Deterministic Asymptotic Projection and Critical Singular Structures for q-Deformed Topological States

We establish a mathematically rigorous framework for deterministic asymptotic projection driven by spatially singular dissipative flows on Riemannian manifolds. Modeling environmental constraints as closed submanifolds, we utilize Hardy–Rellich inequalities to derive the maximal admissible singular structure. By embedding a generalized q-deformed response Φq (η) and a coupling parameter ξ into the quadratic form, we define a norm-preserving semilinear evolution equation. We prove that the induced nonlinear dissipation enforces subspace capture and yields strong convergence of the associated density matrix in trace norm. Crucially, this continuous geometric flow provides a possible dynamical mechanism underlying effective state selection, bridging continuous unitary evolution and discrete structural stabilization.