P vs. NP: From a 2006 Geometric Seed to a Formal 2026 Research Program

“A convincing idea becomes mathematics only when every hidden cost has been made explicit.”

P vs. NP: From a 2006 Geometric Seed to a Formal 2026 Research Program

Hemant Pandey
2026 research revision of a 2006 manuscript

In 2006, I submitted a manuscript titled P Vs NP…. Are they same? to Applied Mathematics and Computation. The manuscript reached the Article in Press stage during the publication process but was withdrawn before final publication. It is now preserved as a historical record on Zenodo.

Historical manuscript: P vs. NP: Historical Manuscript and Zenodo Archive.

Permanent Zenodo archive: Pandey, H., P Vs NP…. Are they same?, DOI 10.5281/zenodo.21770323.

The purpose of this article is not to rewrite that historical document or silently turn a research attempt into a claimed solution. It is to take the original geometric seed seriously, bring back the central ideas—particularly the geometric representation of Hamiltonian paths, convex polygon structure and topological intuition—and then ask what a modern proof would have to establish.

This is therefore a research program, not a claim that P versus NP has been solved. The Clay Mathematics Institute continues to list P versus NP as an open Millennium Prize Problem.

1. The 2006 seed: change the representation

The original research instinct was simple but powerful: instead of looking at a Hamiltonian path only as a combinatorial sequence of vertices, look for a representation in which the same constraint becomes visible through geometry and topology.

A Hamiltonian path asks for a traversal that visits every vertex exactly once. In a conventional graph representation, the difficulty is combinatorial: among a huge number of possible traversals, which one satisfies all adjacency constraints? The geometric approach asks whether the same global condition can be encoded by a simpler spatial object.

The convex-polygon idea

One of the original geometric ideas was to place the graph’s vertices in a convex polygonal arrangement. Convexity is useful because the cyclic order of points is unambiguous and the geometry has no inward dents or hidden reflex regions. Edges or candidate connections can then be viewed as chords or polygonal segments rather than merely as abstract graph symbols.

This does not by itself solve Hamiltonian Path. The important idea is the change of language: a combinatorial traversal becomes a geometric object whose crossings, ordering, separation and topological structure can potentially be studied directly.

The convex polygon is therefore best understood as a seed representation. The research question is not “can a graph be drawn as a polygon?”—that is easy—but rather:

• Can the relevant graph information be preserved without exponential blow-up?
• Can the existence of a Hamiltonian path be expressed by a geometric or topological condition?
• Can that condition be checked in polynomial time?
• Can an actual Hamiltonian path be recovered from the geometric structure?

These questions turn the original visual intuition into a mathematical research problem.

2. From polygon to topology

The deeper intuition behind the geometric construction is topological. Once vertices and allowed connections are embedded into a geometric space, the problem is no longer only about a list of edges. It becomes a question about how a connected structure can pass through the entire set of designated points while respecting the allowed connections.

Topology is attractive here because it studies properties that survive continuous deformation. If a graph property could be captured by a suitably defined invariant, one might hope to replace a huge combinatorial search with the evaluation of a much smaller structural quantity.

But this is precisely where mathematical discipline becomes essential. A visual invariant is not automatically a computational invariant. A topological quantity may be well defined and still be difficult to calculate. Likewise, a beautiful drawing may encode an exponential amount of information.

3. The formal complexity problem

The P versus NP problem asks whether every problem whose proposed solution can be verified efficiently can also be solved efficiently. P denotes decision problems solvable by deterministic algorithms in polynomial time; NP contains problems whose YES certificates can be verified in polynomial time.

Cook’s 1971 work established a foundational polynomial-reduction framework for complexity theory. Karp’s 1972 work demonstrated the breadth of polynomial reductions among combinatorial problems. Hamiltonian problems became part of this central NP-completeness landscape.

Theorem 1 — Hamiltonian-path criterion for P = NP

Theorem. If there exists a deterministic algorithm A that decides the general Hamiltonian Path problem in time O(n^k) for some fixed constant k, then P = NP.

Proof. Hamiltonian Path is NP-complete under polynomial-time reductions. Hence every language L in NP can be transformed in polynomial time into an instance of Hamiltonian Path. If A decides that instance in polynomial time, the composition is polynomial time. Therefore NP ⊆ P. Since P ⊆ NP, P = NP. ∎

The theorem is straightforward. The difficult part is therefore not the final logical implication. The difficult part is proving that the proposed geometric transformation actually gives a polynomial-time algorithm for the general problem.

4. The 2026 extension: geometric compression

The original seed can now be stated as a precise research target.

Theorem 2 — Geometric Compression Criterion

Suppose that for every graph G with n vertices there exist algorithms C, D and E and a polynomial p such that:

• Construction: C transforms G into a convex-polygon, geometric or topological representation T(G) in at most p(n) steps.
• Polynomial representation: T(G) has polynomial encoding size and does not hide exponentially precise coordinates or an exponentially large structure.
• Correctness: G has a Hamiltonian path if and only if T(G) satisfies a formally defined geometric/topological condition.
• Decision: D evaluates that condition in polynomial time.
• Extraction: E reconstructs an explicit Hamiltonian path in polynomial time whenever one exists.

Then P = NP.

Proof. The construction, decision and extraction procedures together yield a polynomial-time solution to Hamiltonian Path. Theorem 1 then gives P = NP. ∎

This theorem is deliberately modest. It does not claim that the required geometric invariant has been found. It states exactly what the original geometric intuition would have to accomplish to become a solution.

5. Where the original idea becomes difficult

The most dangerous possibility is that geometry merely moves the exponential search somewhere else.

• Polygon construction: Can the convex representation be generated without enumerating exponentially many possibilities?
• Coordinate complexity: Do the coordinates require only polynomially many bits?
• Crossing information: Does deciding the relevant intersection or ordering structure remain polynomial?
• Topological invariant: Is the invariant itself computable efficiently?
• Completeness: Does the geometric condition characterize every Hamiltonian path, rather than only a convenient subclass?
• Path extraction: Can the actual path be reconstructed without reverting to exponential search?

A picture can compress our understanding without compressing the underlying computation. That distinction is central.

6. Convexity is a tool, not the theorem

Convex polygons are especially attractive because they remove many geometric complications. Every pair of vertices has a unique straight chord, the cyclic boundary order is fixed, and the geometry can expose crossing and separation patterns cleanly.

But convexity also raises the central question: what information has been preserved? If the original graph is arbitrary, simply placing its vertices on a convex polygon does not make arbitrary adjacency constraints disappear. The graph remains encoded in the permitted chords or connections.

Therefore the interesting mathematical step is not the drawing itself. It is finding a property of the resulting polygonal/topological structure that is simultaneously complete, computable and polynomially bounded.

7. Special cases versus the general problem

Hamiltonian-path problems are efficiently solvable on many restricted graph families. A geometric method may also work beautifully on graphs with additional structure. Such results can be valuable without implying P = NP.

The Millennium-level claim requires the construction to work for arbitrary finite graphs. A method that succeeds only for planar graphs, specially ordered graphs, sparse graphs, or another restricted family does not cross that boundary.

8. A broader complexity-theoretic perspective

Ladner’s theorem shows that if P ≠ NP, there exist problems in NP that are neither in P nor NP-complete. Complexity therefore has a richer structure than a simple “easy versus hard” division. The geometric approach should consequently be evaluated by exact reductions and complexity bounds, not by how visually simple the representation appears.

9. The research program from here

The 2006 seed can now be turned into a sequence of falsifiable mathematical questions:

1. Specify the exact convex-polygon or geometric representation.
2. Define precisely what graph information is encoded by the polygon and its permitted connections.
3. Define the proposed topological/geometric invariant.
4. Prove that the representation has polynomial size.
5. Prove that it can be constructed in polynomial time.
6. Prove the equivalence between the invariant condition and Hamiltonian-path existence.
7. Bound the computation of the invariant.
8. Give a polynomial-time path-reconstruction algorithm.
9. Attack the construction with adversarial graph families and look specifically for hidden exponential work.

Only after these steps have survived rigorous checking would a P = NP conclusion be justified.

10. Why preserve the 2006 manuscript?

The original manuscript should remain untouched. It is the historical seed. The 2026 article is the laboratory in which that seed can be examined using modern complexity-theoretic standards.

Historical record: P vs. NP: Historical Manuscript and Zenodo Archive.

Original manuscript: Zenodo DOI 10.5281/zenodo.21770323.

The historical archive explicitly states that the manuscript is preserved for independent examination and constructive criticism, not as a claim that P versus NP has been solved. Later revisions are intended to remain clearly separate from the historical document.

References

• Cook, S. A. (1971). The Complexity of Theorem-Proving Procedures. Proceedings of the Third Annual ACM Symposium on Theory of Computing, 151–158. DOI 10.1145/800157.805047.
• Karp, R. M. (1972). Reducibility Among Combinatorial Problems. In Complexity of Computer Computations. DOI 10.1007/978-1-4684-2001-2_9.
• Ladner, R. E. (1975). On the Structure of Polynomial Time Reducibility. Journal of the ACM, 22(1), 155–171. DOI 10.1145/321864.321877.
• Clay Mathematics Institute. The P versus NP Problem. Official problem description.
• Pandey, H. P Vs NP…. Are they same? Zenodo (2026). DOI 10.5281/zenodo.21770323.
• Pandey, H. P vs. NP: Historical Manuscript and Zenodo Archive. Historical archive page. Archive page.

“A proof becomes stronger when it tells us exactly where it could fail.”