Z-Plane Neural Networks: Bounded Geometric Activation Replaces ReLU and LayerNorm

The Z-Plane Neural Network introduces a novel architecture that replaces traditional activations like ReLU and normalization techniques such as LayerNorm with a geometric activation function called Radial Bounding. This function maps hidden states onto a 2D hypersphere, ensuring 1-Lipschitz continuity and preventing gradient vanishing, which addresses issues like dead neurons and loss of directional information. A 100-layer Z-Plane Multi-Layer Perceptron achieved 98.34% accuracy on the MNIST dataset, demonstrating that this approach can provide stability and performance in deep learning without conventional activation functions.