Consistency–Defect Gradient Flows and No–Meta Self–Purification in Value–Anchored Multi–Agent Systems

This paper develops a structural “blueprint” for no-meta self-purification in value-anchored multi-agent systems built on entropy–transport (ET) gradient flows. The starting point is an analytic background where macroscopic dynamics of laws on a Polish state space are described by evolution variational inequality (EVI) gradient flows with respect to ET distances such as the Hellinger–Kantorovich and fibered Bures–HK metrics. On top of this existing ET/EVI theory, the paper adds a design layer that introduces a finite-dimensional parameter www controlling an internal law class and defines a windowed consistency-defect functional FT (w) FT (w) FT (w) from the positive part of the EVI residual. In an idealized convex regime, FT (w) FT (w) FT (w) is assumed to be continuously differentiable, Lipschitz, and strongly convex on a compact parameter region. Under these conditions, both the continuous-time gradient flow w˙=−∇FT (w) w = - FT (w) w˙=−∇FT (w) and its discrete gradient descent approximation induce a contraction in weight space with a unique minimizer w∗w^∗. The system is then coupled with a value-anchored EVI flow on law space admitting a distinguished anchor law μ♯^μ♯, which maximizes a value functional among fixed points of the semigroup. A block-diagonal product map on Law×RmLawᵐLaw×Rm, combining the value-anchored law update and the defect-driven weight update, is shown to be a contraction under suitable step-size and convexity assumptions, yielding global convergence to (μ♯, w∗) (^, w^) (μ♯, w∗). Operationally, the framework provides a no-meta self-purification mechanism: at run time the system never receives external labels of “malicious” behavior, and adaptation is driven solely by structural violations of the intended EVI laws, as captured by the consistency defect. The paper discusses how these assumptions can be interpreted as design targets on a compact region of parameter space, sketches a linear Gaussian example where FTFTFT becomes a positive-definite quadratic form, and outlines online stochastic approximation schemes for estimating FTFTFT from finite observation windows. Limitations (nonconvex regimes, fully coupled dynamics, partial observability) and directions for applying the framework to robust multi-agent and AGI-adjacent systems are highlighted.