Theta Closure in Modal Triplet Theory II: Direct Geometric Realization of Nonabelian Overlaps

We present an explicit geometric realization of the nonabelian overlap integrals appearing in the Θ-closure program of Modal Triplet Theory (MTT). Building on the Θ-targets for SU(2) and SU(3) gauge sectors derived in the core Θ-closure paper, we compute the leading-order overlap integrals directly from internal geometry. The SU(2) overlap is evaluated on a constant-curvature lens layer modeled by a round two-sphere, while the SU(3) overlap is computed on a compact Heisenberg nilmanifold with a left-invariant metric. We show that these geometries reproduce the required Θ-targets exactly at leading order and satisfy all spectral-gap and admissibility constraints of the MTT Foundation with large margin. The construction provides a concrete existence proof for Θ-closure using explicit internal geometry, independent of twistor or string-theoretic encodings, and establishes a computable baseline for higher-order corrections or alternative realizations.