“The interesting question is not only how fast we can calculate. It is whether we are searching the wrong space.”

This is an archival research note, not a claim that P Vs NP has been solved. It records the progression of an ongoing investigation into Euclidean travelling-salesman problems (TSP), human visual cognition, clustering, convex geometry and the possibility that an apparently enormous combinatorial search may contain a much smaller structural description.

1. The starting intuition: humans see structure before calculation

Put a large number of apparently random points on a sheet of paper and a human observer does not normally begin by calculating every pairwise relationship. We immediately notice groups, isolated points, boundaries, symmetry and regions of relative separation. The first visual clue is often simply cluster versus non-cluster.

This raises a surprisingly deep computational question: can a subjective visual judgement such as “these points belong together” be converted into an objective mathematical rule? If it can, the resulting rule might be useful not merely for describing data but for reducing the search space of an optimization problem.

2. Published research already formalizes part of this intuition

There is a substantial literature on perceptual grouping. Gestalt psychology identified principles such as proximity, similarity, continuity, closure and symmetry. Modern vision research has gone further and experimentally quantified some of these principles. A major review of the field reports that, for certain dot-lattice grouping tasks, the perceptual organization can be determined by relative distance alone, rather than by absolute distance. This is exactly the phrase that caught my attention.

A Century of Gestalt Psychology in Visual Perception reviews the empirical development of perceptual grouping and discusses quantitative laws of proximity grouping.

Even more directly relevant is the 2021 paper A computational model for Gestalt proximity principle on dot patterns and beyond. The authors construct a computational model for human grouping of dot patterns, use ratio-based neighbourhood information, compare the resulting structures with human observers, and explicitly discuss the relationship with the Pure Distance Law. They also demonstrate hierarchical perceptual grouping.

This is important because it means the bridge we are looking for is not science fiction: researchers have already attempted to convert a human perceptual experience into a quantitative computational representation.

3. Why “relative distance” is more interesting than absolute distance

Suppose three points have mutual separations of roughly 1, 1.2 and 1.5 units, while the nearest point outside the group is 10 units away. A human immediately sees a cluster. If the entire drawing is enlarged by a factor of 100, the absolute distances become 100, 120, 150 and 1000, but the perceived structure has not changed.

This suggests that a useful primitive might be a ratio such as

relative separation = characteristic internal distance / characteristic external distance.

A small ratio means that the points are much more tightly related to one another than to their surroundings. Unlike an absolute threshold, such a measure naturally survives changes of scale. It is therefore a promising candidate for formalizing the human visual judgement that one region is a coherent cluster and another is not.

4. DBSCAN: a different but important formalization

Clustering research has already produced powerful mathematical definitions. The classic DBSCAN work by Ester, Kriegel, Sander and Xu defines clusters through density and was designed for spatial data, arbitrary cluster shapes and noise. It remains one of the foundational approaches to spatial clustering.

DBSCAN — A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise

But density-based clustering is not automatically the invariant needed for TSP. DBSCAN asks whether points form a density-connected region. Our question is stronger: can a recognized group be treated as a structural unit without excluding a global optimum?

5. The TSP experiment: from clustering to convex geometry

This is where the current research takes a different turn. For a spatial TSP instance, once clusters are visually recognized, the next obvious geometric object is the convex hull. Repeating the process naturally produces nested hulls or polygonal layers.

The working hierarchy is:

  • random-looking points;
  • clusters and non-clustered points;
  • convex hulls;
  • nested hulls or polygons;
  • controlled addition of structurally separated points or clusters;
  • small local tour transformations.

The convex polygon is therefore being treated as a catalyst, not as the solution itself. It may reduce the number of meaningful branches that have to be considered.

6. Two elementary operations

The current TSP framework distinguishes two operations that had previously been mixed together.

  1. a+b-c is the insertion/optimization operation when a new point is added to an existing network. It asks where the new point should enter the current structure.
  2. a+b-c-d is a closed-tour transformation. The elementary operation breaks two existing edges and cross-checks the alternative reconnection. If the new tour is shorter, the transformation is accepted and the check is restarted.

The complete final sweep over all admissible two-edge exchanges is essential. If no improving exchange remains, we have established 2-opt terminality. We have not automatically established global optimality; that distinction is one of the important lessons of the current experiments.

7. The one-city addition idea—and its hidden condition

An early hypothesis was simple: if an optimal tour exists for n points, add one point and use a+b-c to extend it. The nested 12-city experiment suggested something stronger: controlled additions repeatedly reached the optimum, and the same fixed dataset could be viewed as a sequence of smaller problems, giving a natural induction ladder.

But arbitrary point ordering exposes the flaw. A newly added point can lie inside the existing convex hull or extremely close to its boundary. Such a point can disturb the established structure in ways that a simple insertion cannot safely capture. The correct response is not to force the induction; it is to re-optimize the affected network and then resume controlled growth.

This leads to a more careful working rule:

A new point should not merely be outside the convex hull. It should be structurally separated enough from the current network that the existing optimized structure remains a useful predecessor.

8. The new clue: separation from the cluster

This is where the phrase relative distance becomes potentially central. “Far enough away” is subjective. But it may be possible to express it relative to the scale of the existing cluster.

For a candidate cluster C, imagine a characteristic internal scale s(C). For a new point P outside the current convex hull, measure its separation h(P,C) from that hull. A natural experimental quantity is then

ρ(P,C) = h(P,C) / s(C).

This is not yet a theorem and may not be the correct invariant. But it converts the visual intuition “this new point is genuinely separated from the cluster” into a testable numerical quantity.

9. What the preliminary computation has already taught us

The computational experiments produced an important negative result: convex-hull externality alone is not sufficient. We found external additions for which a+b-c followed by a complete a+b-c-d sweep reached a 2-opt-stable tour that was nevertheless not globally optimal.

That failure is valuable. It tells us that “outside the hull” is not the whole invariant. The more promising hypothesis is a combination of cluster segregation, geometric hierarchy and structural separation.

Conversely, the controlled sequence in the earlier experiment succeeded through many consecutive one-city additions. The correct interpretation is not “18 successes prove the theorem.” It is that controlled ordering appears to preserve structure much better than arbitrary ordering, giving us a concrete phenomenon to investigate.

10. Why this matters for P Vs NP

The broader question is not yet “have we proved P = NP?” We have not. The useful question is more basic:

Is the apparent combinatorial explosion partly an artifact of representing a structured problem as an unstructured search?

TSP provides a clean laboratory because the input is finite, the objective is objective, the solution is a discrete tour, and small instances can be independently certified. If a universal structural invariant can be found for the relevant class, then an apparently huge tour space may collapse into a sequence of structural decisions plus small local operations.

That would still not by itself prove P = NP. A polynomial algorithm for a restricted geometric TSP class is not automatically a polynomial algorithm for every NP-complete problem. A genuine P Vs NP implication would require the appropriate generality and complexity-theoretic argument.

11. The deeper research question

Perhaps the most interesting part of this investigation is the attempt to convert something humans do almost unconsciously into mathematics.

Human cognition sees:

“These points belong together.”

Mathematics demands:

“Give me an objective invariant that tells me exactly when they belong together, works under scale changes, survives adversarial examples, and preserves the property I care about.”

That transition—from perceptual compression to formal structure—is difficult. But published perceptual-grouping research shows that it is not an impossible category of problem. Relative distance, proximity laws, neighbourhood graphs and hierarchical grouping are already being quantified and computationally modelled.

12. What we test next

  1. Measure relative separation for candidate clusters rather than using arbitrary absolute distance thresholds.
  2. Compare human-like proximity grouping with density-based clustering such as DBSCAN.
  3. Construct convex-hull and nested-hull layers from public Euclidean TSP datasets.
  4. Test controlled external additions against near-boundary additions and hull-interior additions.
  5. Apply a+b-c for growth and the complete a+b-c-d sweep for closed-tour restructuring.
  6. Independently certify small-instance optima so that local terminality is never confused with global optimality.
  7. Search specifically for counterexamples to the proposed structural invariant.

The objective is not to protect the hypothesis. The objective is to find the smallest counterexample if it is wrong—and, if it survives, to understand why it survives.

Further reading

“The hard part is not making the calculation faster. The hard part is discovering the structure that makes most of the calculation unnecessary.”

Research status: ongoing. The ideas, computational observations and proposed invariants in this article are exploratory and should be treated as hypotheses until independently reproduced and mathematically proved.