Beyond the Grid: Probabilistic Expansion of Topological Safety and Asymptotic Utility in High-Dimensional Manifolds

The rigorous deployment of autonomous agents is frequently bottlenecked by the computational intractability of verifying safety constraints in high-dimensional state spaces. While recent advancements in Vector Symbolic Architectures (VSA) have successfully addressed the challenge of semantic drift in Large Language Models, a parallel and equally critical fragility remains in the kinematic control of autonomous decision-making. As agents scale from simple chatbots to persistent control systems, the ``Curse of Dimensionality'' renders traditional deterministic verification methods, such as grid-based Hamilton-Jacobi reachability, computationally prohibitive, with complexity scaling exponentially, $O((1/\eta)^n)$, for degrees of freedom $n>5$. This paper introduces Constrained High-Dimensional Barrier Optimization (CHDBO), a unifying framework that generalizes strict topological safety into the probabilistic domain. We propose three major advancements: (1) The generalization of strict topological definitions into a probabilistic framework utilizing Monte Carlo Barrier Certificates (MCBC) to verify the \textit{boundary consistency} of safe manifolds where $n \ge 128$; (2) A scalable ``Orthogonal Projection'' mechanism that mitigates the exponential complexity of grid checks, reducing verification \textit{sample complexity} to $O(1)$ relative to dimension (with linear computational cost $O(n)$ due to vector operations), by leveraging bounded Lipschitz continuity to guarantee the detection of non-singular failure modes; and (3) The integration of a gradient-based utility maximizer that ensures asymptotic convergence to global optima without violating the safety envelope. By imposing a strict hierarchical priority of the safety constraint over the utility objective, we demonstrate that aggressive performance and high-confidence safety are not mutually exclusive. Simulation results confirm that CHDBO maintains high effective safety rates in dynamic environments while preserving the utility required for high-performance autonomy.