Dimensional analysis applied to physical constants: A theoretical framework for analyzing constant variations

Abstract The stability of fundamental physical constants underpins modern physics, yet distinguishing their intrinsic time variation from systemic drifts in measurement reference frames remains a critical metrological challenge. This work establishes a generalized dimensional analysis framework to categorize constant variations through three distinct scenarios. First, we demonstrate that if dimensionless constants remain invariant, a strict mathematical constraint binds the co-variation of dimensionful constants, preserving the structure of physical laws. Second, we analyze the standard scenario where a variation in a single dimensionless constant maps to a specific non-reference dimensionful constant. Crucially, our results reveal that a drift in a defined SI reference constant (e.g., c or h) does not manifest in isolation but triggers a correlated, "cascading" variation pattern across multiple dimensionless observables. By deriving these specific sensitivity exponents, we provide a predictive metrological signature capable of discriminating whether an experimental anomaly originates from a non-reference parameter or a systemic shift in the reference standards themselves.