The Imaginary Unit as Stability Regulator: A Variance-Coupling Perspective on Quantum Reversibility

This work presents a structural analysis of reversibility, instability, and symmetry in two-channel systems, with particular emphasis on the role of the imaginary unit in quantum mechanics. Using a variance-coupling model, we show that equipartition between channels corresponds to an unstable fixed point under real dynamics, while the presence of the imaginary unit converts real eigenvalues into purely imaginary ones, yielding neutral, oscillatory evolution and preserving reversibility. Building on this framework, we introduce bounded mean-differential diagnostics derived from the arithmetic, geometric, and harmonic means of normalized two-channel partitions. We prove exact identities showing that the arithmetic–harmonic mean difference equals half the squared asymmetry between channels. This quantity provides a symmetric, bounded, and geometrically interpretable measure of deviation from equipartition, exhibiting Archimedean comparability: small deviations remain small, large deviations saturate, and no singular dominance occurs. These diagnostics are shown to coincide exactly with established measures across multiple disciplines, including polarization squared in optics, Simpson diversity in ecology, purity in two-level quantum systems, and normalized enantiomeric excess in chemistry. The results are presented as structural and diagnostic in nature, without introducing new dynamics or modifying existing physical theories.