On the Dynamics of Three-Layer Neural Networks: Initial Condensation

The paper’s condensation theorem (Theorem 3) is essentially correct under the intended regime but its statement omits an explicit dependence on the earlier blow‑up assumption (Assumption 1), which the proof later uses (via Corollary 1) to ensure at least one coordinate diverges. The model’s proof captures the right invariants and obtains clean monotonicity/convexity, and it correctly establishes condensation when T* = ∞. However, in the finite‑time case the model argues that ||b_i|| must blow up because ||ḃ_i|| → ∞ as ||a|| → ∞, which is not justified: an unbounded, even monotone‑increasing derivative near a finite endpoint can still be integrable. The paper remedies this with detailed angle control and an explicit integral lower bound, while the model does not. Hence both are incomplete: the paper in hypothesis bookkeeping, the model in a key step of the finite‑time blow‑up case.