Emergent Quantization from Chi-Structured Deterministic Lattice Modes

Discrete spectral structure in physical systems is commonly introduced axiomatically, particularly in quantum mechanics where discrete energy eigenvalues arise from operator formalisms. This work tests whether such discrete structure can instead emerge constructively from deterministic wave dynamics under spatial confinement, without introducing quantization postulates. We study a deterministic lattice system governed by a Klein-Gordon-type wave equation with a prescribed spatial χ(x) field. Through 23 numerical experiments with explicit, pre-specified falsification criteria, we demonstrate that: (i) reflecting boundaries produce discrete eigenfrequencies matching theoretical box modes; (ii) localized χ-structures support bound states with exponential spatial decay; (iii) parabolic χ(x) profiles generate systematic discrete spectra with curvature-dependent spacing; and (iv) eigenfrequencies track theoretical predictions under adiabatic χ variation. All results converge under grid refinement and satisfy quantitative error thresholds. The study addresses spectral structure only, establishing which frequencies exist under given spatial configurations. Probability distributions, measurement outcomes, state collapse, and the Born rule are explicitly outside the scope of this work. The results demonstrate that discrete spectral structure can emerge from deterministic wave dynamics under spatial confinement, providing a necessary but not sufficient foundation for investigating broader quantum-associated phenomena in subsequent studies.