A Retarded Structural-Fluctuation Mechanism for the Inverse Oxygen-Isotope Effect in Layered Nickelates

Oxygen substitution in layered nickelates poses a deceptively simple question: why can a heavier isotope raise a density-wave transition temperature? In La$_3$Ni$_2$O$_7$, partially exchanged $^{18}$O shifts the charge-density-wave transition upward while leaving the separated spin-density-wave transition unresolved within error. In La$_4$Ni$_3$O$_{10}$, the intertwined charge and spin onsets move by nearly the same amount. We examine this pattern with a minimal structural-fluctuation model. If ionic mass appears only in a quadratic kinetic term, it cannot move a classical static transition. Mass dependence enters instead through quantum fluctuations, delayed response, anharmonicity, or a genuine structural change. For a repulsive biquadratic coupling, oxygen motion acts as an order-destroying fluctuation; heavier oxygen reduces that fluctuation and can therefore raise the charge-ordering temperature. A passive auxiliary structural coordinate adds a causal slow-response channel with a positive spectral measure. More generally, the isotope coefficient is set by the mass derivative of the complete transition kernel along its critical eigenvector. Time-reversal symmetry rules out a bare paramagnetic CDW–SDW bilinear, so isotope sensitivity can pass between the orders only through allowed nonlinear coupling, a symmetry-broken fluctuation Hessian, or a common first-order boundary. A shared-parameter comparison rules out literal $M^{-1/2}$ scaling and the best normal-sign benchmark. The present transition data support the fluctuation mechanism, although they do not isolate a slow memory contribution or establish a unique microscopic origin. **Keywords** Layered nickelates; oxygen-isotope effect; charge-density wave; spin-density wave; structural fluctuations; retarded response; order-parameter coupling.