Minimal Nonperturbative Test of Condensate-Like Behavior in a Reduced Four-Dimensional Lattice Fermion Model

This manuscript presents a minimal nonperturbative finite-volume diagnostic study of condensate-like low-mode softening in a reduced four-dimensional lattice fermion model. The model employs a Wilson-fermion operator coupled to an auxiliary scalar field and analyzes a low-mode condensate proxy, the lowest eigenvalue of the positive fermion operator, condition-number growth, susceptibility-like response, distributional changes, and correlation diagnostics. The study proceeds from an L=4 exploratory baseline to an L=6 validation scan and includes independent-seed checks, fixed-trajectory-length integrator checks, and a minimal L=8 pilot consistency test. The results support an internally correlated finite-volume low-mode-softening diagnostic trend in the reduced model. The manuscript does not claim spontaneous symmetry breaking, chiral symmetry breaking, a thermodynamic-limit phase transition, continuum behavior, or a dynamical Higgs mechanism. The results are intended as conservative finite-volume diagnostics motivating future controlled finite-size-scaling studies.