Foundations of the Meikyo–Shisui Theory

This paper organizes the foundational framework of the Meikyo–Shisui Theory, proposedby the author, and presents a proposal for a translation theory that enables mutually rein-terpreting mathematical objects within a unified formal structure. Although this frameworkdoes not necessarily aim at directly providing solutions, a central feature is that the processof translation itself gives rise to new perspectives and generates new questions. Dependingon the situation, the translation process may also crystallize directly into a solution, in theform of a procedure or a proof strategy.The theory is constructed using a small number of technical terms: the choice of ob-jects (Sui), the rules associated with them (Getsu), their interface (Sui–Getsu), the up-date rule called the Natural Move (NM), and the extracted essence (Shin-Getsu). Here,Sui/Getsu/Shin-Getsu are not literal translations but are used as proper technical terms,and the term “move” in Natural Move is used in the sense of an update or transition step.To facilitate understanding, this paper presents, not representative examples but minimalsketches: a Collatz-type iteration as an example of a dynamic problem, and a Goldbach-typeproposition as an example of a static problem. Through three approaches—retrospectivetracing, emergence, and phase transition—we clarify what should be identified, such as theidentification of Sui or Getsu, or the definition of the Natural Move. This framework naturallyextends to iterative systems, search and optimization, rewriting systems, finite verification,and graph operations, and provides a viewpoint in which the problem itself emerges throughthe process of translation.