Synchronization of High-Dimensional Linear Networks over Finite Fields

The paper defines W1 via (Ai−A1)A1^tα=0 (t=0,…,m−1), sets W= {1n⊗α: α∈W1}, proves A1-invariance of W1 and A-invariance of W (Lemma 3.4), and shows synchronization iff trajectories enter W (Lemma 3.5). Using a basis adapted to W, A becomes block upper triangular with diagonal blocks Q (the action of A on W) and Â22, yielding PA(λ)=λ^{nm−d}PQ(λ) iff Â22 is nilpotent (Theorem 3.2). These steps appear verbatim in the paper’s statements and proofs and match the candidate solution’s structure: invariant subspace construction, quotient/nilpotent characterization, and characteristic-polynomial factorization. The candidate’s use of the quotient map Ā on V/W is equivalent to the paper’s Â22 block; the only addition is an explicit uniform time bound via Cayley–Hamilton, which is consistent with the paper’s nilpotency criterion. See Lemma 3.3–3.5 and Theorem 3.2 for the paper’s statements and proof structure, and the block-triangularization equations (12), (14), (16)–(17) corroborating the model’s argument.