What question did this study set out to answer?

The aim is to demonstrate that P ≠ NP through the non-homeomorphic nature of energy landscapes.

June 21, 2026Open Access

P vs NP: Topological Classification of Energy Landscapes and the Generative Foundation of Computational Complexity

Key Points

The aim is to demonstrate that P ≠ NP through the non-homeomorphic nature of energy landscapes.
Develop a complete solution path for the P vs NP problem based on generative mathematics.
Establish a dual signature theorem linking computational problems to potential functions and energy landscapes.
Connect gradient flow isomorphisms to the classification of convex and non-convex landscapes.
Proved that landscapes with zero topological charge cannot be transformed into those with non-zero charge, implying no polynomial-time algorithm connecting NP to P.
Demonstrated specific correspondence between NP-type problems and topological defects, represented by unsatisfied clauses in 3-SAT.
Unification with other Millennium Problems through a shared core mechanism involving topological charge conservation.

Abstract

Abstract Within the framework of generative mathematics, this paper provides a complete solution path for the P vs NP problem. The core proposition: P ≠ NP is the inevitable non-homeomorphism of the topological types of energy landscapes—landscapes with zero topological charge (I=0) and non-zero topological charge (I≠0) cannot be mutually transformed. Complete argument chain: 1. Dual Signature Theorem: computational problems → potential function V(x)=|F(x)|² → energy landscape2. Algorithm–gradient flow isomorphism: an algorithm is the discrete realization of a gradient flow on an energy landscape (Theorem 2.6)3. Analysis–topology identity: P-type problems correspond to convex landscapes with I=0; NP-type problems correspond to non-convex landscapes with I≠04. Non-zero topological charge of 3-SAT (Lemma 5.2): unsatisfied clauses correspond to topological defects on the constraint surface, I≠05. Conservation of topological charge (Axiom 4): gradient flow cannot change the topological type of a landscape6. Synthesis: I=0 and I≠0 are not homeomorphic → there does not exist a polynomial-time algorithm mapping NP to P → P ≠ NP Steps 1–2 establish the bridge between computational problems and physical systems; the core logic of steps 3–6 is rigorously guaranteed by the Analysis–Topology Identity Theorem (L2 Theorem 4.2), the Dynamic Topology–Algebra Correspondence Principle (L2 Theorem 3.2), and the conservation of topological charge (Axiom 4). Unification with some Millennium Problems: The P ≠ NP of P vs NP shares the same core mechanism—the survivor signature of the external cutting of Axiom 4—with Riemann's Re(s)=1/2, BSD's rank = order of zero, and Yang–Mills' Δ>0. P vs NP is a binary classification of landscape topological types, while Riemann/BSD/Yang–Mills are parameterized signatures of spectral structures. Keywords: P vs NP; energy landscape; topological charge; analysis–topology identity; hole–solution correspondence; generativism; computational complexity; unification of the Millennium Problems

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