THERE IS ONLY ONE FAREY MAP

The paper proves that, up to the natural symmetry action, there are exactly three two-map simplex-splitting algorithms that are contractive, namely the pairs (M0,M1), (M0,M1F), and (M0F,M1); only the Mönkemeyer case (M0,M1) yields a continuous Gauss-type map, and the parity-dependent orientation statements follow from determinant signs. All of this is stated and proved in Theorem 2.1, together with the determinant/continuity discussion and the FM0=M1 symmetry that equates time-t partitions across the three cases . However, the model’s Step 5 claims to have classified all simplex-splitting pairs up to equivalence into exactly three classes, which contradicts the paper’s orbit counting that shows (far) more simplex-splitting pairs exist (only three are contractive): e.g., 160 orbits for n=3 and rapidly growing thereafter . The model’s contractivity argument (Step 7) also relies on an unproven uniform positivity-time bound and an unstated uniform Hilbert metric contraction constant; by contrast, the paper either cites Mönkemeyer’s contractivity and transports it via FM0=M1, or works through incidence-graph restrictions using Lemma 3.2 and Lemma 3.3 to eliminate non-contractive cases . The model’s continuity and orientation conclusions agree with the paper, but its global classification claim is incorrect.