The Γ–Bohr Equation and Torsional Coherence Plateaus

We present an expanded and integrative extension of the Dynamical Growth Parameter framework previously introduced to quantify coherence in physical systems. The growth parameter, denoted by Γ, is here established as the central organizing quantity of the theory and shown to provide a unifying description of coherence phenomena spanning atomic and particle-scale systems. A generalized Bohr formulation, referred to as the Γ–Bohr equation, is introduced as a dynamical extension of the semiclassical energy expression. The mathematical structure of the Bohr equation is preserved, while Γ appears as a dimensionless coherence factor accounting for collective and torsional effects without modifying fundamental interactions or constants. The framework is further applied to muons in the kinetic energy range 5–20 MeV. A predicted Γ plateau emerges in the interval 3.8–4.3 MeV, embedded within a global TDRE coherence limit near 10.6 MeV. This plateau is interpreted as a manifestation of coherence locking, potentially associated with an intrinsic sinusoidal torsional mode of the nuclear core. The fine-structure constant enters the formulation exclusively as a normalization scale for the critical frequency, leading naturally to Γ values of order 1/α in the plateau regime without parameter identification. These results support the interpretation of Γ as a dimensionless, non-fundamental measure of torsional coherence and justify its role as a foundational element within TDRE 4.0.