Notes: Turing Machine and Dynamic Reachability

In these working notes, we introduce the concept of dynamic reachability on an arbitrary functioning space of a Turing machine, based on which an equivalence class called the dynamic component is established. We examine the transition to a numerical submonoid and the construction of its extension to the Grothendieck group. A dynamic contour, formed by the sequential concatenation of linear trajectory segments, is defined. Utilizing a classifying mapping, the contour is lifted to a $d$-dimensional torus. Subsequent analysis focuses on the supertropical layer ($ST$), where the system's behavior is modeled by a rational reachability function consisting of a factored relaxed numerator and a universal constraint denominator of gaps.