Deterministic phase memory III: Spatial fields, domain walls, and continuum information storage

We extend the theory of deterministic phase memory from discrete cascaded systems to spatially distributed continuous fields. Starting from a bistable two-component reaction–diffusion system derived as the continuum limit of a bidirectionally coupled lattice, we establish the following results.First, we prove stability of spatially uniform bistable equilibria. Second, we establish existence and uniqueness (up to translation) of monotone domain-wall solutions using a Hamiltonian structure with an explicit potential and a Maxwell equal-energy condition. Third, we prove spectral stability of kink solutions and show that the spectral gap remains strictly negative in the weak diffusion regime. Fourth, we derive an explicit energy identity for the front velocity and obtain a small-bias expansion for the tanh-type nonlinearity. Fifth, we provide exponential interaction estimates for multiple domain walls and prove existence of multi-wall equilibria. Sixth, we justify the continuum limit of the discrete cascade. Seventh, we derive an information capacity estimate. Finally, we prove a bridge theorem connecting the three scales of the program: single cell (Part I), discrete cascade (Part II), and continuum field (Part III).We also provide a systematic comparison with the scalar Allen–Cahn equation, identifying structural features that are unique to the two-component architecture.This paper constitutes Part III of a three-part foundational program on deterministic phase memory.This work is a preprint and has not yet been peer reviewed. This record is Part III and the concluding paper of the three-part foundational program “Deterministic Phase Memory”, connecting the minimal bistable cell of Part I and the discrete cascade of Part II to a spatially distributed continuum-field formulation. Part I — Deterministic phase memory I: A minimal bistable gate model with explicit switching thresholds https://doi.org/10.5281/zenodo.18663866 Part II — Deterministic phase memory II: Cascaded bistable systems with explicit coupling bounds and quantitative addressability https://doi.org/10.5281/zenodo.18664009