This is a sequel, not a restart. The previous article revisited my 2006 manuscript and began translating its geometric intuition into mathematical language. This article records what happened next: the ideas we tested, the ambiguities we removed, the four-city base case, the triangle-inequality argument, recursive two-by-two reduction, cluster handling, ghost points, selective zooming, closed-tour geometry, and—most importantly—the exact propositions that still have to be proved.

Previous article: P vs. NP: Formal Reassessment of the 2006 Hamiltonian Path Manuscript.

The purpose of this article is partly mathematical and partly mnemonic. I want the research tree outside my head, so that a future discussion can begin from the actual state of the argument rather than reconstructing it from fragments.

Visual reference: The clear P vs NP infographic is used as the article’s overview image. The twelve research figures are now treated as individual figures rather than being compressed into the previous vague reference sheet; each belongs beside the section where its construction is discussed.

1. Start with the 2006 manuscript

The most important decision was not to throw away the 2006 manuscript and invent a new theory around it. The manuscript was crude by current academic standards: notation was informal, the theoretical language was ambiguous, citations were absent, and the complexity argument was compressed. But its geometric seeds were already there.

The original construction revolved around convex polygonal paths, shortest peripheral connections, minimum insertion, independent networks, hypothetical or invisible diagonals, the a+b-c rule, the corresponding a+b-c-d network-joining operation, and a final optimality check. The original intuition was that geometry could replace factorial enumeration by a much smaller sequence of calculations.

The modern work therefore has a very specific role: connect the dots. It does not claim that ChatGPT invented the central hypothesis. The conceptual seed belongs to the 2006 Human GPT manuscript. ChatGPT’s role has been technical assistance: formalization, notation, testing, decomposition, counterexample-oriented thinking, cluster interpretation, recursive formulation, and manuscript preparation.

2. The first ambiguity we had to remove: city, edge, path and distance

A major source of confusion was treating a city order, an edge and an edge length as though they were interchangeable. They are not.

  • City: a fixed geometric vertex Ai.
  • Edge: the unique connection AiAj.
  • Edge weight: dij = d(Ai,Aj).
  • Path: an ordered sequence of edges through intermediate cities.
  • Tour: a permutation of all cities, closed back to its starting city.

This one-to-one mapping matters. Suppose two different edges happen to have the same numerical length. They are still different edges. A path such as 1-5-7-3-2 is also not a triangle side; it is a sequence whose length is the sum of several edge lengths.

Therefore two tours must be compared structurally first. Common edges can be cancelled by identity. Only then do we compare the remaining distances. This removes a surprisingly large amount of ambiguity from the original intuitive argument.

3. Closed tours change the geometry

I am dealing with a closed tour. If the ordered path is A→B→…→K, the actual TSP object is A→B→…→K→A. There is no special endpoint at the end.

For bookkeeping we introduced the idea of a ghost closure edge: the displayed open sequence is closed by adding K→A. This is not a new city. It is simply the final edge required to turn the path into the polygonal tour.

This lets the entire construction be viewed as a closed polygonal network. In Euclidean geometry, crossing edges can be replaced by a shorter non-crossing reconnection, so an optimal tour can be taken as non-self-intersecting. This should not be confused with saying that every optimal tour is convex: the convex hull is the outer boundary, while interior cities have to be inserted into the tour.

4. Triangle inequality becomes the atomic operation

The central relation is elementary:

AB + BC − AC ≥ 0.

This is the original a+b-c rule expressed in formal Euclidean language. It is not an arbitrary score. It is exactly the marginal distance created when a direct edge AC is replaced by A-B-C.

That gives the first operation:

  • Existing edge: AC.
  • Insert B.
  • New path: A-B-C.
  • Incremental cost: AB + BC − AC.
  • Choose the admissible insertion with minimum incremental cost.

The same geometric atom is expected to appear inside more complicated reconnection operations. The key research claim is therefore not that every large exchange is a triangle. It is that a larger exchange may be decomposed hierarchically until its irreducible geometric comparisons are three-city relations.

5. The four-city base case

We deliberately stopped trying to prove the general theorem and chose four genuine points in the plane, with no three collinear. This prevents the argument from becoming a collection of arbitrary symbolic variables disconnected from geometry.

Take the reference tour

T1 = A1−A2−A3−A4−A1.

Now scramble the same four variables:

T2 = A1−A3−A2−A4−A1.

No city has changed. Only adjacency has changed. The common edge A2A3 cancels. What remains is the composite exchange

Δ = d13 + d24 − d12 − d34.

This was an important correction in our reasoning. A four-city exchange is not itself one triangle. It is a composite operation. To expose its geometric structure, introduce an intermediate diagonal and decompose the change into three-city relations.

That gives us the proper hierarchy:

large tour transformation → smaller reconnection → three-city geometric comparison.

6. The important premise about a better tour

Suppose an algorithm has produced a candidate tour T and a shorter tour T′ exists. How can T′ differ from T? The city variables are the same. Therefore the difference must be in their adjacencies.

The smallest nontrivial structural change requires an exchange of city positions or adjacencies. At the geometric core of such an exchange we eventually encounter a path through three cities versus a direct connection, or an equivalent three-point reconnection.

Any improving tour transformation admits a finite geometric decomposition whose irreducible distance comparisons are triangle relations.

This is deliberately narrower than saying “triangle inequality proves the tour is optimal.” Triangle inequality alone does not prove global optimality. The missing step is proving that every possible global improvement must enter the admissible transformation system.

7. Hierarchical naming: the missing bridge between paths and geometry

Another important insight was that tours should be named hierarchically and mapped explicitly. We can write T1, T2, …, but each name must carry its actual permutation and edge set.

If T1 changes into T2, we should not jump directly from one scalar length to another. Instead we construct intermediate valid structures:

T1 → S1 → S2 → … → Sm → T2.

Every intermediate object retains the same city variables. Only its connectivity changes. This gives us a possible mathematical bridge between a global permutation problem and local geometric transformations.

8. Clusters: do not let the monster grow

The cluster problem produced several related ideas. If a group of cities is tightly connected, we should not allow its internal complexity to infect the entire global network. First solve or compress the local structure, then represent it at the next level.

  • Represent a cluster by a centroid or weighted centre.
  • Retain its internal optimal tour cost as a weight.
  • Treat the optimized cluster as a higher-level object in the grand set.
  • Stitch clusters using the same geometric insertion/reconnection principles.
  • Reopen a cluster only when a lower-level optimization is required.

The centroid is therefore not a replacement for the cluster’s internal solution. It is a higher-level representation carrying location plus internal cost.

9. Ghost and catalyst points

A more radical version of the cluster idea was to introduce zero-weight geometric points. A difficult cluster may require an extra geometric vertex to create a useful convex polygon or to resolve a connection. Such a point can be allowed to influence geometry while carrying zero weight in the final cost.

We called these ghost points or catalyst points. The name captures the intended role: they change the geometry without themselves becoming real cities.

This remains a proposed constructional device, not a theorem. Its validity requires precise rules preventing ghost points from artificially changing the feasible TSP problem.

10. Selective zooming and topological imbalance

Another experimental idea arose when a geometric projection did not produce uniform local polygons. Instead of globally rearranging the point set, we proposed selective zooming: locally stretch a problematic region to make its topology easier to resolve.

The proposed compensation is that if a region is zoomed by a factor z, its contribution to the final distance calculation is multiplied by the inverse factor 1/z. In other words, the transformation is intended to alter representation rather than the underlying optimization problem.

This is an interesting geometric hypothesis, but it is currently a representation idea, not a proved invariant. Any formal use must prove that the transformation preserves the ordering of candidate tours after the inverse-distance correction.

11. Mercator-like topological stretching

We also considered a Mercator-like projection: a mild three-dimensional-looking angle produced through two-dimensional stretching rather than arbitrary rearrangement. The objective is to improve cluster separation while preserving the identity and order of the original points.

Again, this belongs to the experimental geometry layer. It becomes mathematically relevant only if we can prove that the transformation has a controlled effect on distance comparisons and does not change the underlying optimization problem.

12. Triangle-pair closure

The earlier manuscript already contained the idea of joining independent geometric networks. The modern formulation turns that into a pairwise closure problem.

Suppose two optimized structures are represented by edges AB and CD at the interface. Two possible reconnections have incremental costs

AC + BD − AB − CD

or

AD + BC − AB − CD.

The minimum admissible reconnection is the two-network analogue of minimum insertion. The critical closure question is whether every globally optimal combined tour can be represented inside this admissible pairwise family.

13. Two-by-two reduction: keep the monster small

This led to a practical algorithmic philosophy: start with the smallest structures and keep them small. Do not allow a difficult cluster to grow until it becomes another brute-force problem.

If pairwise closure is valid, optimized structures can be combined two at a time:

n → ceil(n/2) → ceil(n/4) → …

This creates a pyramid. Each upper structure rests on structures that have already been optimized. The intuition is exactly the same as building a foundation before adding another floor.

The earlier rough complexity argument suggested combinations of C(n,2)-type operations and a final bound around a fourth-order polynomial. But the exact recurrence must still be derived from a fully specified algorithm. Counting operations before proving the move set is complete would put the cart before the horse.

14. The pyramid and the final optimality check

Here the most interesting idea emerged. Suppose the recursive construction has produced an apparently optimal tour on 100 cities. A conventional exact verification would compare it against an enormous space of alternative tours.

The proposed certificate instead asks a different question: where could a shorter tour first differ from the already verified hierarchy?

If the hierarchy is genuinely closed, any competing tour must differ at some lowest unresolved interface. The difference at that interface should create an admissible local improvement. If no such improvement exists, the competing tour cannot be shorter.

No improving admissible transformation remains → no shorter tour exists.

This is the heart of the proposed optimality certificate. It is also the point where intuition must give way completely to proof.

15. What the experiments tell us—and what they do not

The computational work was useful precisely because it was allowed to fail. The early construction reached roughly a 90% success regime. Instead of hiding the failures, we used them to discover where the original intuition was too crude.

  • Failures motivated cluster decomposition.
  • Clusters suggested centroid and weighted representations.
  • Independent structures suggested recursive pairwise stitching.
  • Geometric irregularities suggested selective zooming.
  • The need for a final check led to the closure-certificate formulation.

Later tests reached complete agreement within the small exhaustive hierarchical classes we examined. That is encouraging evidence. It is not a universal proof for Euclidean TSP.

16. What is actually proved?

  • Euclidean triangle inequality is valid.
  • For insertion of B into edge AC, the exact marginal cost is AB+BC−AC.
  • Every tour is a permutation of the same fixed city set.
  • Edge identity is determined by its endpoints, not merely its numerical length.
  • A closed TSP tour includes the final edge from the last city back to the first.
  • Crossing Euclidean tour edges can be replaced by a shorter non-crossing reconnection, so an optimal tour can be chosen non-self-intersecting.
  • The four-city example can be explicitly represented and its changed edges isolated.
  • Larger exchanges can be experimentally decomposed into smaller geometric relations in the examples examined.

These are the secure foundations. They do not yet constitute a proof that the proposed algorithm finds the global optimum for every Euclidean TSP instance.

17. What is evidence rather than proof?

  • The approximately 90% initial success regime.
  • Later complete agreement in tested small-instance hierarchical classes.
  • The apparent usefulness of centroid compression.
  • The apparent usefulness of pairwise stitching.
  • The experimental promise of selective zooming.
  • The observation that difficult structures can often be resolved by working at a lower geometric scale.

These observations guide the proof programme. They cannot replace it.

18. What remains to be proved?

I think the entire problem has now been reduced to a small number of sharply defined mathematical obligations.

Lemma 1 — Three-City Atomicity

Prove that the irreducible local geometric comparison underlying an admissible reconnection can be represented by a three-city path/direct-edge relation governed by triangle inequality.

Lemma 2 — Hierarchical Reduction

Prove that any admissible transformation between two tours can be decomposed into a finite sequence of smaller valid transformations without losing the relevant cost comparison.

Lemma 3 — Local Completeness

This is the dangerous one. If T is not globally optimal, prove that at least one admissible transformation generated by the hierarchical construction strictly improves T.

Lemma 4 — Polynomial Bound

Once the move set is proved complete, derive the exact recurrence and show that the number of distance comparisons and transformations is polynomial in n.

Theorem — Global Optimality

Combine the preceding lemmas: if the construction terminates with no admissible improving transformation, then no shorter Euclidean Hamiltonian cycle exists.

19. The most important warning

We must not make the seductive but invalid shortcut:

“Triangle inequality is true, therefore the current tour is optimal.”

That does not follow. Triangle inequality is a local geometric law. The missing theorem is that our hierarchical transformation system is complete: every possible shorter tour must be reachable through the transformations we test.

This distinction is now central to the research. We are no longer trying to hide the hard part behind the phrase “a+b-c.” We are trying to prove that a+b-c is the atomic certificate inside a complete geometric transformation system.

20. The proof roadmap from here

  1. Prove the four-city closed-tour case completely.
  2. Formalize the elementary exchange and its exact cost change.
  3. Prove the hierarchical decomposition lemma.
  4. Extend the result from n to n+1 using induction only after the base transformation is secure.
  5. Prove that the admissible transformation family is globally complete.
  6. Derive the exact polynomial operation count.
  7. Only then discuss the P versus NP consequence.

This order matters. We should not begin by announcing a solution to P vs NP. We should begin by proving the smallest geometric statement from which the larger statement would follow.

21. Why the four-city case matters so much

The four-city example is not just a toy calculation. It forces us to confront every ambiguity at once: closed tour, permutation of fixed variables, changed adjacency, cancellation of common edges, composite exchange, intermediate geometry and triangle-level decomposition.

If the four-city transformation can be stated and proved cleanly, we have a genuine mathematical object to generalize. If it cannot, there is no reason to hide behind n-city notation.

So the next serious research step is deliberately small:

Prove the four-city closure lemma before attempting the general theorem.

22. The research model in one line

2006 geometric seed → one-to-one mapping → triangle atom → minimum insertion → pairwise closure → clusters → recursive two-by-two stitching → hierarchical certificate → prove completeness → prove polynomial bound.

That is the model I want to carry into the next research chat.

Conclusion

The interesting thing about this research is not that a 2006 handwritten geometric intuition suddenly became a theorem. It is that repeated attempts to formalize it have gradually exposed the exact boundary between intuition and mathematics.

The original ideas—convex geometry, minimum insertion, a+b-c, independent networks, stitching and a final check—were already present. The new work has connected them into a possible hierarchy and, more importantly, identified the place where a real proof must succeed or fail: closure and certificate completeness.

If that completeness theorem can be proved, the rest becomes a conventional mathematical programme: induction, operation counting and complexity. If it cannot, the failure itself will tell us exactly where the geometric intuition breaks.

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.