Dynamic Closure Theory: A Minimal Variational Law for Coherence-Based Self-Consistency

We propose \textbf{Dynamic Closure Theory (DCT)}: a minimal law for coherence-based closure in open systems. Defining closure as a normalized coherence variable $C \in [0,1]$ and parameterizing evolution by invested resources/energy $(E)$, DCT adopts the saturating law:
\begin{equation*}
\frac{dC}{dE} = \lambda C(1-C), \quad \lambda > 0
\end{equation*}
This framework is shown to be a 'meta-law' that unifies two fundamental physical pictures: (1) the \textbf{static equilibrium structure} of \textbf{Fermi-Dirac statistics} (which shares the isomorphic $df/dE \propto f(1-f)$ form) and (2) the \textbf{non-equilibrium dynamic evolution} of \textbf{logistic growth}. The theory's boundary is further defined by its dual, the 'aggregative' Bose-Einstein statistics ($df/dE \propto f(1+f)$), establishing DCT as the governing law for 'exclusive' (Fermi-type) systems defined by competition and saturation.
A gradient-flow formulation reveals an underlying potential $V(C) \propto \frac{1}{3}C^3 - \frac{1}{2}C^2$, ensuring bounded convergence from the unstable state $C=0$ to the stable attractor $C=1$. The framework produces falsifiable predictions via universal scaling laws. We validate DCT through quantitative scaling laws in three empirical domains: (1) Belousov-Zhabotinsky chemical oscillations exhibit predicted inverse scaling (slope $\approx -0.91$, predicted $\approx -1$); (2) protein folding shows consistent inverse denaturant dependence (slope $\approx -0.92$, predicted $\approx -1$); (3) consciousness emergence occurs at a metabolic threshold of $42\%$ of normal activity, within DCT's predicted range ($35$--$50\%$). In the homogeneous limit with constant resource inflow $(\dot{E} = \gamma > 0)$, DCT naturally reduces to the classical time-domain logistic $\dot{C} = rC(1-C)$ with $r = \lambda\gamma$. DCT thus provides a unified framework, grounded in a variational principle, that connects the statistical mechanics of equilibrium (Fermi-Dirac) with the non-equilibrium dynamics of complex systems (Logistic), all while remaining empirically falsifiable across diverse scientific domains. A spatiotemporal field extension is outlined as future work.

 

Aspects of this theoretical framework are subject to pending patent applications by Yunaverse, Inc.