Some research problems do not disappear when we stop working on them. They move into the background, waiting for the right question, the right tools, or sometimes the right technology to bring them back to life.

From 2001 to the present

My personal journey with P vs NP began around 2001. I was approaching the problem independently, without a research team, computational laboratory, or modern AI collaborator. The attraction was simple but profound: if a solution can be verified efficiently, must there also be a way to find that solution efficiently?

P vs NP is one of the foundational questions of theoretical computer science. But my interest was never simply in repeating its standard definitions. I wanted to understand why apparently difficult problems become difficult, and whether some deeper structure might lie underneath that complexity.

The first exploration: 2001

The early work was exploratory. I looked at problems such as SAT, 3-SAT, Hamiltonian Path, Clique and the Traveling Salesman Problem, trying to understand what made them difficult and whether apparently different NP problems shared an underlying structure.

The ideas were primitive compared with today’s mathematical language, but the underlying instinct was important: look beneath the individual problem and search for the mechanism generating the difficulty.

2006: the Elsevier wall

By 2006, I had developed ideas that I believed were worth submitting for serious mathematical consideration. I submitted material to Elsevier through a theory-focused journal. The submission did not lead to the constructive research dialogue I had hoped for. The work was rejected, and the research effectively stopped.

That moment became a dividing line. The problem itself had not been solved, and my curiosity had not disappeared, but life continued. P vs NP went into what I now call the idea box.

The long pause

For many years, professional responsibilities and other interests took priority. Yet the underlying way of thinking did not disappear. I continued to think about patterns, structure, learning, geometry and how apparently intuitive human judgments might eventually be translated into formal rules.

The restart: when ChatGPT became capable enough

The research restarted when modern generative AI became sufficiently capable to function not merely as a search engine or calculator, but as an intellectual testing partner.

This changed the economics of independent exploration. Ideas that previously required weeks of manual checking could be discussed, challenged, reformulated and tested much faster. More importantly, AI made it possible to repeatedly ask the same fundamental question from different directions: what structure does a difficult problem actually contain?

AI has not magically solved P vs NP. In fact, one of the most interesting discoveries of the renewed journey has been seeing exactly where AI is still weak. That weakness has become part of the research.

The current struggle

The current investigation is exploring whether geometric structure in apparently random point distributions can provide a route toward a more general understanding of computational complexity. Human observers can often recognize clustering, relative distance, boundaries and geometric organization almost immediately. Turning that intuition into precise mathematics is much harder.

One developing research direction examines clustering, geometric structure and related properties of point distributions, with the Traveling Salesman Problem serving as an important testing ground. The aim is not to claim that a solution has been found, but to investigate whether apparently difficult search problems contain structural information that can be formally exploited.

The central question is: can the subjective human perception of structure in apparently random data be converted into precise mathematical rules that work on arbitrary instances?

Why the struggle matters

The most valuable result may not ultimately be a single algorithm. It may be a better formulation of the problem. If P = NP, there must be some general reason why the apparent difficulty of NP problems can be overcome efficiently. If P ≠ NP, understanding why certain structures resist efficient discovery is equally fundamental.

Either way, the deeper target is the same: understand the boundary between finding structure and merely verifying structure.

Twenty-five years later

  • 2001: first serious independent exploration.
  • 2002–2005: extended experimentation with NP problems and structural ideas.
  • 2006: submission to Elsevier and the end of that phase.
  • 2007–2022: a long pause while the problem remained in the idea box.
  • 2023 onward: renewed exploration as AI became capable of supporting sustained intellectual experimentation.
  • 2026: a new research direction centred on structure, geometry, clustering and the possibility of extracting useful invariants from apparently random data.

The road ahead

I do not claim that P vs NP has been solved. The current work is a struggle precisely because the gap between an interesting intuition and a mathematical theorem is enormous. Patterns that work on selected examples are evidence worth investigating, not proof of a universal law.

The next stage is therefore formalization: define the problem precisely, establish the objective function, identify the necessary lemmas, attack the assumptions, construct counterexamples and determine exactly where the emerging framework succeeds or fails.

After twenty-five years, the old question has returned—but this time it has a laboratory.

And the laboratory is no longer only a notebook. It is a human mind working alongside a machine capable of challenging, testing and accelerating ideas.

The journey continues: one structure, one experiment and one lemma at a time.

Research note: The detailed active mathematical construction is intentionally not disclosed here while the current investigation is still developing. This archive records the chronology and broad research direction rather than publishing an unfinished research recipe.

P vs. NP: From the 2006 Geometric Seed to a Hierarchical Proof Framework