Gate lattices

The paper proves simplicity of L̂_I by a commutator-and-generation scheme that starts from a nontrivial stabilized automorphism f, produces a nontrivial gate via [f, χ], and then uses EFP plus finite symmetric-group calculus to generate all even gate lattices; hence any nontrivial normal subgroup contains the generators and equals L̂_I. By contrast, the candidate solution relies on a Cantor-homeomorphism-style three-tower/micro-support argument that presupposes the existence, inside L̂_I, of elements supported on a single prescribed clopen cylinder and 3-cycles localized to that cylinder. Such “micro-supported” elements are not among the generators of L̂_I (gate lattices apply copies of a gate simultaneously along a finite-index subgroup), and the outlined localization steps are not justified within the gate-lattice framework. The model’s steps hinge on this unsupported micro-support claim, so its proof does not go through.