Topological Collapse of Complexity: P = NP under the NLS Structural Paradigm.

Published July 18, 2026 | Version v1

Description This document proposes a conceptual and constructive solution to the millennium problem $P$ vs $NP$. It is demonstrated that the asymmetrical complexity barrier between verification (P) and resolution (NP) is a direct consequence of using a deficient algebraic coordinate system (sequential brute force). By introducing the NLS Universe (Discrete Equilibrium Structure), it is proven that combinatorial complexity collapses to constant time $O(1)$. As a proof of concept, the generation algorithm for the NP-Complete Sudoku problem (SDK-NLS) is presented, establishing the Theorem of the Symmetry of Extremes: an empty board (0 clues) and a nearly solved board (80 clues) require the exact same structural computational effort. Furthermore, it is demonstrated that the solution space is not found through search, but generated through topological gears and combinatorial parametrization.

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