Absolute Closure of Point-Defect Operators: Finite-Rank Birman--Schwinger Reduction, Odd Determinant Action, and the Dual Odd/Even Zeta Hierarchy with Empirical Verification

This manuscript provides a closed, operator-theoretic construction of point-defect perturbations on a compact one-dimensional manifold. It mathematically proves that these distributional defects can be exactly reduced to finite-rank matrices using the Birman-Schwinger map, bridging localized physical defects with exact spectral geometry. By constructing the exact periodic Green's kernel for a circular domain, the paper explicitly computes the finite-dimensional Gram matrix. It then introduces an "odd determinant action" that naturally isolates the signed, odd-power trace invariants of the interaction operator. Exact, closed-form symbolic formulas are derived for the third, fifth, and seventh-order couplings, alongside a completely solvable symmetric two-defect configuration. Furthermore, the framework proves a strict structural bifurcation for compact operators exhibiting inverse-linear spectral decay: the odd Riemann zeta values (such as zeta(3), zeta(5), and zeta(7)) are uniquely governed by the compact defect operator itself, while the even zeta values are governed by the positive closure Laplacian. Finally, the theoretical results are supported by a fully reproducible JAX numerical implementation. This empirical layer confirms the closed-form matrix reductions and the symbolic coupling formulas to machine precision, and explicitly demonstrates the odd-zeta trace decomposition—separating the universal Riemann zeta contribution from the convergent spectral remainder.