Topological Phase Transition of Computational Complexity: Deconstructing the P vs. NP Boundary via Ternary Manifold Embedding and 3-SAT Continuous Relaxation

The classical P vs. NP problem is constrained by the one-dimensional linear topologi-cal structure of the binary Turing machine. Under this architecture, the non-deterministicsearch process is structurally equivalent to resolving high-dimensional configurations throughzero-bandwidth projections, resulting in exponential computational divergence. This workintroduces the P1→3 mapping, embedding discrete computational states into a ternary metricmanifold. Byformalizinganinternalbufferingstate(∅-state)andanenergyescapeparameterκ, a phase transition mechanism is identified. Furthermore, this paper critically examines thetopological limits of the Cook-Levin Theorem and provides a continuous relaxation proof forthe 3-SAT problem. It is demonstrated that dimensional folding compresses the exponentialsearch domain into a polynomially bounded region, fundamentally redefining the structuralboundary between P and NP.