On System Operators with Variation Bounding Properties

The paper’s Theorem 2 proves the identity det((ON)_{α,β}) = c̃r Ãr^{t−1} b̃k,r,β for α = {(1:k−r),(k−r+t:k+t−1)} with Ãr = A[r], c̃r = (Or)[r], and an explicit b̃k,r,β built from Ak−r (On)−1 and a selector, and then equates SVBk of O(A,c) to strict external positivity/negativity of the associated LTI triples (Ãr, b̃k,r,β, c̃r) via the SSCk characterization of two-block contiguous minors (Proposition 9 and Corollary 1) . The candidate solution derives the same determinant factorization using the same compound-matrix algebra (Cauchy–Binet, multiplicative compounds) and the same two-block minor scheme, and reaches the same SVBk equivalence. The only substantive discrepancy is a minor remark asserting c̃r ≠ 0 from observability, which need not hold for r < n; the paper does not make this claim. Aside from this, the logical steps, definitions, and conclusion match the paper’s proof and result.