arXiv:2510.00487v1 Announce Type: cross Abstract: The black-box domain adaptation (BBDA) topic is developed to address the privacy and security issues where only an application programming interface (API) of the source model is available for domain adaptations. Although the BBDA topic has attracted growing research attentions, existing works mostly target the vision applications and are not directly applicable to the time-series applications possessing unique spatio-temporal characteristics. In addition, none of existing approaches have explored the strength of foundation model for black box time-series domain adaptation (BBTSDA). This paper proposes a concept of Cross-Prompt Foundation Model (CPFM) for the BBTSDA problems. CPFM is constructed under a dual branch network structure where each branch is equipped with a unique prompt to capture different characteristics of data distributions. In the domain adaptation phase, the reconstruction learning phase in the prompt and input levels is developed. All of which are built upon a time-series foundation model to overcome the spatio-temporal dynamic. Our rigorous experiments substantiate the advantage of CPFM achieving improved results with noticeable margins from its competitors in three time-series datasets of different application domains.