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

Coherent dynamics in deformable quantum systems are governed jointly by geometric curvature,environmental dephasing and the strength of the coherent interaction that mediates populationtransfer. Yet no general principle links these quantities. Here we identify a curvature–coherenceinvariant, the Shield Law, g2 = κ γϕ, which relates the geometric curvature κ of a vibronic gatingmode, the environmental dephasing rate γϕ and the coherent coupling g. Variations of the invariantsatisfy a thermodynamic differential δ(g2) = Teff δScoh, where Scoh is a coherence entropy and Teffan 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 systematicisotopic and chalcogenic perturbations. Transfer rates, coherence lifetimes and vibronic couplingstrengths collapse onto the Shield-Law relation across a broad parameter range. These resultsestablish curvature–coherence invariance as a predictive principle for determining when geometricor vibronic modulation enhances, suppresses or leaves unchanged coherent population transfer incatalytic, molecular and condensed-matter systems. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.