Compressed Manifold Reachability - A Formally Proven Polynomial-Scaling Safety Algorithm for High-Dimensional Robotic Systems ($d \ge 6$)

Real-time safety verification for high-dimensional robotic systems ($d \ge 6$) is constrained by exponential memory growth ($\mathcal{O}(G^d)$) during grid-based Hamilton-Jacobi reachability analysis. We introduce the Compressed Manifold Reachability (CMR) algorithm. This framework integrates local Lax-Friedrichs subspace solvers, a max-plus global synthesis fabric ($\bigoplus$), and tensor-train low-rank projection ($\mathcal{P}_{\text{TT}}$). By sequencing these structural components, CMR achieves tractable polynomial scaling ($\mathcal{O}(d \cdot r^2 \cdot G)$) while preserving absolute mathematical determinism. The complete architectural pipeline is machine-verified in Lean 4 through six foundational proofs establishing monotonicity, bounded tensor approximation, polynomial memory bounds, contraction mapping convergence, Lax consistency, and executable runtime safety contracts. Together, these machine-verified guarantees establish an unbroken bridge from abstract mathematical rigor to verifiable, reliable execution in safety-critical robotic control stacks.