Normalization is the Foundation of Deep Learning

Normalization layers (LayerNorm, BatchNorm, RMSNorm) have long been treated as a training-stability trick, and their mathematical meaning has rarely been examined. Starting from the normalization identity s = h / ||h||, we show that it forces an algebraic split of any hidden state into a hypersphere direction s in S^(D-1) and a magnitude scalar ||h|| --- what we call Normalized State Decomposition (NSD). Substituting into a linear output head yields logit_v = ||h|| * (s^T w_v): direction carries the query, magnitude is only confidence. On this identity we recast inference as deterministic navigation on a fixed hypersphere, treat the LM head as a semantic memory bank, and extract five testable coding laws --- task sparsity, dual voting, structured overlap, weight geometry, and hierarchical differentiation. Without any additional training, the framework unifies three long-standing puzzles --- neuron polysemanticity, superposition, and final-layer linear readout --- explains why NVIDIA's nGPT converges faster as a natural consequence of the spherical constraint, and shows that the sparse autoencoder's "polysemanticity -> superposition -> need a dictionary" chain does not hold geometrically. Experimentally, NSD reconstructs hidden states with zero training and zero error (deep-layer SAE reconstruction error reaches 1.53 on GPT-2); Fisher discriminant ratios locate 254 concept-discriminative dimensions across 20 semantic categories, and editing s along these directions drives 84%--100% of interventions to logit rank 1; adjusting only 20 dimensions redirects "France -> Lyon" from rank 15 to rank 1, with no weight updates. We validate these claims across Transformers, DeltaNet hybrids, MoE, and ViT.