Deterministic Asymptotic Projection and Critical Admissible Singular Structures on Riemannian Manifolds

We establish a mathematically rigorous framework for deterministic asymptotic projec-tion driven by spatially singular dissipative flows on Riemannian manifolds. Modeling en-vironmental constraints as submanifolds, we use Hardy–Rellich inequalities to identify thecritical admissible singular structure. The resulting singular interaction is formulated at thelevel of quadratic forms and embedded into a norm-preserving semilinear evolution equation.We prove that the induced nonlinear dissipation enforces subspace capture and yields strongconvergence of the associated density matrix in trace norm to a unique projector. Thisprovides a deterministic operator-theoretic mechanism for asymptotic state selection.