Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs

TL;DR

This paper develops a formal field theory of synthetic cognition using coupled reaction-diffusion systems on finite graphs, establishing well-posedness, stability, and invariance properties.

Contribution

It introduces a novel RFDE framework for modeling coupled symbolic and geometric fields with delays, providing key formal results on stability and attractors.

Findings

Proved well-posedness of the RFDE under constant input.

Established delay-independent global stability of principal components.

Derived a fast relaxation estimate for the valuative variable.

Abstract

This article develops a field theory of synthetic cognition in which a symbolic field $H_L$ and a geometric field $X_R$, each a section of a vertex bundle over a finite graph, are coupled through a bipartite Hilbert-Schmidt operator with propagation delays. The central object is a retarded functional differential equation (RFDE) on the history space: the reaction-diffusion equation is the operative equation of the theory. Nine synthetic design blueprints specify admissibility conditions for each architectural component; each condition carries a dynamical consequence. The main formal results are: (1) well-posedness of the full deterministic RFDE under constant input $u^*$, (2) existence of a compact global attractor from compact viability and eventual compactness of solution segments, (3) delay-independent global stability of the principal components $(H_L,X_R,P)$ in the closed stability regime with fixed interfield coupling operators satisfying $C_{\mathcal{K}}^2<\mu_L\mu_R$, (4) $\mathrm{SE}(d)$-invariance of the scalar geometric feature dynamics, and (5) an $O(1/\kappa_Y)$ fast relaxation estimate for the valuative variable. The well-posedness and attractor results allow Lipschitz state-dependent attention operators. The stability theorem is stated for the fixed-coupling principal subsystem, with the extra small-gain terms for state-dependent coupling identified explicitly.