Strong stability of linear delay-difference equations

The uploaded paper’s Theorem 11 (scalar case) proves equivalence between: (1) total variation ∫ d|µ| < 1, (2) local strong stability, and (3) strong stability, using the perturbation model via pushforwards x(t)=∫ d(φ_*µ)(θ) x(t+θ) (Definition 9) and the Hale–Silkowski/Melvin criterion for the finitely many point-delay reduction (Theorem 8) . The candidate solution reproduces the same structure: (1)⇒(3) by a variation-based sup-norm contraction estimate, and (2)⇒(1) by a Hahn–Jordan partition and a φ that concentrates mass at rationally independent points so that the scalar Melvin criterion is violated (sum of absolute coefficients ≥ 1) . The only issue is a minor quantitative slip in the constant K for the exponential bound (they take K=1/v, whereas K≥v^{-2} suffices); this does not affect the logical equivalence or correctness. Overall, both the paper and the model establish the same result with essentially the same proof ideas .