Convergent Targeted Spectral Locking and a Counterexample for Adaptive Gating

We study a simple seed-dynamics system that alternates a diffusive linear operator, apointwise nonlinearity, and a spectral “gate.” We prove global convergence with a linear ratefor the targeted variant that projects onto a fixed Fourier band and uses a strictly contractivenonlinearity. In contrast, we show the popular adaptive variant—which reselects the dominantbin each step—is not contractive by explicit counterexample. Empirical results (Python 3.10,NumPy) match theory: an empirical Lipschitz of ≈ 0.024 and a tail-rate ≈ 0.592 under proofparameters, and an adaptive flip witness with ∼9000× output separation from inputs thatdiffer by 10−4. We release a single-file, self-contained prover/disprover script reproducing allnumbers