We present a cosmological framework in which late-time cosmic acceleration emerges dynamically from the time evolution of the χ-field in a deterministic lattice wave equation. Without introducing a cosmological constant, vacuum energy, or modifications to the governing Klein-Gordon equation, we show that slow χ-relaxation produces de Sitter-like expansion with an effective equation of state consistent with w ≈ −1. The emergent scale factor is defined operationally through wavelength stretching and propagation delay within the lattice, yielding an expansion history that is observationally indistinguishable from ΛCDM at the background level. We validate the mechanism through eight independent tests: four theoretical (acceleration confirmation, equation-of-state extraction, energy conservation, null control) and four observational, including 1,048 Pantheon Type Ia supernovae, BAO H(z) measurements, cosmic chronometers, and the present-day deceleration parameter. All tests pass without additional free parameters beyond the χ-relaxation timescale. The results demonstrate that dark-energy-like behavior can arise from the same wave equation responsible for emergent gravity, reframing cosmic acceleration as a kinematic consequence of field relaxation rather than a separate physical energy component. Code, data, and analysis scripts are provided to enable full reproduction of the results. V1.1: Added complete derivations for adiabatic invariant (Appendix D) and friction parameter (Appendix E), pseudocode algorithms, evidence stability table, BBN quantitative bound, and ISW regime clarification; softened abstract wording and removed duplicate paragraph. Fixed LaTeX conversion issues with pseudocode, Unicode arrows, and table vertical bars; build validated with 56-file beta package.