Amplified steady state bifurcations in feedforward networks

The paper proves that, for generic one-parameter steady-state bifurcations in feedforward networks with non-maximal critical cells, every non-synchronous branch has a unique root subnetwork B; cells in B follow the unique synchronous continuation X(λ)=Dλ+Rλ²+…, while cells outside B have asymptotics x_p(λ)=D_p λ^{2−μ_p} (supercritical) or D_p (−λ)^{2−μ_p} (subcritical), where μ_p counts critical cells along paths from B to p. This is stated and developed via Lemma 3.19 (existence of X), the root subnetwork definition, Theorem 3.25/3.26/3.28, and the triangular feedforward induction with explicit 2-jet computations (Ax²+Bλx+Cλ²+…) that yield, for boundary critical cells, transcritical-type behavior x_p(λ)=d± λ+O(λ²), with one branch coinciding with X(λ) and exactly one alternative non-synchronous branch . The candidate model agrees on the existence/uniqueness of B, the use of 2-jet data, and the global scaling law x_p∼λ^{2−μ_p}, but makes a crucial local error at boundary critical cells: it claims a cancellation up to order λ² forcing x_r(λ)∼λ², whereas the paper’s reduced equation is Ax²+Bλx+Cλ² with generically nonzero C, giving x_r(λ)=d−λ+O(λ²) (with the other choice coinciding with X) . The model also omits the paper’s necessary sign inequalities that can block a branch when multiple boundary critical cells impose incompatible λ-sides (Theorem 3.28 and the worked example) . Hence the paper’s argument is correct and complete for the stated generic setting, while the model’s solution contains a material mistake in the boundary analysis and lacks key genericity side-conditions.