Thickness Structure Hypothesis Dynamics - Consistent Dynamical Equation

Description This work presents a compact structural formulation based on the Thickness‑Structure Hypothesis (TSH). The framework introduces three internal variables—p(x), Δf, and γT—and examines how their structural balance can be used to organize different dynamical behaviors within a single covariant equation. The formulation focuses on the structural roles of these variables and analyzes how known limiting behaviors (quantum‑like spreading, classical‑like trajectories, and gravitational‑like localization) can be positioned as different structural regimes within the same dynamical expression. A separate computational simulator is provided as an auxiliary tool. The simulator numerically implements the same structural equation used in the theoretical formulation and visualizes how the internal variables evolve under the structural dynamics. Its purpose is to illustrate the behavior of the model and to support reproducibility of the structural framework. The simulator is not required for the theoretical results but serves as an optional demonstration of the structural dynamics. Overall, this work focuses on the structural form of the consistent dynamical equation and its limiting behaviors, and briefly notes the availability of a companion simulator that implements the same structural formulation. Physics Engine (simulation, games, research) & AI Physics Engine (learning, optimization, universe editing)https://github.com/ababphysics/TSH/ xhttps://x.com/abab162535