When Averaging Amplifies: Stability and Sensitivity in Adaptive Mixing

Positive averaging can converge while amplifying input changes. We study adaptive mixing that retains the original input, as in non-local-means RED denoising and related graph and equilibrium architectures. A class-sharp convergence certificate gives an optimal input-response bound: smooth stochastic mixers approach that bound while their frozen averages (realized weights held fixed) approach identity, and cross-token responses can be negative. A photographic RED/NLM example and ordinary softmax kernels exhibit amplification. Convergence also need not preserve predictions. At a prespecified strong-mixing setting, adaptive mixing of full-dimensional DINO features in 32-image contexts reduces accuracy on 512 matched CIFAR-10 queries from 88.09% to 35.87%; freezing input weights retains 88.02%. An exact score decomposition and context removal identify recoverable interference: the original classifier recovers 87.96%. Finally, intermediate iteration budgets can destabilize downstream feedback even when a longer solve is stable. We give a sufficient condition covering every budget. These results separate guarantees for convergence, input response, task fidelity, and downstream use.