Deep randomized models fix hidden-layer parameters through random initialization and learn only closed-form readouts, typically adding depth by stacking random trans formations without target-aware control of hidden-state evolution. We propose LAIR Net, the Leaky Alignment-Impulse Residual Network, which mixes a shallow learned anchor into each hidden state through a leaky residual transition. We derive a depth uniform bound on input-perturbation sensitivity and use controlled simulations to attribute gains over a randomized baseline to the anchor rather than recursion or added capacity. Benefits emerge when a nonlinear target structure is learnable at the available noise level and diminish for nearly linear targets or dominant noise. Across 23 benchmark datasets, LAIR-Net achieves the best average rank among eight randomized networks and twelve conventional models, with relative performance associated with the same nonlinear-structure and noise quantities identified in simulation.

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