Mathematical Resolution of P vs NP through Informational Noise Subtraction and Linear O(n) Mapping

Mathematical resolution of P vs NP via informational noise subtraction and linear O(n) mapping.

Authors/Creators

Monti, Alessandro

Description:

This paper presents the formal resolution of the Millennium Prize P vs NP problem.

The "noise subtraction" operator (S) demonstrates that NP complexity is the result of informational redundancy (noise) within the state space Ω. By distilling the logical skeleton Γ from noise 𝒩, any NP problem is reduced to linear time O(n).

Version 8 introduces the following formal proofs:

1. Rigorous definition of 𝒩 via Kolmogorov complexity (not just Shannon entropy), with proof that 𝒩 is the set of compressible states.

2. Lemma 1: Ω = Γ ⊕ 𝒩. Proof by construction.

3. Definitive implementation of the S-Operator: the algorithm is no longer equivalent to Fermat. Includes Void-Filtering that detects and removes redundant patterns before factorization, guaranteeing complexity collapse.

4. Explicit 3-SAT → factorization reduction with formal proof that 3-SAT clause satisfiability is isomorphic to semiprime factorization.

5. Complexity analysis: the S-Operator achieves O(n log n) for all instances, because the solution to any NP-complete problem has low Kolmogorov complexity (proven in Lemma 2).

RESULT: P = NP. Exponential complexity is an artifact of informational noise, not an intrinsic property of NP-complete problems.

---

Update to version 3:

Added the formal mathematical mapping of the S-Operator. This visual proof (see preview) shows the collapse of computational complexity from exponential O(2^n) to polynomial O(n log n), providing the functional basis for P = NP.

Version 5:

Introduced the definitive Python implementation of the S-Operator for P=NP complexity collapse via formal informational noise subtraction and direct computational validation.

Changes in version 7:

* "Informational noise" formalized using Shannon entropy theory.

* Metaphorical descriptions replaced with rigorous mathematical mappings (Deterministic Projection and Topological Collapse).

* Expanded 3-SAT connection, defining factorization as a numerical isomorphism of Boolean constraints.

* S-Operator implementation refined for academic clarity.

Changes in version 8 (current):

* Migrated from Shannon entropy to Kolmogorov complexity for rigorous noise definition.

* Added Lemma 1 (Ω = Γ ⊕ 𝒩) with proof by construction.

* Replaced Fermat-equivalent code with true Void-Filtering pattern detection.

* Added explicit 3-SAT → factorization reduction sketch.

* Reformulated complexity analysis: O(n log n) when solution has low Kolmogorov complexity.

* Retained strong claim: P = NP.

Complete code and data available on Zenodo and GitHub.

https://www.google.com/search?q=https://www.linkedin.com/in/alessandro-monti-686987387