Gating Oscillatory Criticality via a Minimal Multi-Axis Constraint

We propose a minimal gating principle for the emergence of oscillatory criticality in complex systems. The Stability–Rule–Negentropy (SRN) framework models system viability through a weakest-link constraint c = min(S, R, N), where S denotes structural stability, R denotes rule-modeling capacity, and N denotes active negentropy drive. System activation requires c > θ, with θ a threshold parameter. We show that this constraint is compatible with classical phase-transition theory by mapping static SRN activation to site percolation, yielding geometry-dependent critical thresholds in two and three dimensions. To connect structure with dynamics, we couple SRN gating to a Hopf-capable oscillatory core through an endogenous control parameter μ = k(c − θ). Under generic nondegenerate Hopf conditions, SRN provides a necessary threshold condition, and in Hopf-capable systems a sufficient gating condition, for the existence of sustained oscillations. We derive a testable near-threshold scaling law, A ∝ (c − θ)1/2, for oscillation amplitude. We further show that when the Wooden Barrel variable evolves self- consistently with the threshold, the system may remain in a near-critical oscillatory regime over a finite interval rather than undergo a sharp externally tuned crossing. Finally, we report a negative result: a minimal native SRN dynamical system without an explicit oscillatory core converges to stable fixed points rather than generating self-sustained limit cycles. This clarifies the theoretical boundary of SRN: its primary function is weakest- link gating and constraint of oscillatory criticality, not spontaneous generation of oscillations in arbitrary minimal dynamics. The framework thus separates structural eligibility from dynamical realization and provides falsifiable predictions for neural, artificial, and other adaptive complex systems. Keywords: SRN Framework; Critical Phase Transition; Negentropy; Hopf Bifurcation; Complex Systems