EFFECTIVE INTEGRABILITY OF LINS NETO’S FAMILY OF FOLIATIONS

Central discrepancy: the candidate asserts the quadratic maps act on the pencil parameter by translations and inversion, namely (Q1)^#(F_t)=F_{t+1}, (Q_τ)^#(F_t)=F_{t+τ}, (Q_{τ^2})^#(F_t)=F_{t+τ^2}, (Q_∞)^#(F_t)=F_{-1/t}. The paper proves (by explicit pullbacks of the defining 1-forms) that the strict transform under Q_i sends Ft to F_{q_i(t)} with q1(t)=−t−1, qτ(t)=−t−τ, q_{τ^2}(t)=−t−τ^2, and q∞(t)=1/t (Proposition 7), and illustrates this on concrete pencils (e.g., Q∞(F∞)=F0 and Q1(F0)=F−1), which contradicts the model’s update rules (F0 ↦ F1 in the model) . The paper’s algorithm reduces any t=m+nτ∈Z(τ) to F1 or F∞ by a piecewise sequence of these involutions and then inverts the word to construct F_{m+nτ}, including degrees and multiplicities; examples and code are provided (Sections 7–9), whereas the model’s single closed-form word W(m,n) based on incorrect “translation” dynamics is not supported by the paper’s proofs or examples . Other aspects (starting pencils F1,F∞; the explicit Qi; the use of strict transforms and gcd-removal; and tracking degrees/multiplicities under Cremona) are in line with the paper’s approach, but the wrong parameter dynamics vitiate the core construction.