [21]. [PERSON] et al. 's focus was on dynamic characteristics, using both static and dynamic finite element analyses to evaluate the stress distribution [22]. Delving into specific applications, [PERSON] et al. 's finite element analysis led to a notable reduction in the contact stress in cycloid gears [23]. Moreover, the acoustic properties of gears are also affected by stress distribution. Fluctuations in stress can alter a gear's vibration and noise levels, implying that a meticulously crafted stress distribution design can diminish these acoustic interferences, thereby improving the gear's acoustic performance [24]. In summation, recognizing and addressing stress distribution in gears, especially on their end surfaces, is vital. A competent design that considers this aspect can not only refine the performance and reliability of the gear but also uplift the acoustic attributes, thereby optimizing the overall functionality of the mechanical system. Despite the extensive research on gear stress distribution, several gaps persist, which our investigation aims to bridge. Firstly, while most of the prevailing studies lean heavily on FEA to interpret stress and fatigue, the wide dimensional span between gear profile modifications and the actual gear geometry poses computational challenges. Mature FEA software, when applied to these scenarios, is computationally intensive, making efficient analysis and optimization elusive. To counteract this, advanced optimization techniques like the Bayesian-enhanced least squares genetic algorithm optimization method (BLS-GAOM) emerge as imperative tools, offering precise and cost-effective analyses. Secondly, the assumption of idealized conditions in many studies often overlooks real-world intricacies like input crank deformation, pin tooth runout, and backlash angles. These variables, aside from their individual impacts, can also lead to altered contact characteristics, affecting the overall gear performance. Thirdly, there is a discernible focus on tooth surface contact in the literature, sidelining the more complex yet crucial aspect of stress analysis on gear end surfaces with intricate shapes. This oversight potentially masks a significant area of stress accumulation, which might be the precursor to premature gear failures or sub-optimal performance. In summary, while the field has made commendable strides, a holistic, integrated approach that amalgamates advanced computational methods, real-world variables, and a deep dive into lesser-explored gear regions is the next frontier in gear stress distribution research. This study is methodically structured to offer a systematic understanding of cycloid gear stress analysis and optimization. Initially, this paper lays the groundwork with a quadratic contact surface analysis and force modeling (Section 2). We then delve into an advanced analysis of end face stress in cycloid gears, introducing a conformal mapping technique that is suitable for multiply connected domains and discussing the geometric intricacies of cycloid gear profiles with pores (Sections 3.1 and 3.2). The novel SACGES-CMMD method is presented (Section 3.3), followed by insights into the global stress scaling factor (Section 3.4). Section 4 propounds an innovative optimization approach, integrating Bayesian analysis with a genetic algorithm.