The Structural Impossibility of P = NP

In this work, it is shown that the P vs NP problem is not merely unsolved, but structurally ill-posed in its classical formulation. Starting from a systemic comparison between prime number recognition and NP-completeness, a common principle becomes visible: global structures that emerge cannot be reconstructed by local procedures. The central thesis is that every previous attempt to solve NP problems using deterministic algorithms of class P implicitly employs the same structural projection that must, in fact, be overcome. What becomes visible is therefore not merely a technical failure, but an ontological boundary: P = NP is not only unlikely – it is structurally impossible.

Experimental Verification

The research is supported by two distinct reference implementations for the newly defined complexity class cNP (Constructive NP). These algorithms are designed as epistemic diagnostic tools rather than efficiency-optimized solvers, proving that the principle of Resonance-Guided Fixpoint Emergence applies to both Number Theory and Combinatorial Optimization:

• MCG (Multiplicative Coverage Generator): A functional reference for prime generation that operates entirely without the modulus operator or trial division. It demonstrates that primality is a structural consequence of additive growth and multiplicative information loss, where primes appear as fixed points of an irreducible process.
• TSP-Fixpoint-Solver (cNP-TSP): An implementation for the Traveling Salesman Problem that replaces the traditional search-metaphor with a resonance-field model. By explicitly traversing the ontological barrier of (n-1)! permutations, it demonstrates that the global optimum emerges as a pre-determined fixed point of the spatial infrastructure, visible only through irreducible globality.

Key Findings of the cNP Class

• Irreducible Globality: The emergence of global fixpoints requires the accumulation of the entire preceding system state, rendering local, stepwise P-class algorithms structurally blind to the solution.
• Fixpoint Emergence vs. Search: The code-bases prove that certain solutions do not exist as "hidden objects" to be found, but emerge as states of maximal structural consistency (fixpoints) that are not locally reconstructible.
• Structural Separation: Since cNP processes are irreducibly global and P processes are locally reducible, the two classes possess incompatible information architectures, necessitating P \neq NP.

Related Theoretical Foundation: Heinz, K. J., The Irreducible Structure of the Prime Distribution, Zenodo, 2025.
DOI: 10.13140/RG.2.2.17117.47846
URL: https://zenodo.org/records/17649211

Reference Implementations: MCG (Multiplicative Coverage Generator): Included in this Zenodo record as MCG.go.

• TSP-Solver (cNP-TSP): Included in this Zenodo record as TSP.py.
• Dataset: The file koordinaten.csv is included as a mandatory input for TSP.py, providing the geographical coordinates (Ontological Basis) for the structural fixed-point identification.

GitHub Repository: https://github.com/syntaris/primes4everybody