The Shield Law: Curvature–Coherence Invariance in a Vibronically-Gated Five-Centre Quantum System. (GQR22)

Coherent dynamics in deformable quantum systems are governed jointly by geometric curvature,
environmental dephasing and the strength of the coherent interaction that mediates population
transfer. Yet no general principle links these quantities. Here we identify a curvature–coherence
invariant, the Shield Law, g
2 = κ γϕ, which relates the geometric curvature κ of a vibronic gating
mode, the environmental dephasing rate γϕ and the coherent coupling g. Variations of the invariant
satisfy a thermodynamic differential δ(g
2
) = Teff δScoh, where Scoh is a coherence entropy and Teff
an effective vibronic temperature.
We demonstrate the invariant in a naturally occurring five-centre CaMn4O5 quantum manifold,
using time-dependent Schrödinger dynamics, experimentally anchored geometries, and systematic
isotopic and chalcogenic perturbations. Transfer rates, coherence lifetimes and vibronic coupling
strengths collapse onto the Shield-Law relation across a broad parameter range. These results
establish curvature–coherence invariance as a predictive principle for determining when geometric
or vibronic modulation enhances, suppresses or leaves unchanged coherent population transfer in
catalytic, molecular and condensed-matter systems.