Research status: This article records an evolving line of investigation into P vs NP. It is not a claim that the problem has been solved. Some technical details of the current construction have deliberately been generalized while the research remains unfinished.
From human clustering to a possible structural invariant
A recurring question in this research is whether the apparent complexity of a geometric optimization problem partly comes from representing structured data as an unstructured combinatorial search. Human observers routinely recognize clusters, separation, boundaries and relative position without enumerating every possible arrangement. The current investigation asks whether some of that structural information can be expressed mathematically.
This line of thought grew out of the older geometric work and became more focused during the 2026 restart. Euclidean TSP provides the experimental laboratory because points on a plane make structural relationships visible while still presenting the global optimization problem that motivates the P vs NP question.
Relative structure rather than absolute distance
One of the most interesting observations is that absolute distance may not be the only useful description of a point’s relationship to a cluster. A point can be close to one cluster and far from another, while the significance of that difference depends on the scale and geometry of the cluster itself. This motivates the search for a dimensionless or otherwise scale-aware structural measure.
The exact candidate definition and its operational use are deliberately omitted from this public version. At this stage it is more important to document the discovery and the question it creates than to publish an unfinished procedure that may later prove incomplete or incorrect.
Convex geometry as a structural laboratory
Convex hulls provide a natural way to separate an external point from the geometry of an existing set. This suggested an appealing hypothesis: perhaps the location of a new point relative to an existing geometric structure can tell us something about how the optimal route should change.
Experiments quickly showed why this cannot be accepted as a theorem merely because the diagrams look persuasive. A geometric property can remain useful while the simplest proposed rule fails. In particular, the behaviour of a newly introduced point depends on the surrounding configuration, and adding a point in an unsuitable region can disturb a previously convenient structure.
Controlled addition and the importance of failure
Controlled addition became an important experimental technique: begin with a known small structure, introduce an additional point under controlled geometric conditions, and compare the resulting optimal solutions. The purpose was not to demonstrate a preferred rule but to discover exactly where a proposed structural principle stops working.
The experiments produced an important methodological lesson. When a new point lies in a position that changes the internal structure of the existing configuration, the previous optimization cannot simply be assumed to remain valid. The structural representation itself may need to be recalculated. This turned a seemingly minor failure into a clue about the limits of hierarchical compression.
Local geometry is not global optimality
The older geometric work contains a useful local observation: a route modification can be evaluated through a small number of distances, and the triangle inequality provides exact constraints on such local changes. During the 2026 investigation this was repeatedly reformulated and tested.
But a local improvement rule is not enough. The central proof obligation is global: if a proposed tour is not optimal, can every possible improvement be guaranteed to reveal itself through a finite family of permitted local or hierarchical structural changes? Without that completeness result, the construction remains a heuristic or conjecture regardless of how well it performs experimentally.
The square, the circle and the cluster
One developing line of thought uses a grid or square as a localization and compression scaffold. The grid is not itself the proposed solution; its role is to help identify candidate regions in a large point set. A geometric boundary can then be used to examine a candidate cluster more faithfully, including points that do not fall neatly inside the original cells.
This distinction matters. The square is a representation device; the geometric boundary is a validation device. The research question is whether such a two-stage representation can reduce irrelevant combinatorial possibilities without discarding information needed for an exact optimum.
From clusters to hierarchy
If meaningful clusters can be identified reliably, a larger point set may be represented through progressively larger structural units. Small units can be examined, compatible structures can be combined, and the combined object can then be treated as a new unit. This is the intuition behind the current hierarchical direction.
However, the research deliberately avoids assuming that clusters are independent. Nearby structures can interact, and a newly added point can alter the geometry of an existing unit. The hierarchy therefore has to preserve the possibility of structural change rather than simply freezing every earlier decision.
The four-point laboratory
Small exhaustive cases have become the main laboratory for testing these ideas. Four genuine points are particularly useful because they are small enough to enumerate completely but large enough to expose alternative closures, reconnections and competing tours.
The current strategy is therefore deliberately conservative: prove or disprove the smallest structural claims first, preserve every counterexample, and only then ask whether the observed behaviour scales. A beautiful diagram is not evidence of a theorem; an exhaustive small case is useful precisely because it can expose the hidden assumption.
What the current struggle really is
The present difficulty is no longer simply finding a clever geometric rule. It is identifying a representation that is simultaneously compact, exact and complete. A representation that works on typical data but misses one adversarial configuration is insufficient. Conversely, a representation that is exact but requires exponentially many alternatives has not escaped the original complexity.
The central unresolved question can therefore be stated more sharply: does there exist a polynomially bounded structural representation of the relevant geometric information from which global optimality can be certified? The answer is unknown.
Research method: try to kill the idea
The most productive part of the current AI-assisted collaboration has been adversarial testing. Instead of asking the model merely to improve the conjecture, the working instruction is to find counterexamples, expose hidden assumptions, test boundary cases and identify where a proposed shortcut secretly reintroduces exhaustive search.
That approach has already changed the direction of the work several times. Failed geometric assumptions led to stronger attention to clustering; clustering raised the question of relative scale; relative structure led back to geometric boundaries and hierarchical representations; and each new representation has had to survive attempts to break it.
Current status
The research has produced a coherent set of questions and experimental directions, but not a proof. The possible structural invariant remains a hypothesis. The role of clustering remains under investigation. The exact conditions under which a hierarchy preserves global optimality remain unresolved.
The next meaningful milestone is therefore not a grand claim about P vs NP. It is a rigorous lemma that survives exhaustive small cases and adversarial counterexample searches, followed by a proof that the same structural principle can be composed without losing completeness.
Why document an unfinished investigation?
The original work began in 2001, reached a major stopping point around the 2006 Elsevier manuscript, and was restarted in 2026 when AI became capable enough to participate meaningfully in formalization and experimental reasoning. Documenting the intermediate stages is therefore part of the research itself. It prevents rediscovering the same failed paths and establishes the chronology of the ideas.
Some current technical details have deliberately been generalized in this public version. The complete working record is maintained separately so that the research history is preserved without turning an unfinished hypothesis into a publicly reproducible recipe.
Ongoing research. No claim of a solution to P vs NP is made.