Spin symmetry is an essential physical principle for both chemical fidelity and scalable electronic structure simulation. We developed and numerically analyzed an algebraic geometric framework for spin-adapted coupled-cluster theory targeting the spin-singlet sector. We provide an explicit description of the singlet many-body states via an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. Enforcing spin invariance through the SU(2) Casimir element that characterizes the spin-singlet subspace, we define spin-adapted truncation varieties via embeddings of graded subspaces of the excitation ring and identify the CCS truncation variety with the Veronese square of the Grassmannian, linking coupled cluster truncations to concrete geometric objects. Compared to the spin generalized formulation, this viewpoint yields large reductions in both dimension and CC degree, which governs the number of homotopy paths required to compute all coupled cluster solutions, and we present scaling studies demonstrating asymptotic improvements. We then exploit this reduction to compute the full manifold, providing systematic algebro computational studies of coupled cluster solution landscapes that were previously out of reach.