We identify a simple second-difference invariant of gauge beta functions that strongly correlates with gauge coupling unification. By analyzing renormalization group (RG) evolution at both one-loop and two-loop levels, including threshold effects, we show that near-unification occurs when this invariant is minimized along the RG flow. We demonstrate this behavior numerically across a broad class of models using a large-scale parameter scan, revealing a strong correlation between the invariant and the convergence of gauge couplings. This provides a predictive diagnostic tool for model building and a new perspective on the structure of gauge unification. Gauge coupling unification is a central theme in high-energy physics, often explored within grand unified theories (GUTs) and beyond Standard Model (BSM) scenarios. Traditionally, unification is tested by numerically evolving gauge couplings under RG flow and observing whether they converge at a common scale.
sree Debasish Dasgupta (Sun,) studied this question.