A Passive Causal Memory Model for the Central Peak in SrTiO₃

The cubic-to-tetragonal transition of SrTiO₃ exhibits a narrow central component near the antiferrodistortive wave vector while the associated soft phonon can retain a finite oscillation frequency. We formulate a minimal dynamical mechanism in which that phonon is linearly coupled to a slow structural coordinate omitted from an ordinary harmonic description. A positive quadratic free energy and fluctuation–dissipation-consistent Langevin dynamics give an exact retarded susceptibility with no independently added central line. We prove causality, passivity, static stability, spectral positivity, and the relevant Kramers–Kronig and moment sum rules. In the separated-time-scale regime the response has one near-zero relaxational pole and an underdamped phonon pair while the complete static Hessian remains strictly positive. Distinct expansions near the thermodynamic instability and in the slow-memory limit yield the central width, pole residues, phonon shift, and redistributed equal-time weight. A positive continuum of relaxation rates extends the construction to nonexponential Volterra memory without sacrificing passivity. Reproducible synthetic benchmarks test pole residuals, spectral positivity, static limits, sum rules, resolution convolution, and model selection. The mechanism is sufficient but not claimed to be unique: its value is the set of joint temperature, momentum, resolution, and time-domain constraints that distinguish dynamical memory from static defects, an independent elastic line, and a completely overdamped phonon. **Keywords** SrTiO₃; central peak; structural phase transition; antiferrodistortive soft mode; causal memory; slow structural relaxation; fluctuation–dissipation theorem; spectral weight redistribution; lattice dynamics; inelastic scattering.