Two new families of finitely generated simple groups of homeomorphisms of the real line

The paper’s arguments for both families are correct and complete. The candidate solution substantially follows the paper’s structure for the hard steps (repetitivity/regularization/advancing for G_λ and proximality/special elements/no-backtracking for H_n). However, it makes a critical error in Task A: it conflates Thompson’s group on the circle with its lifts to the line. It asserts T and T_λ (the lifts) are perfect/simple and uses this to deduce G_λ is perfect and that any nontrivial element of N ∩ T normally generates T inside T. In the paper, simplicity of the lifts is not true; instead, the needed fact is that any noncentral element of the lift normally generates the lift (used explicitly in the proof) . Because the model’s justification hinges on a false claim, its proof of simplicity for G_λ is invalid as written, even though the intended path mirrors the paper’s correct proof via Proposition 2.2.1 and the commutator trick . Task B matches the paper closely (simplicity, finite generation, and faithful lift/left orderability) , but an unnecessary claim about preserving the “marked isomorphism type” of fast n-rings goes beyond what is cited and is not needed.