Elastic Spacetime with Scale-Dependent Coupling (ESSC) v13 presents a structural analysis of when translations between physical descriptions are admissible, and what occurs when such translations terminate irreversibly.This work does not introduce new physical laws, entities, forces, or predictive mechanisms. Instead, it reclassifies familiar descriptions according to their translation behavior. The central concept of v13 is translation termination: the point at which an otherwise coherent representational mapping cannot be continued without irreversible fixation. When translation remains non-terminating—as in unitary quantum evolution or gravitational description—representational non-equivalence can be redistributed, and no privileged boundary emerges. In such cases, the constant π appears only in its conventional numerical role. By contrast, when translation is forced to terminate—most notably in electromagnetic interaction and measurement—equivalence between circular phase descriptions and linear historical records ceases to be admissible. Under these conditions, residual non-equivalence condenses into a minimal structural boundary. ESSC v13 shows that this boundary is consistently marked by π, which is reinterpreted not as a parameter fixing periodicity, but as a structural marker of admissibility limits. The paper further demonstrates that this same boundary accounts for two additional phenomena that are usually treated independently: the fixation of the arrow of time and the localization of the present moment (“Now”). Within ESSC v13, the arrow of time arises from admissibility loss rather than from dynamical asymmetry, and Now is identified as the structural position immediately following translation termination, where irreversible fixation and residual differentiation coexist. ESSC v13 does not claim to solve the quantum measurement problem, propose a theory of consciousness, or elevate π to ontological primacy. Its contribution is diagnostic and classificatory: it clarifies the structural limits of representational equivalence and explains why π, the arrow of time, and Now acquire their apparent centrality only under terminating translation.
umimoto (Mon,) studied this question.