## 1 Introduction  
The accelerating global population aging process is reshaping the supply patterns of urban public services [1, 2]. According to the United Nations 2024 Population Report, the growth rate of the population aged 65 and above has substantially surpassed that of younger demographics [3], with their global proportion projected to reach 16% by 2050. According to the United Nations classification based on the proportion of population aged 65 and above, population aging is categorized into three stages: (1) Aging society (\(\geq\)7%); (2) Aged society (\(\geq\)14%); and (3) Super-aged society (\(\geq\)20%). Global aging demonstrates two key characteristics: First, developed countries show concentrated aging patterns,with Japan (29.1%), Italy (24.3%), and Germany (23.6%) ranking as the top three superaged societies. Second, significant regional heterogeneity exists in aging progression [4], particularly with accelerated growth in developing countries. Asia has emerged as the fastest-growing aging region, with China representing a notable case: transitioning from an aging society (7.1%) in 2001 to an aged society (14.2%) within two decades and projected to surpass 20% by 2032 (United Nations projection). This trajectory positions China to become the fastest nation globally to transition from an aged to a super-aged society, requiring merely 11 years. This process highlights that aging has become a global challenge that transcends development stages. Driven by the concepts of \"active aging\" and \"healthy aging,\" the demand for elderly care services has transitioned from basic survival guarantees to multidimensional quality-of-life enhancements, imposing compound requirements on the spatial supply quality of care facilities [5; 6]. However, existing studies reveal that China's elderly care infrastructure faces the dual challenges of insufficient supply and imbalanced spatial allocation [6; 7; 8]. As a developing country undergoing rapid urbanization, the spatiotemporal mismatch between facility distribution and elderly population agglomerations has become particularly pronounced [6]. In recent years, accessibility studies under the lens of spatial justice have gained increasing academic attention [9]. The planning paradigm epitomized by the \"15-Min City\" advocates achieving walkable access to essential services through compact spatial configurations [10]. How to rationally plan and allocate elderly care facilities within limited spatial resources, balancing efficiency and social equity, has emerged as a critical challenge in urban development [11]. Spatial Accessibility and Distribution Optimization of Elderly Care Facilities from a Spatial Justice Perspective  
Academic research on elderly care facilities has predominantly focused on three domains: management models [12; 13], economic benefits [14; 15], and service quality [16]. However, spatial accessibility--a critical metric for evaluating the convenience of accessing public service resources--remains underexplored [17]. When analyzing spatial accessibility, it is essential to consider not only the geographic distance affecting older adults but also the alignment between facility service provision and the heterogeneous demands of aging populations. Regarding elderly service facility systems, while no unified classification standard exists, existing literature widely acknowledges essential functional requirements for older adults, including community interaction [18], medical care [19; 20], recreational and cultural activities [21; 22], daily living support [23], commercial amenities [24], and administrative services [25]. These facilities play a vital role in fulfilling the daily needs of aging populations. In accessibility studies, [PERSON] et al. [26] examined time-varying accessibility demands of facilities across different daily periods in Wuhan, while [PERSON] et al. [27] identified service accessibility as a key determinant of elderly social engagement. [PERSON] et al.