SHEM 3.0: A Stochastic Effective Model for Fermion--Scalar Dynamics in the Early Universe

This work focuses on the construction, calibration, and first validation of a stochastic effective model for fermion–scalar dynamics in the early universe, rather than on proposing a specific cosmological scenario or fitting observational data.We restrict attention to semiclassical systems in which fermions are treated quantum-mechanically in homogeneous or weakly inhomogeneous backgrounds, and where the scalar field can be evolved efficiently in three-dimensional periodic boxes. Within this regime, we:• Derive time-dependent effective noise amplitudes Γ(t) from controlled comparisons between semiclassical dynamics Structured Hierarchical Emergent Modes (SHEM) and renormalized Dirac field theory (QFT), using the residual discrepancies in fermionic observables as a physically meaningful estimator of quantum backreaction.• Construct and simulate a stochastic extension of the scalar field evolution equation, in which quantum fluctuations from integrated fermionic and short-wavelength modes are encoded in a noise term η(t), with amplitude Γ(t) calibrated from quantum data.• Demonstrate stable and reproducible evolution in three-dimensional periodic volumes up to 256^3 grid points, including computation of the scalar power spectrum P_φ(k) in 3D, and statistical characterization of the ensemble of stochastic trajectories.• Train and deploy a Gaussian Process emulator for the relevant observables, enabling rapid forward prediction and integration into a full Bayesian inference loop based on synthetic data.This defines a modular, numerically implemented stochastic effective field theory architecture (SHEM 3.0), calibrated directly from quantum simulations and fully validated in its regime of applicability. Applications to fully realistic cosmological models, inclusion of gauge dynamics beyond fixed slicing, and confrontation with observational datasets are left to follow-up work based on the SHEM 3.0 / 3.1 framework introduced here.