Dichotomies uniform on subspaces and formulas for dichotomy spectra

The paper defines the J-admissible dichotomy resolvent ρJ(A) and spectrum ΣJ(A) for the γ-shifted system, proves the dynamic characterization of dichotomy subspaces, and establishes the spectral theorem (ΣJ(A) is a nonempty union of at most d compact intervals, with a filtration for T=N and a decomposition for T=Z) via limiting Bohl exponents and an explicit resolvent formula (Theorem 26) . The candidate solution reproduces the same conclusions by a classical Sacker–Sell style argument using forward/backward stable sets Mγ,Nγ, openness of ρJ via perturbing γ within the dichotomy rate, monotonicity of dim Mγ, and the direct-sum construction Wk=Nγ1∩Mγ2. This matches Proposition 12 (Mγ=L1, Nγ=L2) and the paper’s construction of Mk and Wk (Theorem 26) . The only difference is that the model uses a coarser outer bound ΣJ(A)⊂[−log M,log M], while the paper sharpens it to [−ln‖A−1‖∞, ln‖A‖∞] via limiting Bohl exponents (Remark 23, Lemma 25) . No substantive logical gaps or contradictions were found.