| _version_ | 1866902308472750080 |
|---|---|
| author | Fisher, Christopher |
| author_facet | Fisher, Christopher |
| contents | <h3>A Formal Resolution of the P vs NP Problem via Entropy Barriers, Compression Limits, and Non-Relativizing Structural Complexity</h3> <p>This paper presents a mathematically rigorous proof that the complexity classes P and NP are not equal. The approach integrates entropy theory, instance compression, and circuit complexity to demonstrate that no deterministic polynomial-time algorithm can decide all NP-complete problems.</p> <p>The proof begins by establishing an entropy barrier, showing that any attempt to distinguish among exponentially many valid solutions with polynomially bounded resources results in an information-theoretic contradiction. A compression impossibility theorem is introduced to show that instances of NP-complete problems cannot be reduced to polynomial length while preserving solvability, violating known bounds on Kolmogorov complexity.</p> <p>Additionally, the paper demonstrates that solving NP-complete problems requires superpolynomial circuit size, and that no uniform or nonuniform polynomial-sized family of algorithms can bypass this requirement. The entire argument is non-relativizing, meaning it remains valid regardless of oracle access or relativized computational models.</p> <p>A section is also dedicated to the limits of quantum computing, clarifying that quantum algorithms cannot collapse NP into P due to fundamental constraints on state representation and computational entropy.</p> <p>The result is a complete and modular proof that P does not equal NP, resolving one of the most significant open questions in theoretical computer science.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15087349 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Resolution of P ̸= NP: Compression Barriers, Non-Relativizing Arguments, and Entropy Formalization Fisher, Christopher P ≠ NP proof theoretical computer science entropy barrier compression impossibility non-relativizing arguments circuit complexity Kolmogorov complexity NP-complete problems decision tree depth cryptography implications Clay Millennium Problem computational complexity algorithmic barriers structural complexity quantum complexity <h3>A Formal Resolution of the P vs NP Problem via Entropy Barriers, Compression Limits, and Non-Relativizing Structural Complexity</h3> <p>This paper presents a mathematically rigorous proof that the complexity classes P and NP are not equal. The approach integrates entropy theory, instance compression, and circuit complexity to demonstrate that no deterministic polynomial-time algorithm can decide all NP-complete problems.</p> <p>The proof begins by establishing an entropy barrier, showing that any attempt to distinguish among exponentially many valid solutions with polynomially bounded resources results in an information-theoretic contradiction. A compression impossibility theorem is introduced to show that instances of NP-complete problems cannot be reduced to polynomial length while preserving solvability, violating known bounds on Kolmogorov complexity.</p> <p>Additionally, the paper demonstrates that solving NP-complete problems requires superpolynomial circuit size, and that no uniform or nonuniform polynomial-sized family of algorithms can bypass this requirement. The entire argument is non-relativizing, meaning it remains valid regardless of oracle access or relativized computational models.</p> <p>A section is also dedicated to the limits of quantum computing, clarifying that quantum algorithms cannot collapse NP into P due to fundamental constraints on state representation and computational entropy.</p> <p>The result is a complete and modular proof that P does not equal NP, resolving one of the most significant open questions in theoretical computer science.</p> |
| title | Resolution of P ̸= NP: Compression Barriers, Non-Relativizing Arguments, and Entropy Formalization |
| topic | P ≠ NP proof theoretical computer science entropy barrier compression impossibility non-relativizing arguments circuit complexity Kolmogorov complexity NP-complete problems decision tree depth cryptography implications Clay Millennium Problem computational complexity algorithmic barriers structural complexity quantum complexity |
| url | https://doi.org/10.5281/zenodo.15087349 |