The Mathematics of Large Language Models II: Concentration — A Depth Lower Bound for Robust Classification

We prove a depth lower bound for robust two-class separation by deep architectures whose data lies on a normal Lévy family. Let (Xₙ, dₙ, μₙ) be a normal Lévy family of rate cₙ, and let fₐ, fb: Xₙ → ℝ be logits with Lipschitz constants at most Lₙ. Suppose two disjoint Borel sets Aₐ, Ab ⊂ Xₙ, each of measure at least pₙ, are robustly separated by margin Δₙ, in the sense that fₐ − fb ≥ Δₙ on Aₐ and fb − fₐ ≥ Δₙ on Ab. We show that pₙ ≤ 2C·exp (−cₙΔₙ²/ (4Lₙ²) ), hence that Lₙ² ≳ cₙ whenever pₙ and Δₙ are bounded below. For logits realized by depth-kₙ architectures with L₀-Lipschitz primitives, this forces logarithmic depth growth kₙ ≳ log cₙ as a structural consequence of robust separation alone — independent of any specific learning algorithm, loss function, or generalization bound. The result formalizes the intuition that "expressivity is not explanation" on a concentrating ambient space: depth is not optional decoration but the price of binary contrast that survives concentration. We discuss applications to vision and language models, give quantitative numerical estimates for representative settings, and contrast the bound with classical universal-approximation and expressivity results, which can guarantee the existence of arbitrarily expressive logits but cannot make them robust on a high-dimensional concentrating data manifold without paying the depth toll exhibited here.