Adaptive Constraint Cascade (ACC)_Case Study ACC_3.1: Two Illustrative Examples

We extend the Adaptive Constraint Cascade (ACC) framework for quantum field theory, introduced in ACC\_3, to two concrete examples: (1) a one-dimensional crystal lattice (the standard emergence of a phonon field), and (2) the continuum limit of a scalar field on a lattice. The control parameter is the energy cutoff $\Lambda = 1/a$, where $a$ is the lattice spacing. We show how the ACC\_3 framework equation – a deformed commutation relation – smoothly interpolates between the discrete (finite degrees of freedom) and continuous (infinite degrees of freedom) regimes, demonstrating the universality of the ACC approach for field emergence.