Solenoids in automorphism groups of evolution algebras

The paper proves (i) U fits into a split exact sequence U ≅ Diag(A;B) ⋊ AutGrphK(E,w) (Theorem 6.17), (ii) under 2LI one has Aut(A)=U (Corollary 6.19), and (iii) Diag(A;B) ≅ lim C (Theorem 3.10). The candidate reproduces these statements, but its proof of the splitting constructs a “canonical” section by normalizing coordinates to 1 at chosen roots of the initial SCCs via a diagonal element d∈lim C. This normalization is generally impossible on strongly connected components with cycles: d is constrained by relations like d_r=d_r^{2^L}, so d_r must lie in μ_{2^L−1}(K), which need not equal x_r^{-1}. Concretely, for a two-vertex cycle with weights 1 and λ and σ swapping vertices, solutions x to lim F_σ satisfy x_1^3=1/λ, while d∈lim C forces d_1^3=1; unless x_1 is a cube root of unity, no diagonal d can normalize x_1 to 1. Thus the section S defined by the candidate may not exist. The paper avoids this pitfall by providing a functorial section via G(θ) (Lemma 6.9) to split the sequence (Theorem 6.17). The rest of the candidate’s arguments (2LI implies monomial automorphisms; diagonal subgroup as an inverse limit) agree with the paper’s results. See Theorem 3.10 (Diag(A;B) ≅ lim C), Theorem 6.17 (U ≅ Diag(A;B)⋊AutGrphK), and Corollary 6.19 (2LI case), and the construction of Fσ (Definition 6.13) and the EK≃GrphK equivalence (Proposition 6.10) in the paper.