Multi-Regime Structural Frustration in Temporal Compatibility Networks

Abstract We investigate the emergence of global non-integrability in multi-branch temporal compatibility networks within the framework of the Temporal Interpretation (TI). Extending beyond previously analysed three-branch configurations, we perform ensemble simulations for increasing branch numbers and quantify collective behaviour using holonomy-based frustration measures. The ensemble-averaged global frustration F(δτ) exhibits a robust three-regime structure. In the deep microscopic limit, frustration grows linearly with the temporal separation, F(δτ) ∝ δτ, independently of spectral heterogeneity. In an intermediate regime, the local effective exponent satisfies βeff ≈ α, where α is the stretched compatibility exponent governing microscopic dynamics. As the system approaches the operational compatibility scale, frustration saturates toward a universal threshold F∗≈1.96 rad, indicating bounded collective non-integrability rather than unbounded power-law growth. These results show that the previously hypothesised lower bound β ≥ 1/2, introduced in the galactic-scale analysis of TI, arises naturally within the intermediate structural regime of frustration. The growth of collective incompatibility is therefore structurally constrained and spectrally controlled. While the present work does not yield a closed-form derivation of the galactic activation timescale Δτgal from the microscopic compatibility scale Δτcompat, it establishes a quantitatively consistent intermediary mechanism. Microscopic compatibility loss induces bounded collective frustration with a universal saturation threshold, supporting a micro-macro bridge based on projection and integration of structural incompatibility across extended domains rather than direct scale amplification.