Block Balance and Saturation Certificates for Cyclic Root Degenerations

We study boundary points of the reciprocal bilinear system associated with cyclic roots. For an arbitrary actual support pattern, we give a localized elimination that retains all kernel coordinates and identifies the saturated local equations. At the coordinates where both limiting vectors vanish, the two residual linear spaces are orthogonal complements for a signed pairing. Every coordinate block that splits one of these spaces must contain equally many time and frequency coordinates at a genuine boundary point. We prove this condition by constructing an explicit polynomial whose product with the common bilinear parameter lies in the original ideal. A cyclic order-ten example exhibits an unbalanced block despite satisfying the global support counts. Explicit square-order families show why intermediate vanishing rates must be retained.