From the 2006 Geometric Seed to a Hierarchical Proof Framework
Research status: This article preserves the historical bridge between the 2006 geometric work and the 2026 research programme. It does not claim that P vs NP has been solved. Some operational details of the current construction have been generalized while the investigation remains unfinished.
From the 2006 geometric seed to a 2026 research programme
The present investigation did not begin with ChatGPT. Its conceptual seed goes back to my independent work on P vs NP around 2001 and, more specifically, to a geometric approach developed in 2006. That work attempted to understand the Euclidean travelling-salesman problem through geometry rather than treating it only as a permutation problem.
The 2006 manuscript eventually reached the Elsevier publication process and became an important stopping point in the journey. The work then remained dormant for many years. In 2026, more capable AI made it practical to reopen the old ideas, formalize them, test them computationally and, equally importantly, attack them for weaknesses.
The geometric seed
The original intuition was that a collection of points on a plane contains geometric information that a brute-force enumeration of tours does not explicitly represent. Convex boundaries, local distances, insertion effects and the relative position of points can provide a different description of the same combinatorial object.
A particularly simple local observation comes from inserting a point into an existing route. The change in route length can be expressed directly in terms of the relevant distances, and the triangle inequality supplies an exact geometric constraint. This is elementary mathematics, but it suggested a larger question: could a sequence of such small structural comparisons replace a much larger enumeration?
Where the original idea becomes difficult
The central difficulty is the difference between a local improvement rule and a global optimality theorem. It is easy to identify a local change that makes a particular route shorter. It is much harder to prove that every globally better route must be reachable through the permitted family of changes.
This distinction became clearer during the 2026 restart. A geometric rule can work repeatedly in diagrams and computational tests while still failing on a specially constructed configuration. Therefore the research has gradually shifted from asking only “does this rule work?” toward the more demanding question:
“Is the rule complete?”
The four-city laboratory
Small cases have become the most useful bridge between intuition and proof. Four genuine points are particularly valuable because the competing closed tours can be enumerated completely while still exposing alternative reconnections, closure conditions and changes in adjacency.
The four-point laboratory is not being used as evidence that the general problem is solved. Its purpose is almost the opposite: every hidden assumption becomes easier to expose. If a proposed structural principle fails at four points, there is no reason to build a large theorem around it.
From individual points to structural units
The 2026 investigation introduced a stronger emphasis on clusters and structural units. Human observers naturally recognize groups of nearby points and their relative separation from other groups. The research asks whether this subjective visual information can be converted into a rigorous mathematical representation without losing information required for exact optimization.
Convex geometry provides one possible representation. A cluster can be viewed through its boundary and its relationship to external points. A grid or square can also serve as a localization scaffold, while a later geometric boundary can test the candidate structure more accurately.
These are research devices, not established components of a proven algorithm.
Hierarchical reduction
A further development is the possibility of building a solution hierarchically: analyze small structures, combine compatible structures, and treat the result as a larger unit. In principle, this could replace a flat factorial search by a sequence of structurally meaningful operations.
But hierarchy introduces its own danger. Combining locally good structures does not automatically preserve the global optimum. A newly introduced point or a connection between two previously separate regions can change the structure itself.
Consequently, the hierarchy must permit restructuring rather than silently freezing earlier decisions.
The current proof obligations
The research can now be stated as a sequence of proof obligations rather than as a collection of attractive diagrams:
- Identify the genuinely irreducible local geometric comparison.
- Show that larger admissible changes can be decomposed into valid smaller transformations, or identify the obstruction.
- Prove completeness: a non-optimal candidate must expose an admissible improving structural difference.
- Establish that the resulting representation and operations are polynomially bounded.
Only if these obligations survive rigorous scrutiny would the work become relevant to a P vs NP conclusion.
Until then, the framework remains a conjectural research programme.
Experiments and failed approaches
One of the most useful changes in the 2026 workflow has been the willingness to preserve failures.
Early experiments worked well on many configurations but failed on particular geometric arrangements. Those failures forced a reconsideration of cluster boundaries, the effect of adding points, recursive representations and the assumption that a previous local optimum remains valid after the structure changes.
Later computational experiments have provided encouraging results on the tested classes, but they are treated as evidence for further investigation rather than proof.
The research deliberately includes adversarial configurations designed to break the proposed structure.
Construction versus certification
A major conceptual refinement is the separation of construction from certification.
Constructing a good route efficiently is not enough if proving that it is optimal still requires examining exponentially many alternatives. The more ambitious goal is a compact structural certificate from which global optimality can be established.
The unresolved question is whether every hypothetical improvement would necessarily reveal itself through the permitted structural hierarchy.
If a shorter tour can exist while remaining invisible to that hierarchy, the approach fails.
If invisibility can be proved impossible, the research would have crossed a major mathematical threshold.
What AI changed
The 2026 restart is unusual because AI became an active research partner rather than merely a writing tool.
The historical intuition came from the original work; AI accelerated formalization, alternative formulations, computational questioning, counterexample generation and documentation.
The human–AI loop repeatedly moved through:
intuition → formalization → experiment → failure → revised hypothesis → new experiment
That loop has allowed the old geometric seed to be examined at a depth and speed that was not practical in 2006.
But the mathematical burden has not disappeared.
AI can generate a convincing argument that is false just as easily as it can expose a useful pattern. Every important claim therefore has to survive exact definitions, exhaustive small cases, counterexample searches and eventually proof.
The current position
The 2006 geometric seed has therefore evolved into a broader question about whether geometric and combinatorial structure can be represented compactly enough to support exact global optimization.
Clustering, relative structure, convex geometry and hierarchy are possible pieces of that investigation, not established answers.
The immediate mathematical target remains deliberately small:
establish or destroy the relevant structural claims on the smallest non-trivial cases, identify the exact completeness requirement, and only then attempt a general theorem.
The research is deliberately proceeding from the bottom up.
A successful result would require much more than finding a clever geometric shortcut. It would require demonstrating that the structural representation is exact, complete and polynomially bounded.
A counterexample would be equally valuable because it would identify precisely where the current intuition breaks.
The journey continues
The original question from 2001 has therefore not disappeared. The 2006 geometric attempt was not the end of the story; it became the seed for a much longer investigation.
Twenty years later, AI has provided a new laboratory in which that seed can be formalized, attacked and tested.
Whether the underlying intuition eventually becomes a theorem—or becomes another useful failed approach—remains an open question.
Research status: ongoing. No claim of a solution to P vs NP is made.
The 2006 manuscript supplied the seeds. The present research is an attempt to discover whether those seeds contain a theorem.
Technical credit: Human GPT — original 2006 conceptual seed and algorithmic hypothesis. ChatGPT — technical assistance, formalization, testing, concept development and manuscript preparation.