Mathematical Foundations of Constrained $SO(3,3)$ Spacetimes: Part II. Geometric Limits, Constraint Algebra, and Topological Bounds

This paper establishes the rigorous mathematical proofs and algebraic consistency of the constrained SO(3,3) multitemporal spacetime framework. First, we provide the tensor derivation of the extrinsic curvature singularity, demonstrating that continuous diffeomorphism inevitably breaks down at the Wunderlich geometric limit, necessitating topological surgery. Second, we evaluate the Dirac constraint algebra of the transverse gauge, proving strict First-Class closure to guarantee unconditional unitarity and a ghost-free quantization. Third, we apply the Bogomol'nyi trick to the extended non-linear sigma model, establishing the BPS lower energy bound that strictly protects the topological defects from dissipating into the vacuum. Finally, we utilize Wilson Loop integration to prove that the geometric torsion within the transverse-time dimensions directly manifests as the observable Aharonov-Bohm phase shift. Together, these proofs solidify the geometric and quantum mechanical stability of the SO(3,3) multitemporal model.