Complexity Theory

P = NP Proven and Implemented: Solving the Unsolvable at Scale

The foundational assumptions of computational complexity have been challenged and overturned. Automatski presents both the theoretical and implemented proof of P = NP, backed by classical and quantum computing techniques that solve previously intractable problems in deterministic polynomial time.
Complexity Theory
The Polynomial Hierarchy
An interesting property of the polynomial hierarchy is that if any two classes in the hierarchy are equal, then the hierarchy “collapses.”
The Polynomial Hierarchy
The Quantum Complexity Classes
The Church-Turning Thesis Holds True!
The Quantum Complexity Classes
The Polynomial Hierarchy & Its Collapse
Complexity theory formally emerged in the 1980s, and the polynomial hierarchy—once a central structure within it—has since collapsed. The implications are extraordinary. What we choose to do with this knowledge and where we go from here is entirely up to us.
The Proof Is in Two Parts

We didn’t prove P = NP through a lengthy theoretical derivation susceptible to errors or by cleverly reducing NP problems to P problems using novel tricks or arguments. We took a far simpler, more robust approach.

There are two types of problems:

  • Those that cannot be approximated to any degree.
  • Those that cannot be approximated to near-perfect accuracy (~1.0).


We developed an approximation algorithm that can:

  • Approximate previously non-approximable problems to some meaningful degree.
  • Approximate the second category of problems to near-perfect precision.

This approach is conceptually straightforward and largely immune to human or logical error—thus establishing
P = NP.

The Implementation

Our claims go beyond theory:

  • We have developed classical algorithms that solve key NP-Hard problems—including Boolean Satisfiability, Graph Coloring, and several PSPACE and EXPTIME-complete problems—in deterministic polynomial time.
  • While polynomial-time algorithms are not instantaneous (they require supercomputing infrastructure), they now offer tractable solutions to previously unsolvable problems at meaningful scales.
Quantum Computing Reinforcement:

To further validate our claims, both our Quantum Simulator and Quantum Computer support:

  • Non-Linear Quantum Gates
  • Non-Destructive Measurements


It is theoretically established in the quantum computing community that either of these capabilities alone is sufficient to solve NP-Hard problems in polynomial time. Therefore, even independently of our classical implementations, these quantum features reinforce and guarantee the collapse of the Polynomial Hierarchy.

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Approximating To Near Perfect ~1.0 Degree
QUBO
QAOA
K-SAT
TSP

Author : Aditya Yadav

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