Spectral Governance for Dynamical Systems: The Teleken-VÖLUNDR Framework

The Teleken-VÖLUNDR framework is a portable, event-triggered spectral governance law for high-dimensional dynamical systems. It enforces admissibility in the frequency domain by monitoring the spectral threshold functional Δ(t) = U⁻³(t) − X_threshold (where U⁻³ measures high-frequency energy concentration) and applying an analytic governance controller ug(t) = −κ ∇U⁻³ only when the admissible manifold A = {x : Δ(x) ≤ 0} is violated. An O(N log N) closed-form gradient (derived via quotient rule and single IFFT) enables practical deployment at large state dimensions. Formal stability is proven via the discrete Lyapunov equation AᵀPA − P = −Q, confirming positive-definite P and Schur stability with spectral radius ρ(A) = 0.99926212 < 1. The P23 admissibility projection is shown to be contractive (L ≤ 0.95) within the admissible manifold by the Banach fixed-point theorem, guaranteeing a unique attractor. Empirical validation spans 3,600,000 governed steps across three qualitatively distinct regimes—neural-network training, reinforcement-learning policy gradients, and fully chaotic logistic-map dynamics (r = 3.9 with tanh normalisation)—at six stress levels (0.5–20.0). All 18 conditions returned strictly negative Δ(t) and GREEN status with zero governance violations. The framework is system-agnostic and deployable as an external_update wrapper around arbitrary state-update functions. Accompanying files include the full academic paper, detailed empirical results report, and live execution summary of the three required next steps (Lyapunov certificate, chaotic-map fix explanation, and controlled LLM experiment harness). Reproducible with seed = 42. All code and results are open for verification. Keywords: spectral governance, Lyapunov stability, dynamical systems control, AI safety, gradient explosion prevention, chaotic map governance, Teleken-VÖLUNDR, admissible manifold, event-triggered control