Weight Equation as Energy Partitioning Function, the Bridge Equation ξ = f ∗/εD2h, and CL5D as a Universal Scale-Free Coherent Data Bus

This paper establishes three interconnected results. First, the Weight Equation is formally defined as an energy partitioning function: W(f*) = E_sp3_to_sp2 / E_bulk = 255 / 18945 = 0.01346, where f* is the fraction of bulk energy required to drive the sp3 to sp2 rehybridisation at the Phase III to Phase IV transition. Second, the bridge equation xi = f* / epsilon_D2h = 0.2828 is derived, connecting the Weight Equation at molecular scale to the Randall-Sundrum scalar field coupling at field theory scale. This recovers the audit value xi = 0.27 to within 4.7% and gives beta = 0.70, which is naturally of order one. Third, the CL5D architecture is shown to constitute a universal, scale-free coherent data bus. The governing equations G = N-squared times exp(alpha_gain times t), T = P_bulk divided by P_brane, and xi = (1/6)(1 + beta times epsilon_D2h) apply identically to biological systems such as microtubule arrays and ATP synthase, and to non-biological systems such as ZnO nanowire arrays, MEMS cantilevers, and piezoelectric chains. The coherence budget (1 minus f*) = 98.65% is available for lossless information transfer across the molecular-to-macroscopic scale gap via a piezoelectric transducer chain.