Research status: This article documents an evolving research programme, not a claimed solution to P vs NP. The mathematical construction described here is deliberately presented at a conceptual level while the current investigation remains unfinished.
From a question about P vs NP to a structural research programme
The central question behind this work is simple to state but extraordinarily difficult to settle: can problems whose solutions can be verified efficiently also be solved efficiently? My own exploration of P vs NP began long before the present AI era. The 2006 work developed a geometric line of attack, and the 2026 restart has allowed that old intuition to be reformulated, challenged and experimentally examined at a much greater speed.
The important shift in the current phase is from thinking about a combinatorial problem primarily as an enormous collection of possible solutions toward asking whether the underlying structure of the instance can be represented, compressed and manipulated directly. Euclidean TSP remains the main laboratory because its geometry makes structural relationships visible while retaining the essential difficulty of global optimization.
Structure before enumeration
A recurring observation is that points in a geometric instance are not necessarily an unstructured collection. Human perception immediately recognizes clusters, separation, boundaries and relative position. The research therefore asks whether some of the information normally hidden inside a factorial search can be represented by a smaller structural description.
One developing direction uses geometric scaffolds to localize candidate regions and then tests the resulting structures with geometric boundaries. The exact construction, definitions and operational sequence are intentionally omitted here while the work is still under investigation. The underlying question is whether such localization can preserve the information required for an exact global optimum rather than merely providing a useful heuristic.
Local geometry and the global problem
The historical work contains a simple geometric insight: when a route is modified locally, the change in length can be expressed through a small number of distances. Such local comparisons are exact and easy to calculate. The difficulty is not calculating a local change; it is proving that every globally relevant improvement can be exposed through an appropriate sequence of local or hierarchical transformations.
This distinction has become one of the central safeguards of the research. A successful local rule is not automatically a global optimization theorem. Any proposed framework must eventually establish a completeness result: if a candidate solution is not optimal, the framework must guarantee that an admissible structural difference leading toward improvement will be found.
The small-instance laboratory
Small instances provide the most useful testing ground. They are large enough to expose competing tours, closure conditions and alternative reconnections, but small enough that exhaustive computation can be used to challenge every proposed rule. The four-point case has become particularly useful because it exposes assumptions that can remain hidden in larger diagrams.
The research strategy is therefore deliberately bottom-up. Rather than immediately claiming a theorem for arbitrary n, the investigation asks whether the smallest non-trivial structures can be characterized rigorously, whether larger changes can be decomposed into those structures, and whether counterexamples can defeat the proposed hierarchy.
Hierarchical reduction
A second line of thought is hierarchical: represent an instance through progressively larger structural units rather than treating the entire combinatorial object as a flat permutation space. Small units are examined, compatible structures are combined, and the resulting representation is tested again. The hope is that a polynomially bounded description can replace an exponential enumeration.
But this is precisely where the real proof difficulty begins. It is not enough to show that a hierarchy can construct good solutions. One must show that the hierarchy cannot systematically exclude the information needed to recognize a better solution. The current investigation therefore treats completeness as the decisive issue rather than assuming it.
Construction versus certification
The 2026 work has also separated two questions that can easily become confused. The first is whether a structured candidate can be constructed efficiently. The second is whether a compact certificate can prove that no better candidate exists. A fast construction followed by an exhaustive search would obviously not solve the underlying complexity problem.
The more ambitious idea is therefore a structural certificate: if a proposed solution were not globally optimal, its first unresolved difference from an optimum would have to appear through some admissible structural transformation. Proving that statement for the entire class of instances is the central challenge. Until it is proved, the construction remains a hypothesis rather than a theorem.
Experiments, failures and counterexamples
Computational experiments have played an unusual role in this project. Their purpose is not to manufacture evidence for a desired conclusion, but to kill the conjecture if possible. Early failures forced changes in the representation and led to stronger attention to clustering, boundaries, recursive structure and closure. Later experiments have supported the emerging framework on the tested classes, but empirical success is not a proof of universal validity.
The working rule is therefore deliberately adversarial: every proposed lemma should be tested against deliberately difficult configurations, including clustered points, nearly uniform distributions, nested structures, outliers and cases designed to break the assumed hierarchy. A single valid counterexample can invalidate a proposed move set and is therefore valuable research information.
What would have to be proved?
- Identify the genuinely irreducible local structural comparison.
- Prove that larger admissible changes can be decomposed into valid smaller transformations, or identify the precise obstruction.
- Establish a completeness theorem showing that a non-optimal candidate cannot evade the permitted structural framework.
- Derive a genuine polynomial bound on the resulting representation and operations.
- Only after those steps survive rigorous scrutiny should any conclusion about P versus NP be considered.
The current position
The current research has therefore moved beyond the original intuition but has not reached a proof. The most interesting possibility is that what looks like an exponential combinatorial search may contain a lower-dimensional structural description that can be manipulated hierarchically. The most important alternative possibility is that some family of instances will defeat the proposed structural reduction.
Both outcomes are useful. A successful completeness theorem would transform the present conjecture into a serious candidate for a polynomial-time framework. A counterexample would identify exactly where the intuition fails and provide a stronger foundation for the next attempt.
Why the 2026 restart is different
The present phase is also an experiment in human–AI research. The original intuition and historical continuity came from the human side; AI has provided a rapid environment for formalization, alternative formulations, computational questioning, documentation and repeated attempts to find weaknesses. The resulting dialogue has made it possible to move through hypotheses and counterexamples far faster than the original 2006 workflow.
That acceleration should not be confused with proof. AI can help expose structure, generate possibilities and attack assumptions, but the final mathematical burden remains unchanged: definitions must be precise, lemmas must be true, counterexamples must be defeated, and the global theorem must actually follow.
A deliberately unfinished record
This article is intended as a public milestone in an ongoing investigation. Some of the current mathematical details have deliberately been generalized while the research is unfinished. The complete working record, including detailed constructions, experiments, failed approaches and intermediate hypotheses, is being maintained separately as part of the research archive.
The principle remains the same: do not protect the hypothesis. Try to kill it. If it survives sufficiently strong attempts at disproof, formalization can begin. If it fails, the failure becomes the next piece of the map.
Research status: ongoing. No claim of a solution to P vs NP is made.