A Two-Layer Axiomatic Framework for Nonlinearity Nonlinear phenomena appear across diverse domains such as fluid dynamics, quantum systems, discrete maps, and number theory. Despite their differences, these systems share a universal structural feature: local simplicity coupled with global complexity. In this work, we axiomatize this universal structure through a two-layer circular framework (Local Circle / Global Circle) and develop the Generative Theory, Second Generation, a structural theory designed to describe nonlinearity itself. The theory is built upon the interference between a locally closing discrete circle (Local Circle) and a globally continuous circle with a real-valued angle (Global Circle). We present seven axioms (Axiom 1–7) that characterize nonlinear behavior within this framework. These axioms allow us to reconstruct seemingly heterogeneous systems—the Navier–Stokes equations, quantum interference, the logistic map, and the Riemann zeta function—under a single structural perspective. We further introduce the risk indicator R, defined by combining the local–global gap Δ with the global angle Θ, and propose a gating function Γ(R) that modulates nonlinear activation. This leads to a unified nonlinear evolution equation ∂G∂t=LG+Γ(R) BG,Δ, which serves as a general structural form capturing nonlinear activation common to fluids, quantum systems, and number theory. For the Navier–Stokes equations, we formulate a necessary condition for blow-up (structural hypothesis) based on the behavior of R. Numerical experiments using the Taylor–Green vortex demonstrate that R correlates strongly with vortex breakdown, dissipation, and localized nonlinear activation. Finally, we outline several directions toward a third generation of the theory, including a general framework for the gluing map H, a refined definition of the global angle Θ, equivalence classes of the risk indicator R, independence and completeness of the axioms, integration of discrete and continuous structures, and extensions toward a probabilistic generative theory.
Renji Nakayama (Thu,) studied this question.