Deterministic Solution of NP-complete Problems via Inertial Manifold Dynamics within the BSS Model Framework

Ontological Computing Theory (OCT) introduces a framework for resolving NP-complete complexity through continuous inertial manifold dynamics based on the Blum-Shub-Smale (BSS) model. By mapping 3-SAT instances onto a differentiable energy landscape, the system replaces symbolic search with deterministic physical evolution toward global attractors. The mathematical model utilizes Newton-type second-order dynamics and nonlinear adaptive dissipation to enable ballistic trajectories that bypass local minima, ensuring convergence at the global solution. Practical realization is framed through Log-Sum-Exp (LSE) synthesis, providing a path to implement logic-to-potential mapping via mixed-signal CMOS or integrated photonics. Furthermore, the research provides a topological foundation for UNSAT detection: the lack of fixed points in contradictory formulas creates stable limit cycles, detectable through spectral analysis even under significant noise. Simulations involving N=1000 variables show convergence times consistent with Kibble-Zurek scaling laws, validating polynomial-time performance. This work establishes the engineering parameters for a deterministic analog processor, repositioning NP-complete problems within the domain of continuous dynamical systems.