Constraint-Coupled Reaction–Diffusion Systems: Viability, Endogenous Regulation, and Dynamically Generated Localization

We introduce a deterministic class of constraint-coupled reaction–diffusion systems in which an internally generated field modulates transport and reaction processes while being regenerated by the same dynamics that it regulates. This feedback structure provides a minimal mathematical framework for studying dynamically consequential endogenous regulation in spatially extended non-equilibrium systems; the presence of a causal loop alone is not identified with organizational autonomy. The model is formulated as a quasilinear parabolic system on a bounded domain. Positivity, boundedness, viability, equilibrium stability, and spatial localization are treated as distinct mathematical properties. Under structural assumptions on the diffusion and reaction terms, we establish local classical well-posedness, preservation of non-negativity, and global existence under boundedness or forward-invariant-region conditions. Viability is formulated as forward invariance of subsets of an infinite-dimensional phase space rather than as a local pointwise balance. A minimal three-field realization, consisting of a resource field, an organizing field, and a constraint field, is then analyzed. We derive conditions for the existence of positive homogeneous equilibria, obtain sufficient criteria for homogeneous and full modal stability, and characterize stationary and oscillatory spatial instability thresholds through a mode-dependent dispersion relation. We complement the analysis with conservative two-dimensional finite-volume simulations. Below the predicted spatial-instability threshold, perturbations decay, whereas above threshold a persistent heterogeneous state forms with a dominant scale consistent with the dispersion relation. Constraint level sets, occupied fraction, connected components, and concentration indices quantify the resulting dynamically generated localization. Feedback-ablation controls show that, for the representative parameter set, removing either transport or reaction feedback eliminates the finite-wavenumber instability, while alternative bounded monotone constitutive laws preserve spatial patterning. The framework therefore distinguishes externally imposed regulation, passive outputs, and dynamically consequential endogenous feedback, and provides a basis for investigating bounded self-maintenance and localization without equating loop topology with autonomy.