Softmax routing in mixture-of-experts models can concentrate probability on a subset of experts, suppressing access to the others. We study how this concentration forms, why it persists, and how suppressed probabilities recover. For fixed experts, a stationary-loss barrier identifies the face supporting the optimal routed output. In a common-target square-loss model with freely trainable expert outputs, slow adaptation tracks fixed-expert dynamics on finite horizons, while every fixed positive adaptation rate yields an interior routing limit. An exact normal-velocity decomposition separates complementary-expert covariance from relative gradient advantage. Under positive two-sided power bounds along recovery paths, uniform over initial masses, the resulting coefficient's order of vanishing gives logarithmic or power-law recovery times. Integrations of constructed fields reproduce these orders. An exact discrete log-odds identity relates neural updates to threshold crossing. In paired Transformer-MoE training, removing a class-conditional router-gradient contrast delays confirmed recovery and reduces accumulated log-odds on fixed tokens, both with and without capacity constraints.
We introduce NAGO, a noise-aware generative operator for efficient sampling from invariant measures of stochastic partial differential equations (SPDEs) in function space. Our central idea is to treat the source measure as a design variable rather than a fixed Gaussian default. We establish admissibility conditions for well-defined infinite-dimensional flows and show that the finite-band source--target covariance ratio controls the spectral stiffness of the marginal velocity field. This analysis yields an admissibility-constrained spectral-matching rule: choose a source that is sufficiently rough for well-posedness, then minimize covariance mismatch over the resolved Fourier modes. On three representative stochastic-field benchmarks, this source design improves few-step spectral accuracy and several complementary distributional and physical statistics. In a controlled Matern-field ablation, it reduces the accurate RK4 budget from about 30 steps for Gaussian functional flow matching to 3--5 steps, a $6$--$10\times$ reduction in neural ODE evaluations. These gains are strongest for stationary multiscale fields and depend on the metric, resolution, and guidance protocol.