Marginally Unstable Discrete-Time Linear Switched Systems with Highly Irregular Trajectory Growth

The paper proves Theorem 1 rigorously via a two-step strategy: (i) an explicit analysis of special rational angles θ (Theorem 2) yielding either uniform boundedness (marginal stability) or linear growth (marginal instability) of all products for those θ, and (ii) a Baire-category argument which assembles these dense ingredients into the dense Gδ set Ω of Theorem 1, together with a linear-algebraic construction of the exceptional subspace W of dimension ≤ 3. Key technical pillars include: the block-upper-triangular decomposition and a subadditive translation-column functional whose limit exists (Theorem 2(i)) , the identity A1^q = I for a suitable choice of A1 parameters when θ = pπ/q with p even and related geometric estimates that give uniform boundedness across all words (Theorem 2(iii)) , and the continuity/Gδ constructions that yield Ω1, Ω2, and hence Ω, together with the vector-space argument that produces W of dimension at most 3 (proof of Theorem 1) . In contrast, the candidate model’s core bound—asserting a uniform-in-words estimate along convergent denominators q_n based on Denjoy–Koksma/continued fractions—never actually controls the supremum over all words. Specifically, the claimed reduction to a bound of the form max_{|w|=T} ||s(w)|| ≤ C(λ,θ)(||c0||+||c1||)[1 + T||I−R_θ^T||] suppresses an unavoidable linear-in-T contribution coming from many short 1-blocks; the suggested “telescoping” of 0-block contributions does not occur unless c0 and c1 are cohomologous with the same transfer function (the paper’s W-like exceptional case), and even then the interleaving of B1 blocks obstructs a simple telescoping. The paper never relies on such a (false) DK-type uniform bound; instead it proves boundedness at rational angles and uses continuity/Baire methods to reach Ω. Therefore, the paper’s argument is complete and correct, whereas the model’s solution is incomplete and relies on an unjustified, and in general false, uniform estimate.