Elastic Spacetime with Scale-Dependent Coupling (ESSC) v15: Emergent Circular Structure as a Self-Amplifying Deviation System

This work presents a minimal structural framework in which circular geometry emerges as a distributed relational property rather than a fixed constant. In contrast to the conventional interpretation of π as a single invariant value, we show that an effective circular scale (π_eff) forms a finite-width band across astrophysical systems. Using galaxy and cluster datasets, we identify a consistent structural relation in which deviations from circular closure (δ₀) generate secondary structural spread (δ). This spread is not independent but arises through a scale-dependent amplification process. We find that the amplification coefficient k follows a banded discrete structure centered around powers of two: k ≈ 2ⁿ (1 + ε) where ε represents a fluctuation term. Importantly, ε is not externally driven by conventional physical quantities such as halo fraction or acceleration residuals, but instead correlates with the magnitude of the deviation itself: |ε| ∝ |δ₀| This indicates a self-amplifying mechanism in which deviation generates further structure. The resulting system exhibits a hybrid character combining discrete scaling (2ⁿ) and continuous fluctuation (ε). The final structural form can be expressed as: R = π (1 + δ₀) [1 + 2ⁿ (1 ± c|δ₀|) δ₀] 2⁻ⁿ This formulation suggests that observed structural and gravitational-like behaviors may emerge from internal relational geometry rather than requiring purely external mass components. This work does not propose a replacement for standard cosmological models. Instead, it introduces a structural perspective that may complement existing frameworks by emphasizing the role of relational geometry and self-amplifying deviation. v15 introduces: Identification of π as a distributed relational band rather than a fixed constant Empirical confirmation of strong correlation between structural deviation levels Discovery of banded discrete scaling in the amplification coefficient k Identification of ε as a self-generated fluctuation linked to deviation magnitude Final unified structural equation integrating circular deviation, scale hierarchy, and amplification