A balance-mode structure theorem for the unsigned kernel of the sphenic window complex

For the sphenic window complex C at level N (triangles the squarefree three-prime integers pqr in (N/2, N]), we describe the rational kernel of the unsigned boundary ∂ᵘ as a span of explicit balance modes fibered over vertex-pairs, namely Zaslavsky even-circle cycle spaces of the common link graphs, and certify it at an explicit finite perimeter: for five bundled N over ℚ and an eight-prime battery, and over every prime characteristic outside {2,3} on a two-stage integer-certified subset. Consequently the unsigned nullity e is a fibered balance count and the frustration term δ = elem − e becomes a balance count, reducing the canonical density to link-graph balance and connectivity combinatorics. The generators are the classified balanced and balanceable circuits of the oriented-hypergraph program of Rusnak and of Rusnak, Li, Xu, Yan and Zhu; the apparently unstated spanning statement addresses their declared open problem. The mode-span inclusion is proved unconditionally; completeness reduces to a single finite leakage-gauge condition, certified at the perimeter; the all-N form is open, with band localization proved and a richness program stated. This is the fourth paper in a series on the −1-eigenspace of the coprime graph (Papers 1 and 2: doi:10.5281/zenodo.20933665 and doi:10.5281/zenodo.21169867) and the companion to Paper 3. Machine-verifiable certificates are in the linked repository (concept DOI 10.5281/zenodo.21135221).