Dynamics of Self-Referential Production Systems: Topology, Bifurcation, and Operator Dynamics

Important Notice: Unverified Working Memo This document is a "working memo" intended to record the initial thought process of an ongoing study. The information contained herein has not been fact-checked or verified as established fact. > While Google Gemini and Anthropic Claude were used as assistive tools for drafting, the content includes AI-generated hypotheses and data that are currently under rigorous verification. Please refrain from citing this document or treating its contents as finalized findings. We develop a continuous-time dynamical theory of self-referential production systems: networks of artifacts acted upon by bounded operators, where the operators themselves are artifacts within the system. From three axiomsbounded agency (A1), structural coupling (A2), and operatorartifact duality (A3)together with a constitutive principle of diversity (C0) and a methodological convention of tractability (M0), we constructively derive a canonical nonlinear equation of motion for quality decits on continuous networks. We prove a graph-theoretic dichotomy (Theorem A): when the verication-source graph Gδ is a directed acyclic graph (DAG), the system possesses a globally asymptotically stable (GAS) equilibrium, proved unique under weak coupling (∥A∥≪1) and conjectured unique for general A. When Gδ contains directed cycles, catastrophic collapse occurs at the threshold k∈C ρk = 1. A unied spectral criterion ρ(Γ−1M) < 1 handles arbitrary congurations. The bifurcation type is classied (Theorem B): transcritical (α= 1, reversible), saddle-node (α > 1, hysteretic), or collapse-free (α < 1). Operator dynamics (Theorem C) incorporate trust poisoningwith a critical AI accuracy threshold below which improving AI paradoxically increases system uncertaintyand organizational capacity erosion producing secondary hysteresis and a latent regime of hidden degradation. A uctuationresponse duality (Theorem D) generates early-warning signatures under phenomenological Wright Fisher noise: under α = 1, equal-time variance saturates while autocorrelation diverges; under α>1, both diverge. Analytical predictions are veried numerically on heterogeneous networks (n = 5-100) with Monte Carlo SDE validation. Empirical consistency is assessed across 8 domains using exclusively published quantitative results, yielding directional agreement in 7 of 8 domains; we emphasize that this consistency does not constitute causal validation.